Production Note
Cornell University Library pro-
duced this volume to replace the
irreparably deteriorated original.
It was scanned using Xerox soft-
ware and equipment at 600 dots
per inch resolution and com-
pressed prior to storage using
CCITT Group 4 compression. The
digital data were used to create
Cornell's replacement volume on
paper that meets the ANSI Stand-
ard Z39.48-1984. The production
of this volume was supported in
part by the Commission on Pres-
ervation and Access and the Xerox
Corporation. 1991.PRESENTED TO
THE CORNELL UNIVERSITY,	1870,
BY
The Hon. William Kelly
Of Rhinebeck.MATHEMATICAL DISSERTATIONS,
FOR THE USE OF
STUDENTS IN THE MODERN ANALYSIS ;
WITH IMPROVEMENTS
THE PEACTICE OF STURM'S THEOREM,
IN THE
THEORY OF CURVATURE,
AND IN THE
SUMMATION OF INFINITE SERIES.
.N
BY J: R: YOUNG,
PROFESSOR OF MATHEMATICS, BELFAST COLLEGE.
LONDON:
JOHN SOUTER, 131, FLEET STREET.
MDCCCXLI.
!? Lj
A)PRINTED BY C. ADLARD, BARTHOLOMEW CLOSE,PREFACE.
The object which it is the intention of the present work
to promote is briefly explained in the introductory article
prefixed to the first Dissertation. This object, however, is
but partially accomplished in the portion now submitted
to the public; the full completion of the original design is
reserved for a Second Party which will contain Disser-
tations on the doctrine of Angular Sections, on the Theory
of Imaginary Logarithms, and on some General Formulas
of Analysis.
The Part now published contains but four dissertations.
The first of these is occupied with a short discussion of the
theory of the coordinate signs; mainly fqr the twofold pur-
pose of removing the very strong objections advanced
against this theory in the great work of Carnot,* and of
establishing, upon permanent and intelligible principles, the
law which regulates the mutations of sign in the trigono-
metrical secant, and in the analytical expression for the
radius of curvature.
This sign, which is a prefix purely algebraical, ac-
countable for by the common theory of the algebraic signs,
and independently of all convention as to their geome-
* Géométrie de Position.IV
PREFACE.
trical appropriation, is, in the ordinary expositions of the
subject, confounded with that which, by special agreement
in the theory of coordinates, exclusively denotes geome-
trical position. The only principle which governs the
changes of sign in these oblique lines is that, in virtue of
which, algebraic quantities generally pass from positive to
negative. This principle, in the cases referred to, implies
the passage of the line through zero or infinity, the di-
rection of the line being at the same time reversed. The
coordinate conventions are, of course, in perfect harmony
with this essential principle; although, by an unwar-
rantable extension of them, they have sometimes interfered
with its full operation. Thus, the changes of sign in the
secant have been assumed to depend on those of the
cosine ; whereas, in both cases, the changes are inde-
pendent results of a common general principle. The fact,
that the secant is equal to the square of the radius divided
by the cosine, is insufficient to determine the sign of the
secant. It is not enough that the secant passes through
oo when the cosine passes through 0. In order that the
sign of the secant may change with that of the cosine, it
is moreover necessary that the passage through oo be ac-
companied with opposition of direction. If the line called
the secant, upon reaching oo, were to continue to revolve,
meeting the tangent at the opposite extremity of the di-
ameter, the expression above would equally well apply to
it ; though the sign of this secant would never change.
In the second Dissertation—that on the Curvature of
Surfaces—it has been my endeavour in the first place to
present, in small compass, the more remarkable propertiesPREFACE.
V
of surfaces in general; systematically deduced from one very
fertile and beautiful theorem—the theorem of the Indi-
eu trix, an evanescent curve first imagined by Du pin, and
applied by him to very important purposes in the theory
of curve surfaces.
Having deduced these general properties, I have in the
next place attempted to give something like consistency
and stability to the analytical theory of the Lines of Cur-
vature, and of the remarkable points around which they
circulate. This is a topic which seems not to have been
discussed, as yet, with complete success. It is rather a
delicate department of the higher geometry, and requires
for its satisfactory development a very cautious use of the
forms of analysis. Few persons since Monge, the illus-
trious originator of these speculations, have expended any
original effort upon the perfecting of this interesting theory.
With the exception of Dupin, who has added more to the
labours of Monge on this subject than any other person,
and Poisson, who has recently examined the theory under
a somewhat novel aspect, I know of no one who has con-
tributed anything towards a satisfactory elucidation of the
obscurities, or towards a removal of the discrepancies, with
which this theory is still justly chargeable ; yet this is one
of the few of the more elevated speculations of geometers
which find immediate application in the practical affairs
of life.
In the profound work of Dupin,* one main object of the
* Développements de Géométrie.
bVI
PREFACE.
author was to correct the theory which Monge had deli-
vered on this subject ; while the intention of Poisson’s
more recent Mémoire* was to remove the defects which he
conceived to attach to the expositions of both Monge
and Dupin.
In the Dissertation which I have here given, I have ven-
tured to discuss, with some freedom, the views and reason-
ings of each of these distinguished analysts, all of whom
appear to me to have failed in their endeavours to place
the theory of the curvature of surfaces beyond the reach of
valid objection.
I think the chief difficulties and perplexities connected
with the investigation of this theory will be found to lurk
under the form -, the analytical symbol which has been a
fruitful source of embarrassment and controversy in many
other departments of analysis. The error commonly com-
mitted in the interpretation of this symbol appears to me
to arise from not discriminating between the two distinct
meanings, one or the other of which, according to circum-
stances, is really concealed under it.
In connexion with a certain restriction, this symbol is
limited to a single value, or at most to a determinate num-
ber of multiple values. This restriction is, that the values
of ^ must obey the law implied in the general expression
whence the form has arisen, upon introducing into that
expression particular values for the arbitrary quantities.
* Journal de T Ecole Polytechnique, 1823.PREFACE.
VU
This is the condition which connects the particular value
of ^ with the entire series of values implied in the general
expression, and which unites it with them by the common
law which governs the whole. It is this particular value,
to the exclusion of all others, which the Differential
Calculus selects as the main instrument of its operations.
The criticisms, or perhaps more correctly the witticisms,
of Berkeley upon this subject in the Analyst, produced
an unaccountable effect upon the mathematicians of his
time, and shook the confidence of many in some of the
most rigorous conclusions of analysis. Berkeley, how-
ever, appears to have misunderstood, and thus to have
misrepresented, the mathematical argument. He charged
analysts with changing, at the close of the reasoning, the
hypothesis upon which that reasoning commenced; whereas
it was because of a rigid adherence to the hypotheses,
even in the extreme and critical case of the problem, that
the peculiarity of their result arose.
The hypothesis, in such a case as that before us, is, that
each particular value in our general form must come under
the law which binds the whole. The fixing a particular
value at the close of the reasoning upon a quantity which
had all along been considered arbitrary was certainly no
change of hypothesis, since till then no hypothesis, as to
value, had been introduced.
x* a*
Results of the form*  ---—, as briefly noticed in the In-
troduction prefixed to the present volume, furnish the ex-Vili
PREFACE.
treme or critical case here adverted to upon putting x = a.
When it is asked what this result becomes when x = a ?
the enquiry implies the twofold condition,
xn
X
X
But the symbol ^ may be capable of a much wider inter-
pretation ; it may be independent of extraneous restrictions,
and be wholly indeterminate. This, however, it cannot be,
unless the conditions that have led to it be indeterminate
too; because the premises must implicitly contain the con-
clusion. It would be wrong, therefore, to offer an inter-
pretation of such a result till we ascertain the effect of the
hypothesis, to which it is due, upon our original conditions.
The necessity for this examination has however been over-
looked in the theory of the curvature of surfaces, as well
as in some other analytical enquiries.
Another oversight, equally detrimental to the rigorous
establishment of this theory, has also been committed by
all preceding investigators ; and which consists in neglect-
ing the distinction between a certain differential equation,
and the common algebraic equation into which it changes
when the variables take constant values.
The substance of what I have ventured to offer, as cor-
rections of the existing theory, was read before the Royal
Society in December 1838. The publication of the essay
has however been delayed, in the expectation that the
long-promised reprint of the Analyse Appliquée of Monge,PREFACE.
IX
under the able superintendence of Liouville, would soon
be presented to the public; and that it would probably
appear with important modifications from the pen of its
distinguished editor. But the French catalogues for 1840
merely repeat the announcement, so frequently made
during the last five years, that the work is still “ sous
pressed
The Dissertation which follows this is composed of two
articles. The first contains an attempt to prove the in-
commensurability of the circle by help of only the common
elementary formulas of analysis. But very few demon-
strations of this interesting truth have been offered; and
I believe not one that is entirely free from objection either
on the ground of prolixity or inconclusiveness.
The second article is occupied wfith a new and simple
method of finding the sums of certain infinite series whose
values are expressible only in terms of the circular circum-
ference ; and which would thus seem manageable only by
help of the more advanced algebraic theories, or by means
of definite integrals. Their summation is shown in this
article to lie within the powers of common elementary
algebra. It may be proper to add that these investiga-
tions on Series were first published, a few years ago, in the
Philosophical Magazine.
The fourth and last Dissertation in this first Part is
upon a subject which, at the present moment, is ex-
citing considerable interest among analysts, as well in
this country as on the continent. I mean the TheoremX
PREFACE.
of Sturm. I believe that I have already contributed
somewhat to extend the knowledge of this important
theorem among British analysts ; and although it has
been since disparaged and undervalued in certain quar-
ters, I have always entertained the conviction that it
must eventually supersede every other method at present
known for effecting the complete analysis of a numerical
equation. Sensible however of the objections to which it
is fairly exposed, on account of the numerical labour fre-
quently implied in it, I have been anxious to devise some
method of so reducing the work as to render the operation
practicable in moderately high equations.
I think I have, to a certain extent, succeeded in this
very desirable object. At all events, by aid of the modi-
fications recommended in this Dissertation, I have been
able, with but a moderate amount of labour, to analyse
complete equations of the fifth, sixth, and seventh degrees.
Although I have received no assistance from the work,
yet I feel it incumbent upon me to state that I have seen
a publication with the title (cDu Théorème de M. Sturm,
et de ses Applications Numériques, par M. E. Midy ; Paris,
1836;"—in which the general principle, upon which the
operation for the common measure is conducted, is the
same as that which I have adopted in the present Disser-
tation. This principle however is so trifling a modification
of that by which the process is conducted in all the
French books, that I question whether the author himself
lays any claim to originality in the statement of it which
he has given. My own improvements were matured, andPREFACE.
XI
the details even completed, before the work of M. Midy
came to my hands.
These improvements, in so far as they are connected
with the principle just referred to, entirely consist in the
compact arrangement which has been given to the nume-
rical calculation; to which arrangement the superior fa-
cility of the operation is entirely due. M. Midy has not
directed his attention to this circumstance; and indeed his
object throughout has been rather to prove the impracti-
cability of Sturm’s theorem beyond certain very narrow
limits, than to promote its general adoption by the pro-
posal of any method for facilitating its application. Such
a proposal has however been more recently made by Pro-
fessor De Morgan, in his article on “ Involution ” in
the Penny Cyclopedia. It is pretty certain that the method
of M. Midy, that of Mr. De Morgan, and that proposed
in the present work, are all referrible to one and the same
general principle. The numerical labour however involved
in this principle can be economized only by a judicious
arrangement of the elements of the calculation.
Such an arrangement I have endeavoured to devise;
and I have further reduced the labour by the adoption of
an effectual method of restraining the large coefficients,
which encumber and greatly complicate the closing steps
of the operation, within moderate bounds.
In the course of the development of these views, several
interesting points, connected with certain extreme or cri-
tical cases, naturally present themselves for examination.Xll
PREFACE.
But rather than digress into these matters* and thus in-
terrupt the more direct order of proceeding, I have trans-
ferred them to the Notes at the end of the volume ; in
which I have endeavoured to anticipate the objections
which might possibly be made to the efficiency of the
general method in particular cases.
It is probable* from the facilities now introduced into
the solution of numerical equations of the third and fourth
degrees* and unfolded at pages 211 and ‘243* that this class
of equations will hereafter be admitted even among the
rudiments of algebraic science.
I cannot conclude this short account of the book with-
out acknowledging that I have had some difficulty in
fixing upon a suitable name for the miscellaneous collec-
tion of papers* of which it consists. Of the many titles
that presented themselves to my thoughts* each was re-
jected in turn ; because the previous appropriation of if
by some analyst of eminence could not fail to connect
such title with associations unfavorable to a work of such
slender claims to notice as the present. From this con-
sideration the designations Tracts* Disquisitions, Essays,
Researches* &c. were successively laid aside ; and the can-
did critical reader* who well considers the dilemma* will
not I hope conclude* after the present explanation* that
by calling this collection, “ Mathematical Dissertations/’
I invite any comparison of it with the profound and
important investigations of Thomas Simpson.
Belfast College; Nov. 25, 1840.
J. R. YOUNG.CONTENTS
INTRODUCTION.
ART.	PAGE
1.	Object of the present work	.	.	.	.3
2.	Importance and extent of the proposed enquiry	.	.	4
3.	Instances of apparent discrepancies in the conclusions of algebra	.	5
DISSERTATION I.
On the Theory of the Coordinate Signs.
4.	On the signs -f- and — considered in reference to their general in-
terpretation ·	.	.	.	.	.	8
5.	Introduction of these signs into geometry as symbols of direction	.	9
6.	Statement of the objections of Carnot to the geometry of D escartes ) 0
7.	Inconclusiveness of Carnot’s first argument proved	.	.	13
8.	Inconclusiveness of the second argument proved	.	.	14
9.	Specific meaning of + and — in the theory of coordinates .	.16
10.	On the meaning of these signs when prefixed to oblique lines	.	IT
11.	On the algebraic sign of the secant .	.	.	.18
12.	How it happens that an oblique line, in a particular position, may
have claim to one sign rather than to the other	.	.19
13.	On the error of supposing that the algebraic signs prefixed to geome-
trical quantities necessarily refer to position .	.	.	ib.
ib. Objections to the common theory of the secant.	·	.20
14.	On the sign of the radius of curvature	.	.	.21
15.	Proper method of expressing the radius of curvature in reference to
sign .	.	.	.	.	.	.22
16.	Methods of investigating this expression independently of the general
theory of osculation ....	24XIV
CONTENTS,
ART.	PAGE
17.	Explanation of the principle on which the sign of the radius of cur-
vature depends, and of that which gives rise to the double sign · 26
18.	Easy method of obtaining the radius of curvature under a particular
arrangement of the axes .	.	.	.	.27
19.	Remarks on the conventional hypothesis of Lacroix in reference to
the sign of the radius of curvature .	.	.	.28
ib.	Note on the geometrical interpretation	of	imaginary quantities. . 29
DISSERTATION II.
On the Curvature of Surfaces,
20.	On the principal contributors to the theory of curve surfaces	.	33
21.	On the obscurity which still exists in one part of this theory	. ib.
22.	Views of Dupin respecting the interpretation of	-	.	.34
23.	Examination of these views	.	.	.	.35
24.	Fundamental mistake of Monge in reference to the lines of curvature 37
25.	Remarks on the Mémoire of Poisson on the curvature of surfaces . 38
26. Objections to the theory of Poisson .	.	.	.39
27. Investigation of the fundamental properties of surfaces .	. 40
28. Form of the general equation for surfaces of the second order	. 42
29.	On the evanescent curve into which a section of the surface merges
when the cutting plane becomes tangential .	.	.43
30.	This curve is the Indicatrix of Dupin	.	.	.	ib.
31.	Expression for the ultimate ratio of the square of a semidiameter of a
section, to the distance between the cutting and tangential planes . 44
32.	Expression for the radius of curvature of a normal section through
that semidiameter in terms of the aforesaid ratio	·	·	45
33.	Theorem of Euler, expressing the connexion of any normal section
with the principal sections	.	.	.	.	ib.
34.	Sum of the curvatures of two rectangular normal sections shown to
be constant	.	.	.	.	.	.46
35.	The radii of every pair of normal sections through conjugate tangents
shown to be constant ·	.	.	.	. ib.
36.	Sections of mean curvature	.	.	.	.	47
37.	Simple investigation of the theorem of Meusnier	.	.	ib.
38.	Extension of the theorem of Meusnier	.	.	.48
39.	Expression for the radius of curvature in terms of the inclination of
the trace of the normal plane to the axis of x, deduced from the
indicatrix	.	.	.	.	.	. ib.CONTENTS.
XV
ART.
PAGE
40.	Conditions wbich determine whether the indicatrix he parabolic or not 49
41.	Conditions which imply that the indicatrix is hyperbolic ·	. 50
42.	Reason why in the latter case the curvature which is the least possi-
ble is not a minimum .	.	.	.	.51
43.	Expression for the radius of curvature at an umbilical point	. ib.
44.	The indicatrix is only the curve of the second order which approxi-
mates to that in which the parallel sections vanish when the order
of the surface is above the second .	.	.	.52
45.	Consideration of the indicatrix in reference to surfaces of the second
order	·	.	.	.	.	.53
46.	Enquiry whether osculating surfaces of the second order have other
points in common besides the point of osculation
47.	Remarkable property of the osculating paraboloid
48.	General equation of the indicatrix .
49.	Dupin’s proof of the property in (47)
50.	Inferences from the preceding property
51.	Investigation of the differential equation of the lines of curvature
52.	Equations of condition when has innumerable values
dy
53.	Impossibility of the two values of -f- becoming identical.
ax
dy
54.	On the peculiar values of which fulfil more than the ordinary
conditions
ib.
55
56
57
58
59
60
61
62
55.	These values refer to directions of closer contact with the osculating
sphere at an umbilicus .	.	.	.	.64
56.	There is at least one direction in which the contact with the oscu-
lating sphere at an umbilicus is more than ordinarily close, unless
the umbilicus is a vertex of revolution	.	.	.65
57.	At a vertex of revolution the contact is uniform all round the point . 66
58.	Method of ascertaining whether a surface has umbilical points . 67
59.	Method of determining the lines of curvature from the general diffe-
rential equation	.	.	.	.	.68
ib. Nature of the equation when the x and y in it receive particular
values	.	·	.	.	.	.69
60.	Some of the conflicting doctrines about lines of curvature traced to a
neglect of the distinction between the differential equation of the
lines, themselves, and the algebraic equation involving the values of
dy
~ at particular points .	.	.	.	. ib.
dy
61.	Impossibility that the values of as given by the equation of the
third degree, referring to an isolated umbilicus, can be all real .	72XVI
CONTENTS.
ART.	PAGE
62.	Impossibility that any of these values can be imaginary when the um-
bilicus is not isolated* but situated on a line of spherical curvature 73
ib. Remarks on an erroneous inference of Dupin .	.	. ib.
63.	Probable cause of the preceding mistake .	.	.	.74
64.	Observations on the cases in which the differential coefficients r, s, t,
take the form 2 at the point under consideration	.	.75
65.	How to ascertain whether the occurrence of this form actually
implies the existence of points of a peculiar character .	.77
66.	Investigation of the general equation of the lines of curvature on the
ellipsoid	·	.	...	.	.78
67.	Particular equation of these lines when projected on the plane of the
greatest and mean diameters .	.	.	.	.82
68.	Remarks on the determination of the constant in this equation . b
69.	Inference from this determination* namely, that through every point
on the ellipsoid two lines of curvature pass	.	.	.84
70.	These two curves merge into one at certain points : there are four
such points—the umbilici of the surface	.	.	.85
ib. Erroneous deductions of Monge, Dupin, and Poisson on this subject 86
71.	Remarks on the two values of the constant at each of the umbilici . ib.
du o
72.	On the true meaning of -j— ==- at an umbilicus. Errors of pre-
air 0
ceding analysts	.	.	.	.	.	.88
73.	On the actual projections of the lines of curvature : manner of con-
structing the elliptic projections	.	.	.	.89
74. Construction of the hyperbolic projections	.	.	.91
75.	On the change which takes place at an umbilicus in the character of
the surface	......	.94
76.	Equation of the unique line of curvature through an umbilicus
deduced from the general equation	.	.	.	.95
77.	Lines of curvature on surfaces of revolution of thesecoud order .	96
78.	Equation of the projections when the plane which receives them is
that of the least and mean axes .	.	.	.	.97
79.	On the actual construction of these projections	.	.	.100
80.	On the singular solution to the differential equation of the pro-
jections. Geometrical interpretation of this	.	.	.101
81.	On the lines of curvature of the hyperboloid	.	.	.	104
82.	On those of the double-sheet hyperboloid ....	105
83.	On the lines of curvature of the paraboloid	.	.	.	106
84.	Nature of the projections when the paraboloid is elliptic .	.	107
85.	Character of the projections when the paraboloid is hyperbolic	.	108
Note on the sign of the radius of curvature of a normal section of a
surface	.	.	.	.	.	.110CONTENTS.
XVII.
DISSERTATION III.
On the Incommensurability of the Circle, and on the Summation of
Infinite Series·
ART.	PAGE
86.	On the different attempts to prove that the ratio of the circum-
ference of a circle to its diameter is an interminable decimal . 115
87. Analytical proof of the incommensurability of this ratio	.	.117
88.	Formulas for the summation of certain classes of infinite series, with
examples of their application .....	120
89. Formulas for other classes of series, with examples	.	. 128
DISSERTATION IV.
On the Theorem of Sturm, with Improvements.
90.	Some account of the theorem .....	137
91.	Comparative merits of the methods of Lagrange, Sudan, Fourier,
and Sturm .....	.	·	138
92.	Investigation of a general form or type of the calculation	.	142
93. Precepts to abridge the numerical work	.	.	.	.146
94. Other means of facilitating the operation .	.	.	.147
ib. Examples of the method .	.	.	.	.	.148
95.	Consideration of circumstances which sometimes diminish the num-
ber of functions	.	.	.	.	.	.149
96.	Analysis of an equation under these circumstances	.	.155
97.	Example of a method of shortening the numerical process, useful on
certain occasions	......	157
98.	Analysis of Colonel Titus's problem ....	159
99.	Remarks on the preceding analysis, with additional examples of the
method ........	161
100.	Application of this method of analysis to literal equations of the
fourth degree	.	.	.	.	.	.	.165
101.	Application of the method to literal equations of the fifth degree .	167
302. Remarks on the general expressions which result from this analysis 170
103.	Application of the general expressions to a particular example of
the fifth degree	.	.	.	.	·	.	.371
104.	On the general expression for X3 in a literal equation of the sixth
degree ........	172
105.	Remarks on the increase of numerical labour attending the com-
putation of the advanced functions of Sturm .	.	.173xvm
CONTENTS.
ART.
106.
107.
108.
109.
110.
111.
112.
113.
114.
115.
116.
117.
118.
119.
120.
121.
122.
123.
124.
125.
126.
127.
On the means of reducing this labour; which is shown, however, to
be less than that of any other method .	·
Elementary character of the whole operation
Use of the transformations of Sudan in certain cases: superiority
of Sturm's method over that of Fourier shown from the examples
selected by Fourier himself .....
Method proposed for abbreviating the work of computing the
closing steps in Sturm's theorem	.	.	·
Example of the application of this abbreviated method
Remarks on the preceding operation ....
Analysis of the foregoing equation	....
Means of ascertaining whether roots, having several leading figures
in common, are really equal or not ....
Application of the improved method to a complete equation of the
sixth degree .......
Example of the analysis of an incomplete equation of the seventh
degree ........
Application of the method to a complete equation of the seventh
degree ........
Discussion of the case in which the derived functions do not
descend in degree regularly by unity ....
Examples of the modification which the work undergoes in such a
case ........
Example of the work when two consecutive functions differ in
degree by three units ......
Example of the entire operation when the derived functions are
fewer than the ordinary number in the analysis of a complete
equation of the sixth degree .....
Brief summary of the facilities introduced into Sturm's method by
the modifications proposed in this Dissertation .
Observations on the rules of Fourier and Budan	·	·
Proposal of a short method of performing the first step in Sturm’s
method ; that is, of determining the function X2
Investigation of this method	.....
Application of this improved mode of working to the examples given
in the Theory of Equations	.....
Remarks on the extreme simplicity to which the analysis and so-
lution of cubic equations is now reduced, and examples of the pro-
posed improvement in equations of the higher degrees .
Statement of the objects proposed to be accomplished in the Notes
subjoined to this Dissertation	.....
PAGE
174
175
176
180
181
183
185
187
189
191
194
196
199
201
203
206
207
208
209
210
212
216CONTENTS.
XIX
ART.	PAGE
128.	On the connexion between the coefficients in the functions of
Sturm, and those in Lagrange’s equation of the squares of the
differences .	.	.	.	.	.	.218
129.	Lagrange’s conditions for the general equation of the fourth
degree	....... 219
130.	Simplification of one of Sturm's conditions .	·	. 221
131.	Comparison of Sturm’s general conditions with those of Lagrange 223
132.	Application of the method of proceeding, adopted in this Disserta-
tion, to the theory of elimination	.	·	.	. ib.
133.	On equations containing equal roots, or roots nearly equal	· 225
134.	Example of an incomplete equation of the sixth degree with equal
roots	.	..... 227
135.	Example of a complete equation of the sixth degree with equal
roots	...... 229
130. Observations on the infallibility of the improved method, whether
the roots be equal or not	.	.	.	.232
137.	On equations supposed to have roots differing from one another
only at a remote figure .	.	. ~	. 235
138.	Proof that when equal roots exist, the function preceding the last
always has its extreme terms divisors of the extreme terms of the
proposed equation	·	230
139.	On a certain principle of universal application in facilitating the
analysis of an equation, and which has hitherto had but a limited
scope ........ 237
140.	Partial application of this principle by Sturm .	.	.	238
141.	Extension of it proved to be allowable .... 23»
142.	Remarks on the utility of the principle .... 240
143.	Universality of it shown	.....	241
144.	Application of the principle to an equation of the seventh degree · ib.
145.	Observations on the extreme simplicity to which the analysis of
equations of the third and fourth degrees is now reduced	. 243
148. Supplementary Note on an improvement in a certain step of the
investigation of the Binomial Theorem	.	.	.	245The Author of the present work begs to announce that the
ANALYSIS OF
CUBIC AND BI-QUADRATIC EQUATIONS,
according to the Improved Methods adverted to in Note II., and by
means of which this part of the General Theory of Equations is reduced
to a very remarkable degree of simplicity, will form the subject of a
distinct publication.
Also that a New and greatly Improved Edition of
THE ELEMENTS OF
PLANE AND SPHERICAL TRIGONOMETRY,
&C. &C.,
will be published shortly.
ERRATA.
Page 107, line 7, for hyperboloid read paraboloid.
119,	11,... Att*	... Ptt*.DISSERTATION I.
ON THE THEORY OF THE COORDINATE SIGNS; OF THE SIGN
OF THE TRIGONOMETRICAL SECANT, AND OF THAT OF
THE RADIUS OF CUR VATURE ; WITH AN EXAMINATION
OF THE OBJECTIONS OF CARNOT.
IMATHEMATICAL DISSERTATIONS
INTRODUCTION.
(1.) This volume is intended for the use of those who
may be prosecuting a course of study in the modem
mathematical analysis ; and is undertaken in the hope that
it may occasionally supply acceptable assistance and in-
struction supplementary to that furnished by our ordinary
elementary works.
It is usually considered as not within the scope of such
works to present more than a simple exposition of the
elements of science. Controversial discussions and criti-
cal dissertations are professedly, and properly, excluded ;
and indeed, one would think, absolutely forbidden in en-
quiries whose steps, being all conducted by the light of
demonstration, must lead to conclusions necessarily true.
Nevertheless, the fundamental principles of even the most
firmly established analytical theories have been assailed
and repudiated by mathematicians of the highest order.
Discrepancies, still unreconciled, have been detected in
our standard works on pure science ; and even some of
the symbolical expressions most familiar to the algebraical
student, are contemplated by modern analysts with dif-
ference of opinion as to their real interpretation.4
MATHEMATICAL DISSERTATIONS.
(2.) It is quite clear that if these statements be true, it
would be an interesting and, for science, an important
undertaking, to institute such a searching enquiry into
the sources of these discrepancies and conflicting doctrines,
as would ultimately lead to the removal of all ambiguity,
the reconciling of all apparent contradiction, and the
establishing, beyond valid objection, of all those unsettled
points which still divide the cultivators of modern science.
Such an extensive undertaking will not however be
looked for in the present unpretending volume; which is
designed merely to assist the young student in his efforts
to acquire precision of ideas upon some of those more
delicate points of analysis which present themselves even
in an elementary course; but which are usually discussed
without any cautionary hint to the reader as to the re-
strictions under which they must be received in order that
they may conform to the prescribed laws of algebra.
(3.) If one could be quite certain that these governing
conditions would be always present to the mind when em-
ploying the forms of analysis, any such caution would be
superfluous; but, as the preceding statements imply, the
necessary controlling influences are sometimes overlooked;
and thus contradictory theories and disputed doctrines
have arisen, which have tended to bring discredit on the
conclusions of algebraic science; for there can be but little
doubt that every discrepancy in the results of analysis
must arise from some oversight of this kind, however diffi-
cult it may be to detect the precise step in the investi-
gation where the inadvertence has occurred. In the extreme
case of a problem, where the hypothesis is considered as
having reached its ultimate state; and, on the other hand,
in those numerous extensions of analytic formulas by
which they are made to comprehend cases even beyond
the bounds of the original hypothesis, especial caution is
necessary in order to avoid mistakes of this kind; becauseINTRODUCTION.
5
in these critical or novel circumstances, our formula may
be materially affected by some of those general laws of
analysis which, in the ordinary interpretation of the ex-
pression, are not brought into operation; and which are
therefore in danger of being disregarded when our modifi-
cations really render such expression wholly subject to
their influence. To illustrate our meaning by a simple
instance, we may remark that the signs of operation
“J” and — are wholly inefficient when prefixed to zero or 0 ;
so that when an additive or a subtractive quantity has,
in the ultimate state of the hypothesis, been made to dis-
appear, it is, in ordinary cases, of no moment whether we
preserve any trace of this quantity by the actual insertion
of the zero in its place or not; in fact, such insertion
would usually and justly be considered as a useless incum-
brance, and the zero is accordingly suppressed ; yet there
are cases in which this suppression would lead to what,
for the moment, would seem a contravention of the esta-
blished principles of algebra. Thus let the following qua-
dratic equation be proposed for solution, namely,
x2 — 4x = — 4 ;
then completing the square, we have
x2 ™— 4x 4" 4 ~ 0,
and extracting the root
x — 2 = 0
/. x = 2.
Now we know it to be a principle, perfectly general and
incontrovertible, that every quadratic equation has two
roots; whereas we are here presented with only one, and
no trace is exhibited of a second. The apparent discre-
pancy will no doubt be easily accounted for by the reader,
who will see that if the zero which is here suppressed be
inserted, both the roots will present themselves in the6
MATHEMATICAL· DISSERTATIONS.
result. Restoring then this zero, we shall have, after ex-
tracting the root,
x — 2 = + 0 or — 0,
-, and we re-
hence the two roots are
a: = 2, x = 2,
that is, the roots are equal, each being 2.
Again, if we have a fraction, such as—
quire to find the value of it in a particular case, as for
instance when x = 6, b being different in value from a,
we may at once substitute b for r, perform the division,
and the sought value will be exhibited in the quotient.
It is easy to see that this value is to a certain extent inde-
pendent of the form of the general fraction; for it might,
as far as this particular case of it is concerned, be changed
bx1 — a3 b2x — a3 __	_
into---------, or---------. But take the extreme case, that
x — a
of x = a9 then the corresponding value of the fraction will
be entirely governed by its form; and will no longer be
ax3 — a3 a3i — a3
the same under the changes---------and-----:---· Here
x — a	x — a
then is an instance where the law, by which a general
algebraic expression controls the individual cases compre-
hended in it, is rigorously enforced only in the extreme
one of those cases, in which a neglect of its influence
would lead to error, although no error would arise from
such neglect in any other of the series of cases which this
one terminates. The three fractions
a·3 — a3 ax2 — a3 a2x — a3	r 11
,	y	• •••••111
x — a	x — a x — a	■*
furnish the three different quotients
x2 -f- ax -h a2, ax + a2, a2..........[2]
and, for x = a9 the three different values
3a2, 2a2, a2;INTRODUCTION.
7
the first of which is the only value belonging to the series
#3 — a3
of values comprehended in the expression ----, under
the various hypotheses as to the value of x. This is plain,
because whatever be the law implied in either of the ex-
pressions [1], the same must be implied in the identical
expression [2]; this latter being the real thing that [1]
represents; and only the first of [2] furnishes the proper
value of the proposed fraction for x = ft.
Having made these introductory remarks, we shall now
proceed to examine some of those more interesting parts
of analysis respecting which the student may be in danger
of entertaining mistaken views ; either from the subtlety of
the reasonings which they involve, or from the force of the
more formidable objections which may have been brought
against the principles upon which they rest. Where these
principles seem to us to be really defective, or the argu-
ments founded upon them illogical, we shall endeavour to
make the blemishes apparent, and shall supply, as well as
we can, the necessaxy corrections.DISSERTATION I.
ON THE THEORY OF THE COORDINATE SIGNS*
(4.) In the modern application of algebra to geometry a
very important and novel use is made of the signs + and
—, the marks which in calculation usually denote the
common operations of addition and subtraction. We say
usually denote, because, although such is their primary
office, and the purpose for which they are conventionally
introduced into algebraical notation, in the character of
signs of operation; yet, as every student of algebra knows,
these same signs, while they perfectly perform the office
originally allotted to them, are moreover efficient beyond
the limits which this office seems to prescribe to them;
and admit of a wider interpretation than appears to be
implied in their original definition.
This apparent extension of the meaning given to these
symbols at the outset is so naturally suggested to us after
wTe have advanced a few steps, and is seen to be so neces-
sary to the perfection of algebraic language, that it is
usually at once tacitly made, so soon as the need for it
is felt, without formally returning to our definitions for
the purpose of making the modifications apparently ren-
dered necessary. Thus, the meaning of the isolated ne-
gative quantity, whenever it presents itself in the result of
a problem, at once suggests itself; and is dependent upon
no new convention. A subtraction, in the common sense
of the word, is certainly not denoted by the minus in such
a case; yet it must bear to the plus a relation perfectlyTHEORY OF COORDINATE SIGNS.
9
analogous to that which subtraction bears to addition,
otherwise our meaning would be something more than a
simple extension of that already laid down; it would be
an incongruous addition to it. The isolated minus quan-
tity therefore, standing thus in direct opposition to the
isolated plus quantity, has its meaning immediately infer-
rible from that of this latter, which is always at once fixed
by the problem itself. If, for instance, the problem be to
find the amount of gain arising out of any pecuniary trans-
action, this gain will be respresented algebraically by an
isolated quantity assumed to be plus; but should the re-
sult not justify this, but turn out to be minus, the proper
interpretation, in strict conformity to the principle laid
down in the definitions, of the direct opposition of + and
—, is obvious; the result of the transaction in question is
not gain but loss. In like manner if the enquiry respect
the distance of an object from any fixed boundary, and
we start by assuming the object to be upon one side of
the boundary and at a distance from it, represented as usual
by a quantity assumed to be plus, which however appears
in the result with minus, we immediately infer, and in
strict conformity to the principles laid down, that, although
the distance of the object from the boundary was properly
represented, its place is on the side of it opposite to that
supposed.
(5.) It thus appears that the signs + and —, though
attached to isolated quantities, and consequently not im-
plying any addition or subtraction, have nevertheless a
meaning which is perfectly intelligible; and which, though
quite distinct from the signification primarily given to
them, is in strict harmony with that signification, and is
in fact only an extension of it.
It was the observation of this circumstance, coupled
with considerations still more refined, that doubtless led
Des Cartes to his beautiful system of analytical geo-10
MATHEMATICAL DISSERTATIONS.
metry; in which the algebraic letters x, y, being employed
to represent the absolute distances, abscissas and ordinates,
of the points of a curve from two fixed boundaries or axes
crossing one another, the algebraic signs + and —, pre-
fixed to these letters, make known upon which side of
each boundary these several points are situated; and thus
can be expressed in the language of algebra, not only dis-
tance but position.
In which direction from any boundary or axis the dis-
tance of a point shall be marked +, is obviously a matter
of arbitrary choice; but this choice once made, then the
mark —, conformably to what has been shown above, ne-
cessarily implies distance in the opposite direction. The
only thing therefore that is conventional in this appro-
priation of the algebraic signs, is which of the two directions
shall be called plus, the opposite direction being minus as
matter of course. We presume the student to be already
sufficiently familiar with the principles of analytical geo-
metry to be aware of what the ordinary convention is as to
the disposal of the plus sign; that is, in what directions
the positive abscissas and the positive ordinates are usually
taken: he will do well also to remember that, in this appli-
cation of the signs, the terms positive and negative are
merely names given to these symbols of direction, and that
they imply no distinction among the lines themselves save
that of direction.
(6.) Carnot, a mathematician of great eminence, has
strenuously opposed, as inadmissible and absurd, the fun-
damental principles, and consequently the entire system of
the Cartesian geometry. We are not aware that his rea-
sonings on this matter have ever been proved to be nuga-
tory. But we are confident that, if the remarkable work
in which his objections are developed at such length, were
generally accessible to or consulted by students entering
upon the study of analysis, but few would care to wasteTHEORY Or COORDINATE SIGNS.
11
their time on the alleged absurdities of analytical geome-
try. To prove the existence of these absurdities Carnot
reasons as follows :*
Let there be any curve, a circle for example, ACRDBESF,
in the plane of which let two axes AB, RS, be traced per-
pendicular to one another, and cutting at the centre K,
taken for the origin of the coordinates. From any point C
in this curve draw Cp, and prolong it to D. Now, from
the foregoing principles, pD being regarded as positive,
0 \Cp)will be negative, so that we ought to have Cp = — pD;
whence we get Cp >+ pD,'or CDj = 0^ which is absvrcLj
In like manner, if from the same point C, we draw Cm
perpendicular to AB, and prolong it to F, we shall have
mF = — Cm; consequently, Cm + mF, or CF = 0, which
is equally absurd. Moreover, since we have got CD = 0,
CF = 0, we shall have also CD*CF = 0; that is to say, the
area of the rectangle CDEF == 0; from which it would
follow, for example, that the square inscribed in a circle
would be 0. Such are the errors which unavoidably arise
from the foregoing notion. It is then false that Cp is a
negative quantity; it is easy to prove, indeed, that it is
necessarily positive, for if from any point V of the straight
line AB prolonged beyond A, we draw a straight line VT
parallel to KR, it is evident, and admitted even by those
who adopt the preceding notion, that the three straight
* Geometrie de Position—Dissertation Préliminaire, p. x.12
MATHEMATICAL DISSERTATIONS.
lines TC, Tp, TD, all on the same side of the new axis
VT, are positive: now we have Cp = Tp — TC; there-
fore, since Tp > TC, Tp — TC, or Cp, is a positive
quantity.
It may, perhaps, be said that Cp is positive in re-
ference to the axis TV, and negative with respect to the
axis RS. But this would be a new enigma to solve more
difficult even than the first, and that too in a science whose
principal character is evidence.
To demonstrate in a manner still more palpable how by
this principle of negative quantities taken in a direction
contrary to that of the positive quantities, we are una-
voidably led into error, I shall give a new form to the pre-
ceding argument as follows:
Let the circle ARBS be described, let K be its centre,
and draw a diameter AB; we shall have
AB = AK + KB .... [A].
Now through the centre K, I draw the axis RS, perpen-
dicular to AB; then through any point C of the circum-
ference, I draw the chord CD parallel to AB, and which I
suppose to cut the axis RS in the point p. I take R for
the origin of the abscissas: I call the abscissa Rp, x;
the ordinate pD, y; and the radius a. I shall then have
y2 = 2 ax — a2, and therefore y = + V 2ax — x2;
which teaches me, from the admitted theory of negative
quantities taken in a contrary sense to the positive quan-
tities, that y has two values, equal and directly opposite:
one pD represented by the positive root, the other Cp
represented by the negative root; that is to say, we have
p D = + lax — x2 and Cp = — >/ 2ax — ar2
equations which ought to have place whatever be the
value of Rp or x. Let us suppose then that Rp = RK,
or x = a; pD will then become KB, and Cp, AK; there-THEORY OF COORDINATE SIGNS.
13
fore the preceding equations will be KB = -f o, and
AK = — a. Substituting these values of AK, KB in the
equation [A] found above, we shall have AB = 0; a result
which is absurd, though rigorously deduced, or rather
because rigorously deduced, from the theory of negative
quantities being contrary in direction to positive quantities.
This theory then is completely false. It is possible perhaps
to oppose metaphysical subtleties to the above reasoning;
but I do not believe that it can be answered in a manner
clear and satisfactory to a geometrical mind.
(7.) Such are the reasonings and conclusions of Carnot.
The student will do well to examine them with that
attention which the arguments of so distinguished a man
must always deserve; and in proportion as this examination
unsettles his confidence in those theories which he had
previously thought to have been firmly established in his
own mind, will be the necessity of a return to, and a vigi-
lant scrutiny of, the principles whence they have been
deduced, and which are here so unsparingly condemned.
But in all these arguments Carnot has committed the
singular oversight of making assumptions in his premises
which are themselves distinct contradictions of the very
theory he impugns; and, as his reasonings from these pre-
mises are perfectly logical, the conclusions must be absurd
as matter of course, provided the theory contradicted in
the premises be really accurate.
For, referring to his diagram, and bearing in mind the
theory of the directive signs + and —, the two lines pD,
pG, ih reference to the boundary or axis RS, are re-
spectively represented both in absolute length and in
direction by -f pD and — pC. If we wish merely to
consider the absolute lengths of these lines with a view to
ascertain that of their sum or difference, then, as direction
is no element of the calculation, the signs of direction may,
and indeed ought, to be suppressed. But if we choose to14
MATHEMATICAL DISSERTATIONS.
preserve them we recognize their theory; and must of
course conform to it from the outset. Hence, although the
length of pD is equal to that of pC, yet it is not true that
-f- pD = — pC; the sign of equality palpably contradicts
the statement implied in the signs + and —, that the
directions are not the same, but opposite. Carnot, in the
equation Cp = pD, with which he sets out, affects to
express, agreeably to the theory, that Cp is equal and
opposite to pD; and thus the sign — is prefixed to pD
to overrule this contrariety of direction, and so render the
equality complete. Now this would be quite unobjection-
able if the lines Cp, pD of the figure, were really repre-
sented, in the foregoing algebraical expression of them, with
the proper directive signs; but these signs, which involve
the whole matter in dispute, are unaccountably omitted;
and the equation therefore is not that which the theory
condemned would furnish. In accordance with this theory
the proper representation of Cp, as well in direction as in
magnitude, is — Cp; and that of pD, + pD; and either of
these is strictly equal to the other, when the direction of
one of them is reversed, or that reversion assumed by
changing the sign of either. The correct equation is there-
fore — Cp = — pD; from which, of course, no absurdity
can arise. A similar oversight is committed in the state-
ment that rriF = — Cm, where the right direction is given
to Cm and the wrong one to mF.
(8.) Let us now briefly examine the other argument, in
reference to the equation [A], namely,
AB = AK + KB .... [A],
the second member of which does correctly express the
absolute distance between the points A and B: the directive
sign of AK is however not actually exhibited; nor would
this be matter of censure, but rather of approval, since, as
we have already remarked, when the object is to exhibitTHEORY OF COORDINATE SIGNS.
15
merely the absolute sum or difference of lines, the directive
signs are superfluous. As, however, the equation is brought
forward in the subsequent reasoning as if the proper
directive signs were actually inserted, a supposition is
made with respect to that equation which is contrary to
fact: the sign of AK is not there· Introducing this sign,
the proper mode of expressing the interval AB will be
AB = - (- AK) + KB .... [B]
and, as the absolute length of AK, or KB, is assumed equal
to a, we have KB = a, and — AK = — a; and therefore
AB = 2«, as it ought to be.
Now it is worthy of observation, that though the double
minus in [B] really expresses the same thing as the plus
in [A], yet the form [B], of expressing the interval AB, in
conformity to the theory we are here discussing, is immea-
surably superior to the form [A], adopted by Carnot, since
this latter is not general, but requires that the point A be
never situated nearer to B than K. If A were to move up
to K and pass through it, [A] would no longer represent
the interval AB: the diagram would show us that that
expression must be modified before AB could be accurately
represented by it. We should thus have to depend upon
our diagram as a check to prevent errors we might other-
wise commit if guided solely by our algebraical repre-
sentation : that representation then is imperfect; and it is
the paramount advantage of the coordinate method that
from this imperfection it is entirely free; it completely
pictures the diagram in the algebraical symbols, and all its
deductions may be interpreted, with unerring certainty,
independently of any aid from the figure to which those
deductions apply. In the method of coordinates the in-
terval (AB) between the extremities of any two abscissas
x\ xf' is always expressed by AB = ¿r' — x"; where a/, x/'
stand for the lines both in magnitude and direction: in the
particular case, [B], x" has the particular value — AK.
It is thus easy to see, without any more minute exa-16
MATHEMATICAL DISSERTATIONS.
mination of Carnot’s objections, that they are founded in
an erroneous view of the doctrine opposed.
(9.) It may be well to remark here, as it is of consequence
that the student should always bear the fact in remem-
brance, that in thus introducing + and — as signs of
direction into geometry, we recognize the coordinate
directions only; and affirm nothing about the signs of lines
oblique to the coordinates, nor even of lines that may be
detached portions of the coordinates themselves. The
directive signs employed in analytical geometry originate
in the assumption that lines measured from the origin along
either of the fixed axes in a specified direction, are + ; and
in the inference from this assumption, and from the mutual
relation of -f- and —, that lines from the same origin along
the same axes, but in opposite directions, are —. The
directive signs, be it observed, have no other appropriation
in analytical geometry but this; and if any line, not an
abscissa nor an ordinate, be affirmed to be plus or minus,
the assertion cannot be justified as simply a consequence
of the principles already laid down, but only in virtue of
some new assumption or additional principle. The ques-
tion so frequently agitated about the directive signs of
lines oblique to the coordinates is one of great absurdity.
No line has any such sign till we arbitrarily endow it with
one, or, which amounts to the same thing, till we give one
to the line directly opposite to it. In enquiring, therefore,
about the directive sign of a line, the first question to be
put is,—u Is that line an ordinate or an abscissa ?” If it
be one or the other the enquiry is a proper one, admitting
of a definite answer, for a determinate directive sign such
line unquestionably must have; but if the line be neither
an ordinate nor an abscissa, the enquiry is not only useless
but meaningless, no directive sign having been appro-
priated to it.
Whenever, therefore, we have to determine analyticallyTHEORY OF COORDINATE SIGNS.
17
the length of a line oblique to the coordinates, the sign
or signs with which the resulting absolute value may be
accompanied, arise out of the very general processes of
algebra; and in the particular case in question are, as far
as direction is concerned, perfectly meaningless appen-
dages, in the absence, be it remembered, of all special
assumption or purely arbitrary interpretation. It may,
however, be noticed that when such a result appears with
the single sign — it will be found, upon an examination
of the reasoning, that an assumption has at some part of
it been made, and perhaps made inadvertently, of the cor-
relative + ; but for such an examination there is no real
occasion, as the sign has no influence upon the absolute
length, and this is the only thing that is strictly determinate.
(10.) Nevertheless, as respects those oblique lines which,
while they preserve their general relation to certain co-
ordinates, are susceptible of continuous changes of length
and position, and so pass through zero or infinity, we are
not always at liberty to prefix to them in any position
either of the signs -f or — indifferently, nor to deny their
strict claim to one of these signs rather than to the other;
not, however, because these signs refer to the direction of
the varying line in any proposed position; they are not
directive signs of the line at all: but because in the passage
of the line through 0 or oc, its algebraic sign may have
changed. It may be sometimes easy to see whether or
not such changes really take place; if they do take place,
then it is plain whatever sign the line took before, it must
take the opposite sign after the passage; so that knowing
the algebraic sign that is prefixed to it in one position,
and knowing where these changes happen, we can, in fol-
lowing the progress of the varying line up to any other
position, always pronounce upon its algebraic sign.
(11.) These are the considerations which ought to guide
218
MATHEMATICAL DISSERTATIONS.
us in determining the algebraic sign of the secant in ana-
lytical trigonometry. This is not a directive sign as is
usually implied ; its changes arise from mutations of abso-
lute value, and not from variations in position. The pre-
established conventions however, provided these are in
strict conformity to the general laws of algebra, can alone
regulate the sign (otherwise arbitrary) which the initial
value of the secant ought to take. After this initial sign
is thus fixed, the changes take place, irrespective of all
convention, in obedience to the law that the algebraic sign
prefixed to any varying magnitude cannot change unless
the magnitude pass through either 0 or oo.*
In proceeding through the first quadrant the secant, in
virtue of this law, can undergo no change of sign ; and
that it may at the same time conform to the previous con-
vention, when it coincides with the coordinate axes, it is
necessary that this unchanged sign be plus. Arrived at
the extremity of the quadrant it is -foo, the opposite
secant being — oo. It is this latter that now moves, in-
tersecting the negative tangent, diminishing in length up
to the initial position, and thence increasing to infinity.
Throughout all this range the law forbids a change of
sign ; the secant therefore continues minus.	The oppo-
site secant, taking of course the opposite sign, now re-
volves, and so on continually ; the coordinate conventions
as to which directions on the axes shall be plus, being
regarded only so long as the fixed laws of algebra, para-
mount to all conventions, permit. Similar reasoning
applies to the chord. Its initial position, corresponding
* The converse of this however is not necessarily true; for although a
change of sign can arise from no other cause than a passage through Oor ®,
yet a varying quantity may pass through either of these states without under*
going any change of sign ; and on this account, as remarked by Caonolt,
(Trigonométrie, seconde édition, p. 12,) the principle stated in the text is
not alone sufficient to establish the mutations of sign in the varying trigono-
metrical lines. The passage through 0 or oo must be accompanied with op-
position of direction to necessitate a change of sign.THEORY OF COORDINATE SIGNS.
19
to the arc + 0, has the direction of the plus tangent ; the
revolution of this tangent will give all its other positions
round the circle ; and as it continues finite till the revolu-
tion is completed the values are all plus. At the end of
the revolution, however, the value is again 0 ; but since it
is still plus, and as the series recommences in the di-
rection opposite to that where this terminates, it recom-
mences with a minus. The value therefore in passing
through 0 changes its sign from + to — and continues —
till it again passes through zero, and so on.
(12.) It is easy to see how it happens that, in the case of
such oblique lines as we have been here considering, the
value in any particular position has claim to one sign
rather than to the other. Every such line is but one of a
continuous series of lines, united together by a common
law, and commencing at a common fixed origin, and each
is estimated in reference to this union and to the common
origin. The original sign is fixed by the coordinate con-
vention, the changes by the common law of the signs.
(13.) It is a mistake to suppose, as Carnot uniformly
does, that the particular appropriation of the signs + and
—, to mark the coordinate directions, necessitates the in-.
terpretation of these signs as marks of direction in general.
On the contrary, they always preserve their purely alge-
braic character in every analytical enquiry, implying di-
rection only when prefixed to coordinates in conformity
to the original convention, and which convention is quite
consistent with their common algebraic import.
Instead, therefore, of seeking for an explanation of the
minus sign prefixed to any isolated line in the particular
direction that line may take, unless indeed the line be an
ordinate or abscissa, we ought to regard such sign as having
no reference to position, but simply as a consequence of
some previous supposition of plus as to value, in the course20
MATHEMATICAL DISSERTATIONS.
of the investigation whence our minus result has arisen.
Cases of this kind have nothing to do with the coordinate
theory, wThich they neither confirm nor oppose ; their con-
sideration belongs to the common theory of the algebraic
signs in their ordinary character of signs of operation.
It may be proper to observe, that although, in speaking
of the signs of the trigonometrical lines, wTe have con-
sidered these lines only in reference to the positive arcs,
yet the very same arguments hold wdien the arcs are
negative, proceeding from the origin in the opposite di-
rection. At the origin itself the secant belongs indif-
ferently to the first of the positive, or to the first of the
negative series of arcs; and, as in passing through this
state from either side, no change of sign can take place, we
infer that, whether the arc be positive or negative, the sign
of the secant of it is the same.
R2
The common expression for the secant is —, which
truly represents it both in value and algebraic sign; for
the secant not only passes through oo when the cosine
passes through 0, but it so happens that it also changes
sign after this passage. The theory which connects
the sign with the direction of the secant must, to be
consistent, give the same sign to the radius whose pro-
longation that secant is. The R2 is not the square of
this radius, but the combination of the initial radius with
it. Hence, when the secant is negative, ^t2 ought to
be negative too; and this would lead to absurdity.
Although at the origin the secant is the Same, whether
that origin be viewed in connexion with the positive or with
the negative arcs, yet it is not so with the cosecant, which
is -f oc in the former case, and — oo in the latter. A like
ambiguity is observable in the cotangent.
But the ambiguity here noticed, although more pal-
pably felt than in the other trigonometrical lines, is never-
theless not peculiar to the cosecant and cotangent. TheseSIGN OF THE RADIUS OF CURVATURE.
21
others are all subject to it at their limits, although all
situated in the co-ordinate directions. For take any finite
ordinate, say a plus ordinate, and let it continuously
shorten, till its extremity reaches the axis of abscissas ; its
representation is then strictly + 0, while the opposite or-
dinate, diminishing in the same manner, and arriving at
the same limit, would be — 0. To be sure, as the value
here actually vanishes, attention to this difference of sign
is commonly of no moment ; yet the analytical fact of the
existence of such difference ought not to be overlooked ;
and indeed, in the doctrine of vanishing fractions, the
neglect of it might unquestionably lead to error.
ON THE SIGN OF THE RADIUS OF CURVATURE.
(14.) In the general theory of curves, founded upon
that of coordinates, the differential expression for the ra-
dius of curvature is usually written thus :
the double sign being introduced because of the extraction
of the square root, and the proper sign to be taken, in
any individual case, is usually supposed to be such as to
mark the direction of curvature. But the radius of curva-
ture, being in general a line oblique to the coordinates,
has, in strictness, no directive sign; and therefore to
d*y
dx222
MATHEMATICAL DISSERTATIONS.
designate it as positive or negative, in reference to direction,
is improper. The only thing that is determinate is the
absolute length of the radius. That which determines di-
.	.	. d?y ·
rection of curvature is the algebraic sign of ; for if a
linear tangent be drawn through the point of osculation
(x, y\ an ordinate to it, corresponding to the abscissa
x + A, is
V +
•m
and an ordinate to the curve, corresponding to the same
abscissa, is
dy d2y h2
9 + £h + ?s 7T+&C.......[2]
so that taking a sufficiently small value of h, [2] will obvi-
d2y
ously exceed or fall short of [1] according as — is posi-
tive or negative; in other words, the curve will be convex
<Py
to the axis of x at the point of contact, if — be plus, and
concave if it be minus.*
(15.) Because in the common applications, it is the part
of the curve concave to the axis of x that is usually sub-
mitted to the above formula, it is sometimes recommended
to preserve in that formula only one of the two ambiguous
signs, the sign minus, in order that the radius belonging to
* All this however is on the supposition that the point under consideration is
itself situated above the axis of x, or has a positive ordinate. The general in-
ference from [1] and [2] is that the curve will be convex to the axis of x if y
and —■ have the same signs, and concave if they have different signs.SIGN OF THE RADIUS OF CURVATURE.
23
a concavity may come out plus.* But these subsequent
adjustments of a general formula can never be necessary
unless there be some defect in the analysis, or a mistake in
the result itself; and accordingly, by paying strict attention
to the algebraic signs throughout the general investigation
whence R is derived, it will be invariably found, whatever
method of reasoning be employed, that enters into the
resulting expression for R preceded by the sign minus;
so that, altogether disregarding the double sign ±, which
is a meaningless appendage to the above form, the proper
value of R is
dx2
If this formula be applied to any particular point in the
convexity of a curve we shall have a negative result, but
this is merely one of those common algebraical cases
already adverted to (13), and is to be explained in the
same way; namely, we are to conclude that in the parti-
cular example before us, a negative quantity, or a negative
remainder, has in some part of the investigation been
tacitly considered as positive. Upon examination we
accordingly find that y — /3, which occurs in every step of
the reasoning, and which is positive for the concavity,
* “ Le double signe est relatif à la position de R j si la courbe tourne sa
d~y
concavité vers 1 axe des æ, alors sera négatif, et pour qu’alors R se déter-
dsr
mine positivement, nous le prendrons avec le signe négatif, et nous écrirons
d%y
dx*
(Boucharlat, Calcul. Diffp. 112.)24
MATHEMATICAL DISSERTATIONS.
changes to negative for the convexity; so that if the
change take place for two points in the same curve, R, in
proceeding from one to the other, must pass through co .
We may with propriety affirm, therefore, that when, in
any individual example, R comes out plus, not that this is
in consequence of R taking a particular direction, but that
the curve is concave to the axis of x at the point; and that
when it comes out minus the curve is convex.
(16.) The following methods of investigating the dif-
ferential expression for the radius of curvature, independ-
ently of the general theory of osculation, may perhaps be
acceptable to the student.
The trigonometrical tangent of the inclination of a
curve at (x, y) to the axis of x is
and, consequently, the tangent at the point whose abscissa
is x + h is, by Taylor’s theorem,
tan «' = H- ^ h + &c*
dx dx4
Hence, by the known trigonometrical form for the tangent
of the difference of two angles we have, calling a! — a, a a
tan Aa =

dy2	dy d2y
The angle a a is that between the two linear tangents at
the points whose abscissas are x and x + h; or which is
the same thing, the angle between the normals at those
points. Call this latter angle a q ; then, observing thatSIGN OF THE RADIUS OF CURVATURE.
25
tana and tan# have contrary signs, and increase or di-
minish together, we must have
tan a o
h
d2y 0
dtf dy
dx2 dx
d2y
d*h + &c-
Consequently, in the limit,
d2y	d2y
dO____ dx2 ___________ dx2
dx	dy2	ds2
^ dx2	dx2
Now — expresses the radius of curvature,* and this is the
ds de	_
same as	5 consequently,
Otherwise,
dy . _ ds
tan a = ~r~; and R = —
dx	do
Differentiate the first of these, and we have, observing as
,	, da	do
above that - = —
dx	dx
da
1 dx
= — sec
do __d2y
dx dx2
* For A$ being the arc of the curve subtending the angle A# between the
A s
normals at its extremities, it is plain that —- continually approaches to, and at
A0
length arrives at R; as this ratio proceeds towards, and finally arrives at its
ds
ultimate state,
dO'26
MATHEMATICAL DISSERTATIONS.
	æy	<£y
do _	dx2 '	dx2
dx	1 + tan * a	i+Ê? + dx*
dx	(-)3 do v dx'	
	dx d2y	
dx2
as before.
(17.) To whatever particular example this general formula
for R be applied, if the result come out plus, we may infer
that, whatever tacit supposition as to this sign has been
apparently introduced into the general investigation of R,
the particular case in question confirms that supposition;
but if the result come out minus, then we are to conclude
that the supposed plus is in our particular example minus.
In the general investigation given in the Differential
Calculus, page 164, the quantity subject to this change of
sign, and yet apparently assumed plus, is y — p. In the
first investigation above it is a' — a · and it is virtually this
in the second process, the assumption being that ~ is plus.
Let either of these suppositions be reversed, and R will
change its sign accordingly. We may of course leave the
hypothesis distinctly open throughout the general in-
vestigation : this will merely require that we write ± {y— (5)
fory— P; or ± Aa for a a ; or ± —for — ; we shall then
ax ax
arrive at the double formula
■ ds"
R:
(—)
VdxJ
_ d?y
dx2
[0]
the upper or lower sign having place according as the
upper or lower sign has place in the hypothesis, as to the
quantities selected above; or, which implies the same thing,SIGN OF THE RADIUS OF CURVATURE.
27
according as the curve is concave or convex to the axis of x.
The double sign therefore does not denote ambiguity, but
distinguishes the results of the two hypotheses·
The double sign in [A] arises, as all such superfluous
appendages must arise, from the analytical premises being
capable of a wider interpretation than the geometrical con-
ditions strictly warrant. The problem really solved by
the analysis is this, namely: To determine the length
of a straight line R, drawn from a point (a, /3), to a
point (x, y); this line being perpendicular to a certain
line (the tangent) through (x ,y). The analysis furnishes
two points (a, /3), one on each side the tangent, as we have
already explained at length elsewhere. (See Anal. Geom.
Part I., p. 42.) The formula [B], when employed in any
particular case, will involve a distinct hypothesis as to
y — /3, and consequently a distinct hypothesis as to x — a;
this so qualifies the analytical conditions that the double
sign before the numerator of [B] would be not merely su-
perfluous but unjustifiable.
There is some confusion of signs in the reasoning of
Cauchy, {App. du Calcul Infinitesimal a la Geometrie,
p. 89,) in reference to the radius of curvature. He repre-
sents the simultaneous increments of the arc s and an-
gle a, by dt as and ± A a; and obtains the equation
± as=±Ra« (in the limit); and thus, under the
appearance of the greatest generality, does really assume
an hypothesis, implied in the signs prefixed to the incre-
ments ; and the equation
~ = ± (in the limit)
R As
which he finally deduces is not in conformity to that
hypothesis. The double sign should not have been pre-
fixed to a s, since the proper sign of it is really implied
in that of a a.
(18.) If we were to take the axis of ordinates in the di-28
MATHEMATICAL· DISSERTATIONS.
rection of the normal at the proposed point, and conse-
quently the axis of abscissas parallel to the tangent at that
point, the radius of curvature would he obtained with
remarkable simplicity. Thus: since, in the proposed hy-
pothesis as to the axes, the abscissa a of the centre of
curvature, is 0 ; and the radius obviously y — j3, the equa-
tion of the osculating circle will be
z2 + (?-/3)2=R2,
and, by differentiating,
y-p=-y- = R
ay
dx
(Lit
But, as at the proposed point x = 0, and also —= 0, to
get the true value of this fraction we must differentiate
both numerator and denominator: we thus obtain
R -------1-----[D]
dx2
(19.) We shall merely add, in conclusion, that Lacroix
is one of the very few writers who regard the double sign,
usually prefixed to the expression for R, as meaningless in
reference to the direction of R. But yet, in common with
most others, he considers the single minus sign as arbi-
trarily introduced purely for convenience; whereas this
sign, as we have seen, necessarily accompanies the correct
result. The words of Lacroix on this subject are as follow :
tc The value of R being susceptible of the double sign
±, it may be asked, which of the two we ought to em-
ploy; for it is evident, that in general for one point of the
curve there is but one radius of curvature; and since this
radius has not, except at some particular points, the same
direction with the ordinate or abscissa, it cannot properly
be said to have any peculiar sign with respect to those
lines. The determination therefore of that by which weSIGN OF THE RADIUS OF CURVATURE.
29
commonly affect it must depend upon a convention pre-
viously established on the direction of the curvature, with
respect to the normal. If we agree to consider the radius
of curvature as positive, for those curves whose concavities
are turned towards the axis of the abscissae, as the value of
d2y
is, in these cases, negative, we must affect the ex-
pression for R with the sign — ; and the same assumption
will also make the radius of curvature negative when the
concavities of the curves are turned from the axis of the
abscissae, since it changes its sign at the same time with
d2y
In conformity to this convention, we shall always
in the application of the formulae, assume
{dx2 + dy2f:,
dxd2y
(Lacroix, Diff. Calc. English Trans, p. 117.)
It has been seen however that, without any assumption
at all, this minus sign is a necessary part of the true re-
sult. It has no reference either to the double sign of the
radical, or to the geometrical position of the radius. It
refers solely to the hypothesis, before adverted to, as to
y — j3; an hypothesis which implies that the point under
examination is in the concave part of the curve. If this
supposition be right, the minus confirms it; if wrong, the
same minus corrects it.
NOTE.
Some advance has been of late made towards the esta-
blishing of a kind of analytical geometry, in the funda-
mental principles of which fewer conventions appear to
enter even than in the geometry of Des Cartes. Only a
single fixed axis is here employed, an axis of abscissas;
along which, from a proposed origin, the only positive and30
MATHEMATICAL DISSERTATIONS.
negative directions are taken ; and a directive sign, different
both from + and —, although perfectly consistent with
them, is then employed to mark the positions of lines
oblique to the axis. Thus $ representing the inclination
of any such oblique line drawn from the origin, the sym-
bol (cos 0 -f sin 0 V — 1) prefixed to it will mark its di-
rection, and will conform to the original conventions
when the line by revolving comes into coincidence with
the fixed axis. If the absolute length of a line from the
origin be a, then (cos 04- sin e s/ — 1) a, in the proposed
notation, represents it in length and position : if it come
to coincide with the axis, taking the positive direction,
then cos 0=4-1, and sin 0 = 0, so that the representa-
tion is 4- a ; if it take the negative direction, upon coin-
ciding with the axis, cos 0 will be — 1, and sin 0 = 0, as
before; so that the proper representation will then be
— a. If it take a direction at right angles to the axis it
will be denoted by a s/ — 1, because in that position of
the line a, cos 0 will be 0, and sin 0 = 1.
In this manner expressions, which in their common alge-
braic character denote imaginary or impossible quantities,
are introduced into geometry to indicate linear direction.
Whether or not this appropriation of them is perfectly
consistent with all the other conventions of analysis, it
must be left for further experience to determine. The
author of the present work has certainly his own objec-
tions to it ; but as it is advocated by one of the ablest of
modern analysts, it must not be hastily opposed.*
• Professor Peacock. See Report to the Third Meeting of the British
Association. Dr. Peacock has, we believe, expounded this theory at length
in his Algebra, a work that has been long out of print, and which I have not
been able to procure. The student may also consult, on this subject, Mr.
Warren’s papers in the Phil. Trans, for 1829 ; as also a work on the Geo-
metrical Representation of Imaginary Quantities, published by the same
gentleman in 1828.DISSERTATION II
ON THE CURVATURE OF SURFACES, WITH AN EXA-
MINATION OF THE THEORY OF UMBILICAL· POINTS,
AS DELIVERED BY MONGE, DUPIN, AND POISSON.DISSERTATION II.
ON THE CURVATURE OF SURFACES.
(20.) If we except some of the recent researches of
M. Chasles we may truly assert that to the labours of
Euler, Monge, and Dupin we are almost exclusively
indebted for a comprehensive and systematic theory of the
curvature of surfaces. Our obligations to Monge are
especially due. The elegance and originality of his inves-
tigations, and the remarkable beauty of the results to
which they conduct, justly entitle him to the preeminence
in this difficult department of research; and render the
work which he has bequeathed to posterity under the
title of Application de VAnalyse à la Géométrie, although
one of the most profound, yet at the same time one of the
most attractive performances which analytical science can
boast.
The Développements de Géométrie, by his celebrated
pupil Baron Dupin, is intended to form a supplement to
the work of Monge. It is unquestionably entitled to a
place by the side of its precursor ; as well on account of
the consummate analytical skill which it displays, as
because of the valuable contributions it furnishes to the
same important object—a complete and general theory of
curve surfaces.
(21.) But there is one interesting subject connected
with this theory, which these illustrious authors have left
in considerable obscurity : the subject of “ ombilics,” or
334
MATHEMATICAL DISSERTATIONS.
those remarkable points on a surface at which the normal
sections all have the same radius of curvature.
The obscurity here complained of will be found upon
examination to arise in a great measure from these two
causes ; namely, from overlooking the change which takes
place in the nature of a certain differential equation, when,
by giving constant values to the coefficients, it becomes an
equation of common algebra; and from the inaccurate
conceptions, entertained both by Monge and Dupin of
the import of the symbol^-, in the analytical discussion
referred to. Dupin {Développements, p. 178,) clearly saw
that Monge had misinterpreted this symbol in his inves-
tigation of umbilical points ; yet, as it appears to us, he
himself failed to perceive its true character in the same
enquiry, and is in consequence led, in the course of his
reasonings, to certain positions that a close examination
will prove to be untenable.
(22.) At page 160 of the Développements, the author,
after exhibiting the general differential equation of lines of
curvature, observes respecting it, that a simple inspection
shows that its terms all vanish together when the two radii
of curvature become equal; and that then, the equation
presenting itself under the form
number of different values.
He remarks, however, that we ought not hastily to
conclude that from the points where the two curvatures of
the surface are equal, and directed in the same sense, there
issue an infinity of lines of curvature. For, he adds, we
know from the theory of the differential calculus that there
becomes indeterminate, and capable of an infinite
dx-ON THE CURVATURE OF SURFACES.
35
are cases in which certain magnitudes present themselves
under an indeterminate form without being so in reality;
where the general equation implies either more or less
than the particular case under consideration: whence we
might immediately conclude that in general, in the singular
points with which we are at present occupied, there are
more or fewer than two lines of curvature; there must be
then only one, or three, or else a greater number; but these
inductions, however excellent to guide us in the search of
truth, ought never to be regarded as absolute proofs.*
(23.) Now this language evidently implies a doubt as to
whether the values of	which in the case before us
and reference is made to the subsequent determination of
the differential calculus for the removal of this doubt.
* “ Nous reconnâitrons, à sa seule inspection, que ses termes s’évanouissent à
la fois quand les deux rayons de courbure deviennent égaux, et qu’alors cette
équation s’offrant sous la forme
devient indéterminé, et susceptible d’une infinité de valeurs différentes.
“ Cependant ne nous hâtons pas encore d’en conclure qu’aux points où les
deux courbures de la surface sont égales et dirigées dans le même sens,
viennent de croiser une infinité des lignes de courbure. Car nous savons par
la théorie du calcul différentiel, qu’il est des cas où certaines grandeurs se
montrent sous une forme indéterminée sans être telles en effet ; ce sont ceux
où l’équation générale dit plus où dit moins que le cas particulier que l'on consi-
dère, d’où nous pourrions déjà conclure qu’en général, dans les points singu-
liers dont nous nous occupons actuellement, il y a plus ou il y a moins de
deux lignes de courbure ; il y en aura donc une seule, ou trois, ou un plus
grand nombre; mais ces inductions excellentes pour nous guider dans la
recherche de la vérité ne doivent jamais être regardées comme des preuves
absolues.”
appears under the form are really innumerable or not;
{Développements de Géométrie, p. 100.)36
MATHEMATICAL DISSERTATIONS.
Dupin is accordingly guided by such determination, and
concludes that the values sought are not necessarily innu-
merable. (See Développements, pp. 127, 185.)
We submit, however, that in no case is the differential
calculus competent to decide whether the form which a
general analytical result takes in certain particular hypo-
theses, as to the arbitrary quantities entering that result,
has, or has not, innumerable values. All that the subse-
quent application of the calculus can do, in such a case, is
to impose upon those values, be they innumerable or not,
a new condition or restriction; and to bring out from
among them all those which conform to this new condi-
tion, as well as to the original restrictions implied in the
fundamental equations.
The values which satisfy these equations may be innu-
merable ; although out of these the calculus may select but
one, which one however will be distinguished from all the
others by the peculiarity implied in the new condition.
The calculus is not, therefore, to be resorted to in order
to test the admissible values concealed under the form
but only for the purpose of bringing out from among them
all those which are singular or unique ; those, in fact,
which more than fulfil the original conditions.
It is a general analytical principle, too often overlooked
certainly, but which will be assented to as soon as stated,
that those values of the arbitrary quantities (and none but
those) which render the equations of condition indetermi-
nate, must also render the final result to which they lead
equally indeterminate. When such a result therefore
assumes the form jj, its true character is to be tested by the
equations which have led to it, after these are modified by
the hypotheses from which that form has arisen. The
perplexities and mistakes in the theory of umbilici, as wellON THE CURVATURE OF SURFACES.
37
as in many other analytical enquiries which terminate in
ambiguous expressions, are mainly attributable to a neglect
of this obvious principle.
(24.) The fundamental mistake however in the article
from Monge, criticised by Dupin in the Développements,
page 173, is of a different and indeed of a more remarkable
kind. It consists in the supposition of an equality between
the two roots of the equation of the lines of curvature,
although the very form of that equation is such as to ren-
der an equality of roots absolutely impossible; an over-
sight which, however palpable, has escaped the detection
of Du pin and subsequent analysts. The author of the
interesting paper on 66 Lines of Curvature,” published in
the fourteenth volume of the Transactions of the Royal
Irish Academy, has fallen into the same error ; and has thus
given a theory of umbilical points which is fundamentally
incorrect.
The equation of the lines of curvature being of the
second degree, with its third term always negative, cannot
possibly have equal roots ; so that these roots, generally
two, can never merge into one and coincide, although such
a coincidence is frequently affirmed to be the distinguish-
ing peculiarity of the umbilical points, and therefore the
criterion of their existence upon a surface.*
It is nevertheless quite true, that at every such point,
not the vertex of a surface of revolution, the lines of curva-
ture, generally two, do really merge into one, and become
# Th us Monge remarks, “ les ombilics sont les points dans lesquels les
lignes des deux espèces de courbure se changent Tune en l’autre, et par con-
séquent pour chacun desquels les deux lignes de courbure se confondent. Ces
points sont donc ceux pour lesquel les deux valeurs de que fournit Péqua-
(IX
tion générale des lignes de courbure, sont égaies entr’elles.”—Application
de VAnalyse, art. ix. See also Leroy, Analyse appliquée à la Géométrie des
Trois Dimensions, p. 308, edit. 1835.38	MATHEMATICAL DISSERTATIONS.
confounded; but this uniting of the two lines of curvature
in one is, as we shall endeavour to show, a thing quite
distinct from the confounding of the two roots of the
quadratic equation above adverted to, when this equation
is rendered determinate by substituting in it, for x and
yi the constant coordinates of the umbilicus.
Could such a coincidence of the roots take place, we
should be presented with a very remarkable circumstance,
one altogether unique in analysis, an instance of two quan-
tities becoming equal in a certain extreme hypothesis, as to
the disposable magnitudes, without their making the least
approach to such equality during the progress of those
magnitudes towards the ultimate state.
(25.) In a Mémoire sur la Courbure des Surfaces, pub-
lished in 1832 in the Journal de P Ecole Polytechnique,
tome xiii., Poisson has considered the disputed question
as to the number of lines of curvature passing through an
umbilical point ; and he has arrived at the conclusion, that
these lines are infinite in number, and issue from the umbi-
licus in all directions; and that those particular lines,
selected from among these by Dupin, differ from the
others only by their satisfying an additional analytical
condition. But this is not the correct theory. The con-
clusions of Dupin on this subject, although we think some
of his reasonings to be not only obscure but unsound, and
shall accordingly venture to animadvert on them in the
course of this dissertation, do really involve the true doc-
trine of the lines of curvature ; and those of these lines,
which Dupin has shown to issue from an umbilicus, are
not selections from an infinite number of such lines, but
do really comprise all that can possibly proceed from that
point.
Moreover, the mode of investigation pursuedby Poisson,
in order to detect from among the infinite number of lines
which he affirms to pass through an umbilicus those dis-ON THE CURVATURE OF SURFACES.
39
tinguished by Dupin, is peculiar; and, as appears to us, not
only far-fetched, but wanting in that analytical elegance
which generally marked the researches of that eminent
philosopher.
His professed object seems to have been to reconcile the
conflicting doctrines of Monge and Dupin on the subject
of umbilici; the former appearing to think that an infinite
number of lines of curvature might proceed from these
points, and Dupin distinctly maintaining, that their num-
ber must always be limited, except at a vertex of revo-
lution : he agrees with Monge that the number is infinite,
and so far with Dupin that a limited number of them are
distinguished by peculiarities. But even supposing this to
be the true theory, which it is not, his analytical process is
so entirely different from theirs, that it could be of no
assistance whatever towards removing the obscurities and
discrepancies of his predecessors, the causes of which
should be sought, not in their conclusions, but in the rea-
sonings which have led to them. We suspect that the novel
mode of investigation employed by Poisson was adopted
by him only because of his concurring with Dupin in the
opinion, that at the umbilical points tfC la méthode ordi-
naire pour trouver les lignes de courbure est en défaut.”
(Développements, page 359.)
(26.) Poisson, in common with all subsequent writers,
infers, that when the coefficients of the equation which
determines the direction of curvature all vanish, the infinite
number of roots of that equation are all admissible, an
inference not necessarily true ; and one which, as we have
seen from the quotation at page 35, Dupin was careful to
avoid. Passing by this, however, it does not clearly appear
from Poisson’s investigation, nor from any that has
hitherto been given, what the geometrical peculiarity of the
surface in the directions of Du pin’s lines of curvature at
an umbilicus really is. These directions are certainly dis-40
MATHEMATICAL DISSERTATIONS.
tinguished from all other directions from the point by
peculiar analytical conditions: to what geometrical circum-
stances do these refer? Poisson, indeed, replies, that the
approximation of the consecutive normals, in the directions
which the analysis points out, is infinitely more close than
in any other direction; a statement which, although it
really does involve an interesting geometrical fact, we
cannot but regard as too vague to convey any satisfactory
idea as to the peculiarity of the surface in the directions
referred to.
In the present dissertation it will be our endeavour to
remove the obscurities connected with this subject, and at
the same time to develope some of the more important pro-
perties of curve surfaces in general; adopting a mode of
investigation somewhat different from that usually em-
ployed in such enquiries, and which will conduct us to
those properties by a very short and easy route.
(27.) The general equation of any surface is
Z = F (X, Y) .... [I]
Let the coordinates X, Y, Z, take the increments x, y, z;
then we shall have for z the value
dZ dZ <PZ x2	d*Z	<fZ y2	0
Z~dXX+dYy+dX!> 1-2 + dXdY Xy + rfY* l-2+ C'
This equation may be equally regarded as the repre-
sentation of the surface [I]; the coordinates x, y, Z\ of any
point in it, being now measured from the point X, Y, Z, on
the surface, as the origin; the axes in the one case being
parallel to those in the other.
If we suppose the axes to be rectangular, and that of z
to coincide with a normal to the surface, the axes of x and y
will be in the tangent plane; and, under this arrangement,ON THE CURVATURE OF SURFACES.
41
the coefficients of x and y must vanish, so that the equation
of the surface becomes simply
z
æz x2 æz	æz
dX3 1*2 +dXdYXÿ + dY3 1-2
[1]
When z is a rational and integral function of x and y, the
series on the right terminates with those quantities having
the same dimensions as the proposed equation. But when
2 is not a rational and integral function of x and y9 the
series extends indefinitely, the higher powers of x and y
arising merely from developing the irrational part of the
expression for £. When the equation is of the second
y2
order, the terms beyond — obviously represent merely the
function pz2 + qxz + ryz.
The equation of any plane, parallel to that of xy, and at
the distance h from the origin, will be
z = h_____[2]
Hence, for the curve of intersection made by this plane
with the surface, we have the equation
d2Z x2
dX2 2
d2Z	æZ y2 0
7v xv +	— + &c· = h *
dXdY
dY2 2
and, consequently, the equation of the curve of the second
order to which this indefinitely approaches as h (or z)
diminishes; that is, as the plane of intersection approaches
towards the tangent plane, is
æz x^_ d2z
dX3 2 + dXdY
f	L
*y+wY = h----w
which equation represents in general only a point, namely,
the point of contact, when the moving plane [2] actually
reaches the tangent plane; that is, when A, and conse-42
MATHEMATICAL DISSERTATIONS.
quently x and y, become zero. Nevertheless, before the
moving plane attains this ultimate position, the section
made by it with the surface will obviously be a curve
approaching nearer and nearer to coincidence with the
curve [3] as the plane advances towards coincidence with
the tangent.
(28.) If the surface under consideration were itself of
the second order, the form of our general equation would
then be
d27j x
:dX*~2
+
æ z
æz
dXdY Xy + dY2
!>+ z(pz + qx + Ity).
•[4]
as already remarked; and the curve of intersection would be
æz x2 æz
dX* 2 + clXdY
d?Z v2
xy + df2 H + ^(p^+Q x+*y)=k-[5]
which curve, we know, always continues similar to itself
during the progress of the intersecting plane, (Anal. Geom.
Part II. p. 210,) however much h may diminish. The simi-
larity being thus undisturbed by the variation of A, con-
tinues then up to the limit; so that when this limit is ac-
tually reached, and h becomes zero, whatever ratio any two
diameters may have all along preserved, in that same ratio
must they vanish when A, x, y, become zero, or when the
section becomes a point; the same point, in fact, as that
on the original surface, and represented above by [3],
when h = 0.
This ultimate section however is not necessarily always
a point; whether it be or not will depend upon the re-
lation among the three constant coefficients. If this rela-
tion be such that
( d2Z 2d2 Z d?Z
'dXdY' ~dX*' dY2ON THE CURVATURE OF SURFACES.
43
the contact of the tangent plane will be, not a point
merely, but a straight line, (Anal. Geom., Part II. p. 26;)
and if the relation be
, d2Z .2 d2Z d2Z
> ¿X» · dY*
the surface will be touched by the plane along two straight
lines which cross at the point under consideration, (Anal,
Geom. p. 20.) These particulars will be otherwise esta-
blished at art. (39.)
(29.) Such ultimate intersections as these will imply
that the preceding sections [5], which have finally merged
into them, are all parabolas or hyperbolas, and that the
surface is not uniformly convex or concave about the point
under examination; nevertheless, agreeably to what has
already been shown, whatever diameters or chords of the
similar curves of the parallel sections of the surface [4],
may have vanished at this limit, they must have vanished
in the ratio that they all along had. So far therefore as
such ratios only are concerned, we may infer the properties
of the evanescent curve, or tangential section [3] of our
proposed surface, h being zero, by discussing the proper-
ties of the finite or real curve [5], h being any finite value;
or, which is the same thing, we may speak of the diame-
ters, &c., of the evanescent curve, as far as their ratios are
concerned, just as if these diameters were really of finite
length, and not so many zeros.
(30.) This evanescent curve, the limit of the inter-
sections of the plane [2] with the proposed surface, is
that which Dupin calls the indicatrix of the surface at the
point X, Y, Z. Its equation, when agreeably to the usual
practice r, s, t is put for the three differential coefficients
in [3], will be
rx2 + 2sxy + ty2 = 2h ... . [6]44
MATHEMATICAL. DISSERTATIONS.
understanding of course that the relations among x9 y,
and h, are, for this indicatrix, those which these lines have
in the limit.
If now any normal section through our point X, Y, Z
be made, h or z will represent, what may be called, the
versed sine of the section; when h becomes evanescent the
corresponding sine will be that semidiameter of the indi-
catrix through which the section passes. The ratio of the
square of this evanescent sine to the corresponding versed
sine is a common expression for the radius of curvature of
the section at the proposed point, an expression imme-
diately deducible from Euclid, (Prop. 36, Book III.,)
in conjunction with the doctrine of limits. This radius,
however, being an important element in the theory of
curvature, we shall establish its value in a manner more
strictly analytical.
(31.) We have already seen (page 28) that the absolute
length of the radius of curvature at a point in a plane curve
at which the tangent is parallel to the axis of abscissas, and
which axis we may, for the moment, call the axis of x is
dx2
and since in this case y = 0, and = 0, the ordinate A,
of the curve, due to an abscissa a9 is
h
1	(Py
2	dx21
+---
The ratio of this ordinate to a2 is
h	1 d2y
a2	2 dx2
I d2y .
- m the limit.ON THE CURVATURE OF SURFACES.
45
(32.) Hence, calling the semidiameter of the indicatrix,
through which a normal section passes,«, we have
[7]
In like manner for the radius of curvature R', of a normal
section through another semidiameter o', we have
It ___
R' “ a'2
so that the radii of curvature of normal sections at any
point are to each other as the squares of those diameters
of the indicatrix through which the sections pass.
(33.) If a, a! be the principal semidiameters of the
indicatrix, and a" any other semidiameter, then (Anal.
Geom., pp. 104, 150,)
where a is the inclination of a" to a. Hence [7] calling
the radius of curvature of the section through a", R", we
have the well-known property of Euler, namely,
¿,= -1 sin*. +-^cos»a.........[8]
showing that the curvature of any normal section at a
point is equal to the sum of the principal curvatures at
that point multiplied respectively by the squares of the
cosines of the inclinations of the proposed section to the
principal sections.
Since46
MATHEMATICAL DISSERTATIONS.
the above expression may be written in the form
2RR'
R' + R — (R' — R) cos 2 a
(34.) If, instead of the principal semidiameters, we take
a, ato represent any rectangular semidiameters whatever,
then, as it is a general property of the conic sections that
1	1
— H—- = constant,
a*	a12
it follows that
1 1
~ + —= constant,
that is, the algebraic sum of the curvatures of any two
normal sections, at right angles to one another, is a con-
stant quantity; observing that will take a positive or a
negative sign according as a, a! belong to an ellipse or to
an hyperbola. That the sections of greatest and least
curvature are always at right angles to one another follows
immediately from the fact that the greatest and least
diameters are at right angles to one another.
(35.) If a, a! be any pair of conjugate semidiameters
then
a8 + a'2 = constant,
.·. R + R' = constant;
so that the sum of the radii of curvature of the normal
sections passing through a pair of conjugate tangents to
the surface at any point is a constant quantity, namely,
the sum of the principal radii, or the radii of curvature of
the surface at that point.ON THE CURVATURE OF SURFACES.
47
(36.) The radii of curvature of the sections through the
equal conjugate diameters being themselves equal, it fol-
lows that each of these is an arithmetical mean between
the two radii of the surface. Such sections therefore might
with propriety be called the sections of mean curvature of the
surface at the point: a sphere whose radius is that of
a section of mean curvature, would not have contact of the
second order with the surface in the directions of these mean
sections, yet the sections themselves would osculate, and the
sphere would represent the mean or average curvature of
the surface at the proposed point. It might be called the
sphere of average curvature at that point; and when the
curvature is uniform, this sphere would be the osculating
sphere.
(37.) As respects the normal sections, we see that h
remains the same for each, while a, that semidiameter of
the indicatrix through which the section passes, differs
with the direction of a; that is, these semidiameters are
not in a ratio of equality. But if through any proposed
semidiameter oblique sections be made to pass, then,
obviously, the h in those sections would differ, whilst the a
remains the same for all. In fact, if were the inclination
of any oblique section through a3 to the normal section
through the same, we should evidently have the re-
lation h! =	, calling the oblique A, A'. Hence the
radius p of the oblique section would be
a8 a2
‘,==2Â'==2ÂCOS*=RCOS*
which is the theorem of Meusnier ; and from which we
see that the radius of an oblique section is always a shorter
line than the radius of the normal section through the
same linear tangent. But a more interesting property is48
MATHEMATICAL DISSERTATIONS.
immediately deducible from this theorem, as will appear
from the following considerations:
(38.) It is plain that two surfaces must mutually oscu-
late at a point when the normal sections all osculate at
that point. Now in order to this it is only necessary that
the principal sections be in the same normal planes, and
have respectively the same radii of curvature. For the
indicatrix at the point of contact must then have its prin-
cipal diameters the same for one surface as for the other,
so that the same indicatrix must be common to both; and
therefore a normal plane through any diameter, 2a, of the
indicatrix must give sections, having a common radius,
a1 2
namely, ^r. Such being the case it follows from the the-
orem of Meusnier that
If two surfaces touch at a point, and their principal
sections lie in the same planes, and have respectively the
same radii of curvature, then whatever plane be drawn
through the point, whether normal or oblique, the two
sections made by it will mutually osculate at that point.
(39.) It is useful to express the value of R, the radius
of any normal section, in terms of the inclination 9 of the
trace of that section to the axis of x. To obtain such an
expression substitute for a2 in [7] its value in terms of
x', 3/ the coordinates of the extremity of that semidiameter:
we shall thus have
= *'2 + y'2
2h
1 +
x'2
x xn
1 + tan2 9
r + 2s tan 9 + t tan3 $
[8]ON THE CURVATURE OF SURFACES.
49
agreeing with the common expression usually deduced by
more operose methods.
If for 1 -f tan2# we put its value sec2#, and then divide
numerator and denominator by this, the form for R will be
R=-------=----—^--------------r-i- ......[9]
rcos2e -f- 2s sine cose + t sure	L
If it be possible to draw a straight line upon the surface
through the point under examination, then for the section
through this line we ought to have R = oo; and conse-
quently from [8]
r -f- 2s tane -f f tan2# =0....[10]
so that the direction in which such line lies is
Hence, that only one straight line may pass through the
point and lie upon the surface, there must exist the con-
dition already adverted to at page 42, namely,
s2 — tr = 0,
and in order that two straight lines through the point may
lie upon the surface, we must have
s2 — tr — +,
a condition which must always have place when, at the
proposed point, the normal plane by turning round passes
from convex sections to concave, or vice versa; the two
intersecting lines being the boundaries which separate the
convex compartments around the point from the concave.
(40.) When the former of these two conditions holds,
the indicatrix is a parabola, or rather that extreme variety
450
MATHEMATICAL DISSERTATIONS.
of this curve which takes the form of two parallel straight
lines, ultimately merging into one (See Anal. Geom.
Part. II., p. 25.) The parallel, midway between these, is
the line of centres; that is, every point in this line, bisect-
ing all the chords through it, may be regarded as a centre;
and each chord a diameter; the line of centres and the
perpendicular chord through any centre being the princi-
pal diameters in reference to that centre. Calling the finite
principal semidiameter a, any other semidiameter a", in-
clined to it at an angle «, will obviously be a" == a ;
consequently
— = — cos«
a,u a*
" R" “ R C0S a>
an inference that we might at once have obtained from
Euler^s formula [8] ; since the diameter a', at right
angles to a, being infinite, , and consequently It'’is
zero.
From the expression just deduced we have for the ra-
dius R" of any oblique section, in terms of the radius of
the section perpendicular to the straight line, and the
inclination between the two, the value
R" = R secV
(41.) When the latter of the two conditions above has
place, the indicatrix is a hyperbola, having the intersecting
straight lines through the point for its asymptotes. The
directions of maximum and minimum curvature, that is,
of the principal diameters of this hyperbola, will therefore
lie in the lines bisecting the adjacent angles formed by
these asymptotes. As the squares of the principal diame-
ters 2a, 2a', have, in this case, opposite signs, we infer
that the two principal sections have curvatures of oppositeON THE CURVATURE OF SURFACES.
51
kinds; and the same may be said of every pair of sections
through conjugate diameters: the extreme pair of these
diameters, the asymptotes, give sections neither convex
nor concave, but rectilineal.
(42.) It maybe useful to observe here, that it is because
of the change of curvature that takes place at these recti-
lineal sections that they are not regarded as the sections
of minimum curvature. For although in reality the cur-
vature of every such section is — = 0, yet this value, under
the circumstances, does not fulfil the condition of a maxi-
mum or minimum. In developable surfaces however, for
which the relation s2 — tr = 0 everywhere holds, the rec-
tilinear section through any point does fulfil the analytical
condition of a minimum, since the curvature upon both
sides of such a section is of the same kind; so that —,
R
in passing from one side to the other, through the value
0, undergoes no change of sign.
(43.) If in the expression [8] we put s = 0 and r = t,
then
r _	1 + tan2 e __ 1
r (1 + tan2 g) ry
so that at the point where those conditions have place,
the radius of curvature is constant for every normal sec-
tion. This point, therefore, is an u ombilic,” and the
indicatrix a circle.
(44.) Thus far we have limited our examination of the
surface to an isolated point taken upon it; and have de-
duced the more important particulars respecting the cur-
vature of the surface there. For such an examination the
indicatrix, as we have seen, offers great facilities; as it52
MATHEMATICAL DISSERTATIONS.
readily leads to the correct expressions for the radii of the
normal and oblique sections, and thus indicates the cha-
racter of the surface at any proposed point. The student
ought, however, to be reminded that, however complicated
be the surface, the indicatrix, from the properties of
which our several deductions have been obtained, is an
evanescent curve of only the second order; and that, con-
sequently, it is not rigidly the actual curve in which the
series of parallel sections, made in a surface of a high
order, vanishes when the tangent plane is reached; for
strictly speaking this will generally be a curve of the same
order as the surface; although to no curve of the second
order does it so closely approach as to that which is called
the indicatrix. This approximate indicatrix however is
all that is necessary for the accurate determination of the
character of any surface at a proposed point; so that any
correction of it, by introducing into its equation terms of
a higher order, would, for the purpose to which it is ap-
plied, be altogether useless. The expression [7], from
which the subsequent ones have mainly been deduced,
involves only the semidiameter a of the indicatrix of the
second order:—a fact which deserves especial notice.
When the surface under consideration is itself of the
second order our indicatrix is, of course, rigidly correct;
we shall briefly consider it, in reference to these surfaces,
before proceeding to discuss the lines of curvature of sur-
faces in general.
(45.) A surface of the second order osculates any other
surface when the indicatrix of the former, at the point of
contact, is also the indicatrix of the second order of the
other surface ; as is plain from what has been proved at
(38), or from the equations [1] and [4]. (See Diff. Calc.,
p. 225.) We may therefore say that two surfaces osculate
at a point when they have a common indicatrix there.
Taking the axes of #, y, and z as before, we have for twoON THE CURVATURE OP SURFACES.
63
surfaces of the second order, osculating at the origin, the
equations
z = ax2 4- bxy + cy2 -f mz2 + nxz 4- pyz
z = ax2 4- bxy 4- cy2 4- m'z2 4- n'xz 4- p'yz,
the equation of the common indicatrix being
ax2 -f bxy 4- cy2 = 0.
If, however, the axes of x and y be made to coincide with
the principal diameters of the indicatrix, then b in this
equation must be zero; so that the two osculating surfaces
may be rather more simply represented by the equations
z = ax2 -f cy2 4- wz2 4- nxz 4- pyz......[11]
z = ax2 4- cy2 4- «i'z* 4- n’xz 4- pryz..[12].
The equation of the common indicatrix being
ax2 4- cy2 = 0,
it will be a point provided a and c have the same sign, a
pair of intersecting straight lines if they have opposite
signs, and only a single straight line if one of these be
zero. If a = c the surfaces osculate at an umbilical point.
(46.) Besides the indicatrix, the surfaces may have other
points in common; these will be discovered by deter-
mining all the values of the coordinates for which the
equations [11], [12] exist simultaneously. In order to
this subtract [12] from [11] and we shall have
(m — mr) z2 4- (» ■— »') xz 4- (p — pr) yz = 0,
an equation which resolves itself into the two
z = 0 and {m — m!) z 4- (» — n!) x + (p — p') y = 0.
The first has already been considered; it represents the54
MATHEMATICAL DISSERTATIONS.
indicatrix; the second represents a plane passing through
the point of osculation. This plane however coincides with
the tangent plane if n = n' and /?=//; in which case
therefore the surfaces are entirely separated, except at
their point of osculation.
The two surfaces [11], [12], so long as this equality has
not place, have contact of the second order only; for con-
tact of the first order requires merely that, in the develop-
ments of 2, the terms containing the first powers of x, y,
be identical: contact of the second order requires, more-
over, that the terms containing the second powers of these
should also be the same in the two developments. (See
Diff· Calc. p. 225.) These conditions are fulfilled by the
common indicatrix. But, if n = nf} and p = //, the con-
tact will be of the third order; because then, in the deve-
lopments spoken of, the third powers of x and y, or the
terms involving these in three dimensions, would be the
same in both. This is plain; for, putting the term con-
taining z2 last, the equations may be written
z == ax3 -f cy2 -f (nx + py) (ax2 + cy2 + ....) + mz2
z = «x3 -b cy2 + {n’x -f p'y) (ax2 + cy2 + . . ) -f m!z2;
which, when n = n', and p = p;, become identical, as far
as terms in x and y of the third order are concerned.
It follows, therefore, that the equations
z = ax2 + cy2 + mz2 + nxz -{- pyz
z = ax2 -f cy2 + m' z2 -b **xz + pyz
represent surfaces having contact of the third order, and
that such surfaces cannot intersect as well as touch. If n
and p be each zero, the point of osculation will be the
common vertex of both surfaces.
(47.) If in the equation [12], m', ri, and//, be each zero,
the surface will be a paraboloid, whose vertex is the pointON THE CURVATURE OP SURFACES.
55
of osculation with the other surface. The equation of the
plane of the curve in which the surfaces then cut is
mz -f nx + py = 0 .... [13]
Substituting this value of z in the equation of the parabo-
loid, we have for the projection of the curve of intersection
upon the tangent plane, the equation
ax2 + cy2 -f — x + —y = 0.
m	m
The coordinates of the centre of this projected curve are
___ n	p
a	2am'	^	2cm *
and consequently, when the curve is referred to its axes
and centre, by substituting for x, y, in the above equation,
the values
y+L·'
the equation is
ax2	cy
=U—,+^d
4 Kam* cm J
Hence the curve of intersection is a similar curve to the
indicatrix, or to the sections parallel to the tangent
plane,—a remarkable property of the osculating para-
boloid.
(48.) This theorem is investigated nearly as above by
M. Olivier in the Journal de Mathématiques for May
1838, who gives it as a new property of the paraboloid.
But it was established long efore by Dupin in his
Développements, and indeed in a form still more general.
It may be deduced in this form, as follows:
Let û, 5, cy be the coordinates of any proposed point56
MATHEMATICAL DISSERTATIONS.
upon a curve surface, the axes of coordinates being rect-
angular, but in other respects arbitrary: then, X, Y, Z,
being the coordinates of any point near the former, in
passing from one to the other, the increments of the coor-
dinates will be X — a, Y — 6, Z — c; and the development
of the latter increment in powers of the former will, as at
p. 40, be
Z-c=j>(X-a) + 9(Y-6) + lr(X - a)*
+ S(X - a)(Y - b) + I f(Y - by + &c....[14],
where /?, q, r, s, t, &c. represent the values of the several
partial differential coefficients of Z for the point (a, 5, c).
The equation of a tangent plane through (a, fi, c) is
Z - c =p(X - a) + q (Y - b) .... [15].
A plane parallel to this, and passing through a point on c
prolonged, so as to become c + h, will therefore be repre-
sented by
Z — c — h=p(X — a)+ q(Y — b) .... [16];
and as [14] and [16] exist simultaneously for the points
X, Y, Z, of the surface, and the curve of intersection with
the plane, we have, by subtraction, the equation
A = ir(X-a)a + s(X-a)(Y-i) + 11 (Y - bf + ....
for the projection of this curve on the plane of XY, which
curve becomes the indicatrix of the surface at the point
(«, 5, c), when //, by the approach of our secant [16] to
the tangent [15], becomes evanescent. Hence the pro-
jection of the indicatrix of the second order has for
equation
r (X — a)2 + 2s(X — «)(Y — b) + t(Y -5)*=2A;
h being zero; and this is the projection of the completeON THE CURVATURE OF SURFACES.
57
indicatrix when the surface is itself of the second order,
and h may then be of any finite value, without affecting
the character of the curve.
(49.) The general equation of the paraboloid, when the
coordinates originate at the vertex, and have their planes
coincident with the principal planes of the surface is
(Anal. Geom. Part II. p. 181,)
where P, P are the parameters of the parabolic sections
made by the planes of YZ, XZ.
Now for every point (a, i, c), on the surface, we have, as
appears by taking the differentials of the second order,
Hence the equation of the projection on the tangent plane
through the vertex, of the indicatrix at any point (a, 5, c),
whatever on the surface, is by last article,
±(Y-6y=h;
which obviously becomes the same as the indicatrix at the
vertex itself, by simply removing the origin of X and Y to
the point (a, 6), on the tangent plane: the two curves,
therefore, are similar and similarly placed on the tangent
plane through the vertex; and as, in surfaces of the second
order, parallel sections always give similar and similarly
placed curves, we infer generally with Dupin, that
All the plane sections of a paraboloid, projected upon a
plane perpendicular to the axis, are similar curves, and
similarly placed, to that given by the perpendicular plane
itself. It is clear, that if the cylinders which project these
sections be cut by any oblique plane, similar sections will
also result, so that the sections of the surface will always53
MATHEMATICAL· DISSERTATIONS.
be projected into similar and similarly situated curves,
whatever be the plane of projection, provided only the
projecting lines be parallel to the axis of the surface.
The property at p. 65 is obviously only a particular case
of this general theorem.
If the paraboloid be one of revolution, then the pro-
jection of any section parallel to a tangent plane, upon the
plane perpendicular to the axis, will be a circle. But it
may be remarked that, whether the paraboloid be of
revolution or not, in the extreme case, where the tangent
plane is at a point on the surface infinitely remote from the
vertex, the sections parallel to it would seem to be straight
lines, when projected upon a plane perpendicular to the
axis. Every such line will, however, only be the ultimate
form of a series of similar curves, whose axes, preserving
a constant ratio, continually increase, till they finally
become infinite in length.
(50.) When a paraboloid osculates a surface of re-
volution of the second order, at the vertex of the latter,
the two can never intersect. For since in the equation [11]
of a surface of revolution, both n and p must be zero, the
equation [13] of the plane of intersection would be mz = 0,
or z = 0; which represents merely the tangent plane at the
common vertex. Hence the sphere, osculating the para-
boloid of revolution at its vertex, will lie wholly within the
parabolic surface, and so also will an osculating ellipsoid.
But the hyperboloid of revolution which is osculated at its
vertex by a paraboloid, will wholly comprehend the latter
surface. This is obvious, because the tangent plane to the
paraboloid approaches nearer and nearer to parallelism with
the axis of z as the point of contact becomes more remote;
so that if the parabolic surface could, around the common
vertex, lie without the hyperbolic, a tangent plane to the
former would at length cut the latter, and, a fortiori, the
intermediate surface would cut it.ON THE CURVATURE OF SURFACES.
59
(51.) Let us now resume the consideration of surfaces in
general, for the purpose of determining their lines of curva-
ture, or those curves at every point in each of which a
normal to the surface is intersected by a consecutive normal
in the direction of that curve.
Let
A =0
B =0
[17]
be the equations of a normal at any point (x, y, z\ of the
surface z = f (x,y), the axes being arbitrary. The equa-
tions of a second normal through (x + ai, y + Ay,
z + A z) will be
* d\
A + n
AX + -7- Ay + &c. = 0
_	d B
B + — AX +
dx
dA
dy
dB
dy
Ay -f &c. = 0
s
[18];
and if these intersect in a point (x', y', zf), the equations [17],
[18], must exist simultaneously. As these equations are
four in number, the three quantities, x', y', z', involved in
A and B, may be eliminated from them, leaving for the
result a single equation, involving, besides the fixed coor-
dinates x, y, z, of the proposed point, the increments a x,
Ay. These increments therefore always have a depend-
ence upon each other; which dependence, when the normals
are consecutive, wrill mark out the direction from (x,y, z), in
which ,the consecutive intersections takes place. Fore-
seeing, then, that the increments are dependent, the four
equations above, when subjected to the restrictions neces-
sary for consecutive intersection, reduce to
dA	dA	dy ^ ^
dx	dy ’ dx	f
dB	dB	dy	i-"
----1---. — = 0 1
dx	dy	dx	J
.. [19] ;60
MATHEMATICAL DISSERTATIONS.
or, substituting for A and B their known values, the equa-
tions of condition are
i + p (.p + q + (* — *')(»· + * -^) = 0 · · · · t2°]
dy
dx
+ q(p + q-^;) + (* — *')(* + <‘^) = 0—t21l·
These equations, however, are not immediately fit for
determining seeing that the indeterminate quantity s',
dy
which is actually dependent upon still enters them.
Eliminating this, therefore, we have finally
f!±J
l t

rt
dy
dx
{
1 +P2
rs = 0 .... [22] ;
an equation of the second degree, showing that, ordinarily,
there are lines of curvature proceeding from the same point
in only two distinct directions.
(62.) Suppose the point to be such that for the coordi-
nates of it there exists the double equation
1 +p2
P9_ 1 +72
..[23];
the equation [22] is then satisfied with any value for
and the inference from this usually is, that under these
conditions, lines of curvature issue from the point in every
direction. But this conclusion is at least premature,ON THE CURVATURE OF SURFACES.
61
because the innumerable values of given by the equa-
tion [22], are not all admissible, unless in the proposed
hypothesis the conditional equations themselves are equally
indeterminate. It is necessary, therefore, to examine these
before the above conclusion can be legitimately drawn,
since we know that the indeterminate form ^ sometimes
arises from the introduction of foreign factors giving
solutions which are inconsistent with the premises. That
our conditional equations [20] and [21] may be really
indeterminate, their coefficients must satisfy the con-
ditions
1 4- p2 4- (z — zf) r = 0
pq + (z — z') s = 0
and
1 + q2 + (z - z') t = 0
pq + (z — z') s = 0
}
and these evidently resolve themselves into the double
condition [23].
We infer, therefore, that at the proposed point the di-
dxt
rections, 5 in which the consecutive normals intersect
ax
are innumerable: they intersect in every direction.
dii
(53.) It is impossible that, the values of ordinarily
two in number, can ever coincide and become one; al-
though this is affirmed by Monge, Du pin, and more
recent authors; for by placing the rectangular coordinate
planes, which have hitherto been arbitrary, so that the62
MATHEMATICAL DISSERTATIONS.
plane of xy may be either coincident with, or parallel to
the tangent plane at the point under consideration, p and
q will both be 0; and the equation [22] will consequently
become
djfl	r — t dy _
dxa s dx
0
[24]
which can never have equal roots, since the absolute term
is negative; nor can the roots ever be imaginary, for a like
reason. As under this arrangement of the coordinate
planes the conditions [23] become identical with those
(page 51) which characterize an umbilical point—since it
is impossible that p and q can be 0 in [23] without s being
so too—it follows that from such a point the consecutive
normals intersect in every direction ; yet whatever single
direction may be fixed upon either capriciously, or because
of any peculiarity in that direction; in other words, what-
ever particular value be given to the coefficient of
in [24], a second direction, at right angles to that chosen,
is always furnished by the same equation.
(54.) That some of these innumerable directions may be
distinguished from the others by certain peculiarities is
highly probable; they may for instance be directions of
closest contact with the osculating sphere; the question
arises then, what is the analytical character of such par-
ticular directions, or of the particular lines of curvature
issuing from the point in these directions ?
In order to answer this question let us take a normal in
the vicinity of the umbilical point. Its equations ob-
tained by changing x> y into i + ai, y + Ay in [17]
may be written in the following form, h being put without
the braces for a x9ON THE CURVATURE OP SURFACES.
63
. f dA dA a y 1 ,
A + {^ + ^r dr} h +
2{ +2 é/% z? +dd? (dD}h*+&c·=0
_ rrfB ¿B ai/ i ,
B -j- J 4-   f A +
t dx dy Ax J
KS+*A^+?0>+fc—■
- - [25]
Now this normal approaches nearer and nearer to that
through the umbilicus, as h diminishes ; and the point in
which they intersect approaches nearer and nearer to the
centre of the osculating sphere. This point is actually
reached only when h becomes zero, and the normals coin-
cide ; and we have already seen that a necessary condition
accompanying this coincidence of the intersecting normals
is that the coefficient of A in both equations, as well as h
itself, must become zero; that is, we must have
dA dA dy
dx dy dx
dB dB dy
dx dy dx
There is no necessity that the next coefficients, those of
A2, should also become zero in the ultimate position of the
moving normal, in order that it may intersect the fixed
one at the centre of the osculating sphere. It is sufficient
that all the subsequent terms, of the form PA2 + QA3 +
&c., tend towards and become zero when that position is
reached; and this they do simply by the diminution and
ultimate evanescence of A, though P, Q, &c., should con-
tinue finite. Nevertheless, the more of these coefficients
which tend to zero and vanish with A, the more nearly will
the moving normal, for a small value of A, approach to the64
MATHEMATICAL DISSERTATIONS.
centre of the osculating sphere, the point of ultimate in-
tersection ; inasmuch as the terms PA2 ■+■ QA3 -f &c., for
such small value of A, are then more nearly evanescent
than under other circumstances; and although when we
say that two normals, be they ever so close together,
actually cross at the centre of curvature, we commit error,
yet it is plain that that error is necessarily diminished
more and more the nearer the coefficients of A2, A5, &c.,
approach to zero with A.
It follows, therefore, that when, besides the necessary
conditions [26], the moving normal approaches the fixed
one in such a direction that the following conditions also
have place at the same time, namely, the conditions
ax~ ax ay ax ay~ ax
this normal, when near to coincidence with that through
the umbilicus, passes closer to the centre of the osculating
sphere than it would do if the coefficients [27] were finite.
In other words, this normal is more nearly a normal to the
sphere than any other in the vicinity of the umbilicus.
.	dy
(55.) We infer, therefore, that when the direction — of
any line issuing from an umbilicus is such as to fulfil the
conditions [27] (those marked [26] are fulfilled for every
direction), the surface in the direction of that line will lie
more closely to the osculating sphere, in the immediate
vicinity of the umbilicus, than in any other direction not
satisfying the same equations.
If besides these equations the additional conditions
arising from equating to zero the coefficients of A3, when
these attain the limit, also have place for any particular
[27]ON THE CURVATURE OF SURFACES.
65
values of —■
dx9
then such values will refer to directions in
which the surface will lie still more closely to the sphere,
as is evident from the foregoing reasoning.
(56.) It is of importance to observe that the conditions
[27] arise from differentiating the preceding conditions
[26] with respect to x and y considered as dependent,
regarding as constant ; and as the conditions [26] are
ax
equivalent to the single condition [22] at page 60, it will
be sufficient to differentiate this latter under the re-
strictions just mentioned, in order to obtain a new condi-
tion equivalent to the conditions [27]. And as this new
condition will appear under the form of an equation of the
third degree in there will in general be at least one
direction of proceeding from the umbilical point, in which
the surface will approach with more than ordinary close-
ness to the osculating sphere ; and, it would seem, there
may be three directions of more than ordinary intimacy of
contact. If, indeed, this equation of the third degree
should, like that of the second from which it is deduced, be
identical for the coordinates of the umbilical point, it is
obvious from the preceding investigation, that to ascertain
the particular directions of closest contact we must proceed
to another differentiation, and so on till we arrive at a de-
terminate equation, the real roots of which will make
known the number and directions of the lines most nearly
in contact with the osculating sphere.
The equations of the lines of curvature themselves will
be obtained by integrating the differential equation in-
volving the real component factors of this determinate
equation before substituting for the general coordinates
those of the umbilical point, and then introducing these
latter for the purpose of fixing the different values of the
566
MATHEMATICAL DISSERTATIONS.
constant; these values being the same in number as the
component factors referred to.
(57.) When, however, the umbilicus is such that, pro-
ceed in whatever direction from it we may, the lines are
equally close to the osculating sphere or to the tangent
plane at the point, then these successive differentiations,
dy
since they can lead to no particular values for must
either at length exhaust the coefficients differentiated, and
thus conduct to no determinate equation, or else they will
furnish an equation whose roots are all imaginary. It is
obvious that one or other of these circumstances must
always have place at the vertex of a surface of revolution;
since from this vertex we at once perceive that the lines of
curvature must proceed in all directions, and with equal
closeness to the osculating sphere. But in such a case as
this we can never be led to roots that are all imaginary,
dy
because whatever values of — are concealed in the differ-
dx
ential equation [22], under the form no imaginary value
can be implied; since, as already shown (53), the roots
are necessarily real. At a vertex of revolution, therefore,
the differentiations referred to must always eventually ex-
haust the coefficients differentiated; and this must always
happen at a point where the curvature is perfectly uniform
in all directions.
It is plain that this uniformity cannot exist all round
the point except at a vertex of revolution ; since, when it
does exist, each normal section may be regarded as merely
the successive positions of the same section in turning
round the normal; and it obviously matters not to how
small a distance beyond the umbilicus the uniformity may
have been proved to extend. An interruption in this uni-
formity is the only thing that can distinguish an ordinary
umbilicus from a vertex of revolution.ON THE CURVATURE OF SURFACES.
67
In the ordinary umbilicus, as the foregoing investi-
gation proves, there will be one direction at least in which
the approaching normal, immediately before coincidence
with the fixed one, will be more nearly identical with this
latter than in any other direction; but if, as in the case here
supposed, the analytical conditions show that there is no
distinction of this kind among the approaching normals,
then, immediately before coincidence, the fixed normal
must be symmetrically placed with respect to all the sur-
rounding ones; so that they must all meet it in the same
common point, which point ultimately becomes the centre
of curvature. It is clear that a point so circumstanced
must be a vertex of revolution.
(58.) In order to determine whether a surface admits of
umbilical points, we must deduce from the equation of that
surface the general values of p, q, r, s, t, and substitute
them in the double equation
.........[28];
r	s	t	L
which is, in reality, equivalent to two distinct equations.
These, combined with the equation of the surface, will fur-
nish three equations, which, solved for x, y, z, will make
known the coordinates of the umbilical points whenever
such points exist; or, by presenting these coordinates
under impossible values, will show that the surface has no
umbilici.
But if it should happen that the two equations [28]
involve only one distinct condition, each equation being
virtually no more than a repetition of the other, then the
combination with the equation of the surface will be, not
a point, but a locus; namely, a curve traced upon the sur-
face, every point of which curve will be an umbilicus: this68
MATHEMATICAL DISSERTATIONS.
curve, at every point of which a sphere osculates the sur-
face, is called a line of spherical curvature.
Lastly, if no condition at all be implied in the equations
[28], each side being virtually nothing but a repetition of
the other, then every point on the surface will be an umbi-
licus; and since, in consequence of the identities here
supposed, whatever point be taken, the subsequent dif-
ferential coefficients can never become determinate for this
point, it follows that lines of curvature proceed from it, not
only in every direction, but with equal closeness of contact.
Each point, therefore, must be a vertex of revolution; and
hence the surface itself must be a sphere, so that the
elaborate processes of Monge and Poisson, to prove that
the surface, every point of which is an umbilicus, fulfilling
the identical conditions [28], must be a sphere, would
appear to be unnecessary. Uniform closeness of contact
about a point, with the osculating sphere, or with the
tangent plane, marks a vertex of revolution; and if such
uniformity be proved to exist about two points only, pro-
vided these be not diametrically opposite, it is sufficient to
warrant the inference that the surface is a sphere.
(59.) The equation [22] is generally referred to as the
differential equation of the lines of curvature, or rather of
the projections of those lines upon the plane of x, V· To
determine the lines themselves for any given surface, the
expressions for p, q, r, s, t, in terms of x and y, must be
deduced from the equation of that surface, substituted in
[22], and then the integration of the resulting differential
equation effected. The integral equation thus obtained will
represent indifferently either of the lines of curvature be-
longing to the individual surface under examination ; and in
order that any one line in particular, or rather one pair of
lines, may alone be represented by this integral equation,
the arbitrary constants, introduced into it by integration,ON THE CURVATURE OF SURFACES.
69
must be so determined as to fix a particular point in the line
sought.
But if the point be fixed first, before integration, by
giving particular values to the x, y involved in the coeffi-
dy* dy
cients of and and in the absolute term of [22], that
equation will no longer be the differential equation of a line
of curvature through (x9y, z); it will be simply a common
quadratic equation, the unknown term in which is repre-
sented by which expresses the tangent of the inclination
to the axis of x of the projected directions, according to
which consecutive normals intersect that at the given point.
Whether or not lines of curvature proceed in these di-
rections ; that is, whether or not a continous series of points
proceeds from that proposed in the initial directions marked
du
out by the values of taking different routes, is a question
that cannot possibly be determined at this stage of the
enquiry.
(60.) It is, we suspect, from not attending to this dis-
tinction—the distinction, namely, between the general
equation [22], when, by applying it to particular points,
it becomes a common algebraic equation of the second
degree, and when, by leaving the points arbitrary, it is a
differential equation of the first order and second degree,
—that the principal perplexities and conflicting doctrines
about the lines of curvature issuing from umbilici have
arisen. Thus, the common mode of expression in reference
to this subject is, that when the point under consideration
is an “ ombilicf the equation of the lines of curvature
relatively to this point will take the form 0 = 0; which
form shows, it is said, that from an ombilic there issue an
infinity of lines of curvature, the direction of the first ele-
ment of each of them being arbitrary. This is the lan-
guage of Leroy, the latest and most perspicuous writer on70
MATHEMATICAL DISSERTATIONS.
this subject.* Poisson, too, in his Mémoire, [Journal de
P Ecole Polytechnique, tome xiii.] entertains a similar
view.f But nothing can be legitimately inferred merely
from the roots of the quadratic equation [22], whether its
coefficients be fixed, or take the indeterminate form jj, be-
yond the simple fact of the number of values of the in-
clination not of a curve to the axis of xy but of the
direction in which the consecutive normal intersects the
fixed one. What, in the language of Leroy, is the first
element of the line of curvature may, for aught the equa-
tion referred to can tell us, constitute the entire line; the
so called line may be, in fact, but a point.
All that can be safely inferred from what has preceded,
as respects umbilical points, is that at such points conse-
cutive normals intersect in every direction; and that, in one
of these directions at least, the surface is in closer contact
w ith the osculating sphere than in any other direction.
If we wTere to use the language of the infinitesimal cal-
culus, we should say that, as in this particular direction
the normals intersect for two successive elements, if any
line of curvature at all proceeds from the umbilical point,
it must be in that unique direction. And as the general
integral of [22] shows that two lines of curvature always
pass through every other point of the surface, the law of
continuity would authorize us in pronouncing that, at an
umbilical point, the particular direction referred to must
be the path of one line of curvature ; and which must be
* Analyse appliquée à la Géométrie, p. 296. 1835.
f u Lorsque le point M sera ombilic, l’équation [14] sera identique; quelle
que soit la direction déterminée par le rapport de dy à d%, la normale MN au
point M sera donc rencontrée par la normale consécutive, et les droites menées
par ce point dans le plan tangent seront toutes tangentes à des lignes de
courbure. Mais parmi toutes ces lignes en nombre infini, il yen aura une ou
plusieurs que se distingueront des autres par une propriété particulière.”—
Journal de l’Ecole Polytechnique, tome xiii., p. 218.ON THE CURVATURE OF SURFACES.
71
that into which two distinct lines, through a neighbouring
point, have merged when the point has coincided with the
umbilicus.
The complete integral of the differential equation [22],
for any proposed surface, will, we know, from the principles
of the integral calculus, be an algebraical equation, in-
volving an arbitrary constant C in its second power; so that,
solving this quadratic in C, for any particular values of x,y,
we shall thus have two constants, and therefore twTo dis-
tinct equations, representing the curves passing through
the point whose coordinates have been used to determine C.
If it be possible to choose these coordinates so that C may
have equal values, or be zero, then it is plain that the two
equations will become confounded; and we shall be pre-
sented with the peculiarity just stated. If the coordinates
can be such as to render C indeterminate, or of the form jj,
then the lines will be innumerable; and it is obvious that,
on every other supposition, the quadratic in C must give
two distinct values. Hence, through an isolated umbilicus,
there must pass either but one, or else an infinite number of
lines of curvature.
But it must not be inferred, as is customary with writers
on this subject, that the merging of two lines of curvature
into one implies a coincidence of the two roots of the
quadratic equation [22] or [24] in the extreme case sup-
posed, and in which the coefficients are replaced by the
symbol : the form of the equation forbids the inference.
The single value of belonging to the line of curvature
at the umbilicus, is the real root of the cubic equation
obtained by differentiation, as already directed (56), from72
MATHEMATICAL DISSERTATIONS.
the quadratic [22], and which real root is the only one
fulfilling the two conditions [22] and [29] for the proposed
coordinates; although there is in [22] or [24] a com-
panion root to the value thus determined of and which,
if [24] be employed, is minus the reciprocal of that value.
It is this accompanying value of which here, as at
every other point of the surface, marks one of the two
directions of principal curvature, the other direction being
that of the unique line of curvature through the umbilicus.
(61.) In order to remove from the mind of the student
all suspicion, as to whether at an isolated umbilicus the
roots of the cubic equation [29] can in any case be all real,
it will be sufficient that he reflect, that this equation in-
volves only those values of ^ which enter into the equation
[22], when the coordinates of the umbilicus are substituted
in both; not indeed all the innumerable values of which
CLX
in the case before us [22] furnishes, but those only which
obey the general law implied in [22] ; in other words,
those which belong to lines of curvature ; and we have just
seen that the passage of three lines of curvature through the
point is impossible. We are justified therefore in affirming,
that at an isolated umbilicus two roots of the determinate
equation [29] are always imaginary.
But when the umbilicus is not isolated, but is merely
one of the continuous series which make up a line of sphe-
rical curvature, then equation [29] holds for every point
in that line. It is therefore the differential equation of it;
and, being of the third degree, furnishes three integrals
corresponding to every particular point in it. It is this
differential equation of the third degree that then supplies
entirely, and not for a particular point merely, the place of
the equation of the second degree which suffices for linesON THE CURVATURE OF SURFACES.
73
of curvature in general; and just as in the case of these
ordinary lines of curvature we infer that two pass through
every point, so here we infer that through every point on
a line of spherical curvature three lines in general pass,
one being of course the line of spherical curvature itself.
If in the general integral of the equation [22] the con-
stant C be determined, for one of the points on the line of
spherical curvature, the two values of C will obviously be
in general unequal; for to render these values equal, the
irrational part of the root of the quadratic that determines
C must be made zero; by which condition particular
values only of the coordinates will be determined; and
even these may be forbidden by the equation of the line of
spherical curvatures, which we have seen is given by the
combination of [28] with the equation of the surface.
Hence the two values of C will, in general, be unequal.
These therefore will imply two lines of curvature at right
angles to one another, two lines uniting with the general
series of lines of curvature which cover the surface. Con-
sequently through every point on a line of spherical cur-
vature there passes in general besides that particular line
two others, of the ordinary series, at right angles to one
another.
The reasoning by which Dupin (Developp., pp. 165-6,)
arrives at a similar conclusion we must confess ourselves
unable to comprehend.
(62.) It is contrary to the spirit of analysis to conclude
from the peculiarities of an umbilicus that the greatest and
least curvatures at such a point, instead of being at right
angles, become confounded in direction. Whatever di-
rection be taken at an umbilicus for that of one of these
curvatures, the direction perpendicular to it will alone in
strictness be that of the other. The constant and uniform
law that governs these directions is, that from whatever
point they proceed they preserve their perpendicularity to74
MATHEMATICAL DISSERTATIONS.
one another; and to admit that when the point reaches
an umbilicus this law is abruptly annulled, and that the
angle changes at once from 90° to zero, is to admit so pal-
pable a violation of the principle of continuity, a principle
which essentially connects together the individual cases of
every general result of analysis, that it is remarkable such
an admission should have ever been made. The direction
dy
~ at an umbilicus is undoubtedly expressed under the
form jj, which, as we have seen, indicates innumerable
values; these values however necessarily occur in pairs,
obeying the general law referred to above; so that instead
of saying with Dupin,* that the two principal sections
there become confounded, the proper inference is that
these sections are innumerable. One of these, however,
and only one, unites with the continuous series of values
dv
which — passes through during the progress of a point
Or, y, z) along the surface up to the umbilicus. This
unique direction is, as we have seen, peculiarly a direction
of principal curvature ; that perpendicular to it is there-
fore the direction of the other curvature.
(63.) It is probable that the mistake just adverted to
may owe its prevalence to the following circumstances,
which appear to confirm the doctrine : 1st, it is admitted
that the greatest and least curvatures at any point have
the same directions as the lines of curvature issuing from
that point; and 2dly, it is also admitted that at an umbi-
licus the lines of curvature, usually two, merge into one.
At a hasty glance it does indeed seem that, if the two lines
of curvature have always the same directions as the sections
of maximum and minimum curvatures, when the former
* “ Dans ce cas la plus grande et la moindre courbure, au lieu d’etre con-
stamment a angle droit, se confondent dans leur direction.5’
Dtvchppcmeats, p. 100.ON THE CURVATURE OF SURFACES.	75
lines coalesce these directions must coalesce too. But
the lines of curvature coalesce only in the sense in which
an ellipse,by the continual shortening of its minor diameter,
collapses into a straight line, the major diameter, and be-
comes confounded with it ; whereas the directions of the
fixed diameter, and the curve, at the vertex of it, never
become confounded; but always preserve their mutual
perpendicularity.
An examination of the lines of curvature of the ellipsoid,
an examination usually presented by analysts to illus-
trate and confirm the ordinary doctrines on this subject,
will fully support what has been now advanced. But pre-
viously to entering upon this it may be instructive to offer
a few reflections on the more general theory, suggested by
the foregoing investigation.
(64.) It will of course be understood by the reader,
that throughout that investigation it is supposed that the
several differential coefficients r, $, t, derived from the
equation of the surface, do not themselves become suscep-
tible of multiple values, for the particular point under
consideration. This, however, they may do when for that
point they take the form ; and, in the Mémoire already
referred to, Poisson has entered at length into the ana-
lytical investigation of this peculiarity. We may remark,
however, that it is obvious without any analytical reason-
ing, that if there exist a point upon a surface such that
the tangent plane at that point touches the surface along
fly straight lines passing through it, a normal section exists
between the angle formed by every pair of these lines, of
which the radius of curvature is a maximum or a mini-
mum. For as the curvatures of the sections through the
sides of the angle are each 0, and as the surface between
these sides is continuous, it follows that within these
limits there is a continuous series of normal sections, of76
MATHEMATICAL DISSERTATIONS.
which the extremes have each the curvature 0; and con-
sequently, in proceeding from one extreme to the other,
the curvatures must increase from 0 up to a certain limit,
and then must decrease down to 0 $ so that there must
necessarily be a maximum state of curvature between
every two successive lines of contact.
But it must be borne in mind that the tangent plane, in
which these lines of contact are situated, is regarded as
the only one passing through the point in which the lines
meet, as in the ordinary theory of curvature; where the
tangent plane through the point under examination is
always considered to be unique, those singular points
which admit of multiple tangent planes, and therefore
multiple values of p, 9, being excluded from the general
theory. This restriction requires, in the case before us,
that the tangent plane cut the surface in the several straight
lines ; dividing it into compartments about the point alter-
nately convex and concave; just as every tangent plane
divides the surface of the single-sheet hyperboloid.
It is easy to see that the portions of surface intercepted
between every pair of opposite vertical angles will have
curvature of the same kind when n is even, as in the
hyperboloid referred to ; and curvature of opposite kinds
when n is odd. In this latter case the sections of princi-
pal curvature will each have two radii, since the normal at
the point will be cut by a consecutive normal on one or
the other side of the surface, according as this consecutive
normal approaches the fixed one from the concave or from
the convex region. The number of lines of curvature,
or more accurately the number of directions in which
consecutive normals intersect that at the point, may there-
fore be said to be n or *2w, according as n is even or odd.
With respect to the two radii of curvature of each of the
sections at the point under consideration, it will be seen
that they are infinite, because the section has an inflexion
at the proposed point. At every such point in a planeON THE CURVATURE OF SURFACES.
77
curve the radius of curvature may always with propriety
be said to be twofold, because of the twofold character of
d^y
at the point of inflexion. On one side of the fixed
«	d?i/
normal the algebraic sign of is plus, and on the opposite
side it is minus : of these signs that is of course preserved
at the limit which accompanies the moving normal up to
that limit. As two normals or radii with opposite signs
thus terminate in the normal at the point, it follows that
the proper expression for this latter is ± oo; the opposite
signs denoting not any particular positions in reference to
the axes, but mere opposition of direction.
(65.) It has already been noticed that in surfaces pre-
senting points of this kind, one or more of the differential
coefficients r> s, t, derived from the surface, must for every
such point take the form ^; although the mere occurrence
of such a form will not necessarily imply that the point
under consideration has the character above described.
To determine this wTe must ascertain by the ordinary me-
thods what are the particular values, concealed under 5,
which unite continuously with the preceding values of
the general function whence ~ has arisen : there may pos-
sibly be but one such value.
We may lay it down as a general analytical principle
that when we cannot make reference to preceding states
of the function which presents itself in the form 5 y and
yet know that innumerable values are inadmissible, our
mode of proceeding, in the singular instance before us, is
inapplicable; or that, as the French analysts express it,
“ la methode ordinaire est en defaut.’>78
LINES OF CURVATURE OF THE ELLIPSOID.
(66.) The equation of the ellipsoid, when referred to its
principal diameters as axes of coordinates, is
ar
a2
+
Differentiating this with respect to x and y, we have
dz		C2X
dx	= P =	~!Tz
dz		c2y
dy	=? =	b2z
d2z		C4 (y2 _
dx2	=,· =	a2b2z3
d2z		cAxy
dxdy		aWz*
<Pz		 t 		cV—a2)
df		a2b2z^
And these values, substituted in the general differential
equation,
f1 + W\ dy* . f1 +	1 + P21 , dy
~ "Tr'S + l"---------------~)rtTx~
CURVATURE OF THE ELLIPSOID.
79
furnish the particular differential equation
b2—c2	dy2	f a2—c2	2 b2—c2 2 dy
t? + la2(rt2—b2) X ~ b\a%-bi)y ~ )~dx
CL2 — <?
a\a2—b2)
xy = 0 .... [31] ;
which equation* we may remark* will belong to the single-
sheet hyperboloid if the sign of c2 be changed* and to the
double-sheet hyperboloid if the signs of a2 and b2 be
changed* that of c2 remaining.
For abridgment* put
a2(a2—ò2) B__b2(a2—b2)
b2-c2
....[32],
and then the equation will become
xy dy2 ƒ x2
B è2 + \A
.1 dy xy
B J dx A
= 0.... [A];
or finally
df f_B_ x^y
dx2	\ A y x
B 1 dy
xy ƒ dx
±=o..
[33 ]
To determine the integral of this in the most convenient
manner* it will be best to deduce from it the differential
equation of the second order* and then to combine the two*
for the purpose of obtaining* through the elimination of
one of the constants* a single differential equation more
easily manageable than either.80
MATHEMATICAL DISSERTATIONS.
Differentiating then the equation just given, we have
dy2
'¿S'-*-2
_B_
T
dy
dx
x —
_B
A
(s* + y)=°.
or
5Ì(2ij,£+4i’-!'’-b) +
(£+x)(4-»)---pfl
The constants entering [33] and [34] may be cònsi-
g
dered to be j and B ; and if we eliminate the latter from
[33] and [34], there results
which would be satisfied by equating to zero either of the
two factors on the left; but the first factor cannot possibly
be zero, since A and B are quantities essentially positive.
Hence it is the second factor that must be zero; that is,
d2y	dy2	dy 

dx
= 0;
or dividing by .r’,
x^y
y <Py	dy	dx
x dx'2	dx 3?
= 0 .... [36] ;LINES OF CURVATURE OF THE ELLIPSOID. 8i
and multiplying by dx, with a view to the integration of
this differential, we have	
	y ddy dy xdy—y dx x dx dx x2
that is,	y ', dv ,dv ,y x dx + dx X
which is obviously the differential of the product
V dy_
x * dx
C,
C being an arbitrary constant.
Multiplying again by dx, preparatory to a second inte-
gration, we have
ydy = Cxdx
of which equation the integral is
2,2 = Co:2 + C',
C' being another constant, not wholly arbitrary however,
since this integral equation must verify the differential
equation [33], to which it belongs. To ascertain then what
x dv
relation exists between C and C', put first C - for in
y dx
[33], and we get
+	=o
y2	y x xyy y A
or
,r ,	BC B
(C+ A)Cf~y~{~À + C)
0,
or
<c+ |) (Cx* — ys ) - BC = 0,
682	MATHEMATICAL DISSERTATIONS.
that is, putting now for y2 its value Cx2 -f C', as given by
the integral,
(C + |)C' + BC = 0,
.·. C'= —
ABC
AC + B
Consequently the correct integral is
ABC
y2 = Cx2 —
AC + B
[37]
in which C is the only arbitrary constant.
We thus infer that the projections of the lines of cur-
vature upon either of the principal diametral planes are,
for central surfaces of the second order, all curves of the
second degree,
(67.) We shall suppose first that the plane of these pro-
jections is that of the greatest and. mean diameters of the
surface; that is, we shall suppose that
a > b > c.......[38]
and shall confine our attention for the present to the
ellipsoid as at first proposed.
Now, in order to determine the line of curvature passing
through any proposed point, (xy', z'), the constant C
must be so determined that the general equation [37] may
apply to this particular point; that is, we must have
Or'2-
ABC
AC + B’
which furnishes the following quadratic equation in C,
namely,7
LINES OP CURVATURE OF THE ELLIPSOID. 83
Aar'2 C2 - (Ay'2 - Bar'2 + AB) C - By'2 = 0,
from which we get
Ay'2 _ Bar'2 + AB ±s/{(Ay'2 — Bar'2 + AB)2 + 4ABx'2y'2\
2 Aar'2
Now both terms under the radical in this expression
being essentially positive, it follows that the two values of
C will always be real and of opposite signs. Moreover,
ABC
for the positive value of C the constant ——g wdll ne-
cessarily be positive, from the conditions [32] and [38].
For the negative value the same constant will be negative,
because AC + B is always positive, inasmuch as this ex-
pression, taking the negative value of C above, and multi-
plying it by the positive quantity 2a72, is the same as
Ay'2 + Bar'2 + AB - J { (Ay'2 - Bar'2 + AB)2 + 4ABar'y2}
or as
Ay'2 + Bar'2 + AB - ^ { (Ay'2 + Bar'2 + AB)2 - 4ABV2}
which is obviously positive.
(68.) It is necessary to show that the two values of C in
[37] give two equations of different forms, before we can
conclude that the two curves passing through a point (xf,«/),
are really distinct in kind. The method above given is from
Leroy (Analyse Appliqnee, p. 304.) But the same re-
sults may be obtained more easily without actually solving
the quadratic equation in C, and thence, by means of the
artifice adopted in the final expression above, inferring the
sign of AC + B.
For the sign of the last term in that quadratic shows us84
MATHEMATICAL DISSERTATIONS.
at once, that the two values of C must have opposite signs;
and from the known composition of the coefficient of the
second term, after division by that of the first term, we
infer that
y* J3______J3
is the sum of the positive and negative roots. Also,
B yri .
since - is the product of the same roots, we conclude
ft. x
that the negative root cannot be equal, in absolute magni-
g
tude, to for then the positive root would be equal
to H—£, and thus their product would surpass its pro-
g
per value; nor can the negative root exceed — in absolute
A
value; since, in that case, the positive root would be
greater than on the former supposition, and consequently
the product of the two would differ still more largely from
the true product. Hence, the negative root must be less
B
in absolute value than —, or A times this root less than B;
A
so that, whether the positive or the negative value of C be
taken, AC -f- B will always be positive.
(69.) We infer therefore that, in general, through every
point (x', y, zr), on the surface of the ellipsoid, there pass
two fines of curvature, whose projections on the plane of
the greatest and mean diameters are represented by the
equations,
ƒ =-Cx2 + C'
y= + <v-c/.
The former of these equations is that of an ellipse; theLINES OP CURVATURE OP THE ELLIPSOID. 85
latter that of an hyperbola concentric with it, the common
centre being that of the principal elliptic section of the
surface on the plane of xy.
(70.) If, however, the point (x/> y, zf) be such as to
render the irrational part of the expression for C zero;
that is, if it satisfy the conditions
Ay’2 - Bz'2 + AB = 0,
and
Xr yf = o,
these two curves must merge into one, since the two values
of C will then be identical. If such a point exist on the
surface, it must have either x' = 0, or yf = 0. It cannot
have the former, since for xf = 0 the first condition ren-
ders y imaginary; but for y' = 0, that condition gives
xf = ± >/ A. Hence there are four of these points on the
ellipsoid, the coordinates of them being
+ n/A,0,z,;-v'A,0,z'; + n/ A, 0, — zf; — V A, 0, — z\
in which
.	a2—b2
v' A — “ V -¿¡—¿I·
These are the four umbilical points of the surface: they
would have been as readily made known by equating to
zero the coefficients and absolute term in [A], the equation
then giving an infinitude of values for	only one of which
values however points out the direction of a line of curva-
ture. The line itself having for its projection the equation
y = 0, the ~ belonging to it is everywhere —■ = 0.
ax	ax86
MATHEMATICAL DISSERTATIONS.
It thus appears that the conditions which render the
values of at the point determined by them, innu-
merable, are the very same as those which cause the two
lines of curvature to coalesce and form but one. It is
therefore incorrect to say, with Monge and Poisson, that
the lines of curvature through such a point are innu-
merable, and that the particular line here adverted to
differs from all the others only by a certain peculiarity.
And it is equally incorrect to say with Dupin, that the
value of at this point is unique, and determined by the
direction of the single line of curvature passing through that
point; since, as already shown, there is no limit to the
dv
number of admissible values for when the conditions,
dx
here supposed to exist, have place.
The manner in which the single line of curvature passing
through each umbilicus has been determined, has nothing
in it that is peculiar, or that amounts to any departure from
the method of obtaining the lines of curvature in general,
passing through any other point on the surface. In all
cases the coordinates of the proposed point are employed,
like as we have employed those of the umbilicus, to deter-
mine the constants, and thus fix the equations of the twro
lines through that point; and the occurrence of equal values
for these constants is all that is necessary to apprize us that
the point to which they refer is an umbilicus; so that the
statement of Dupin, that the ordinary method of deter-
mining the lines of curvature through such points fails,—
unless, indeed, they be upon a line of spherical curvature,
—appears to have been hastily made.*
(71.) We have just seen that there are four points on the
ellipsoid, in reference to which the two values of the con-
See Développement» de Géométrie, p. 359.LINES OF CURVATURE OF THE ELLIPSOID.
87
stant C, in the integral equation of the pair of lines passing
through each, become identical, these values being each
zero. From the mere form of the quadratic equation which
determines C, or indeed from that of its final term only, we
might have predicted that if such identical values were pos-
sible, they must necessarily be zero; for as that final term
is essentially negative after dividing by the coefficient of C2,
it is impossible that equal roots can exist except in the
extreme case of C = 0; and even then the roots, strictly
speaking, are C = 0 and C = — 0.
(72.) The evanescence of the irrational part of the ex-
pression which exhibits the two roots of a quadratic equa-
tion is the ordinary indication of the equality of those
roots. But in extreme cases—those, namely, in which the
entire numerator vanishes with the irrational part of it—we
must not hastily conclude merely that two roots have
merged into one; for it is necessary to such conclusion that
the denominator be not, like the numerator, evanescent.
Our inference, as to the two values of C being zero, when
the numerator is so, is valid, because the denominator
remains finite. But if we take the equation [A], namely,
xy_ df — t	o
B ¿cl+U B Jdx A
and equate to zero the irrational part of the expression for
—£; or which is the same thing if we take the ordinary
criterion of equal roots, namely,
4 x2y2
AB"
0;
and infer that when this is satisfied, the two values of
dy
dx88
MATHEMATICAL DISSERTATIONS·
which in the absence of this condition are distinct, will then
become confounded, we shall commit error. For this con-
dition cannot hold unless we have the two conditions
xy = 0
inasmuch as the sum of two squares cannot be zero except
each be zero separately. Now these two conditions can-
not have place, for real values of x and y, unless y = 0 ;
and since the denominator in the expression for the root,
and which denominator has been disregarded, is 2 the
true inference is, not that in the supposed circumstances
the two roots have become one, but that these roots have
become innumerable ; the expression for them being
dy _ A B __ o
dx	xy	0
1 B
The two conditions above, being identical with those at
(70), determine the umbilical points, as this last indeter-
minate form for ^ would imply. These same conditions
dx
however have been erroneously supposed to indicate equal,
•	,	dy
instead of innumerable values for The umbilical points
have thus been correctly determined, although not as a
consequence of the true interpretation of ~ in reference
to them.
The mistake here adverted to was committed by Monge,
the illustrious originator of these speculations; and al-
though the error was pointed out by Dupin, (Développe-LINES OF CURVATURE OF THE ELLIPSOID. 89
merits, p. 185*) who does not however supply the correct
conclusion* it has been repeated by later writers. (See
Transactions of the Royal Irish Academy* Vol. XIV.
p. 89.)*
(73.) The line of curvature through the four umbilici*
having for its equation y = 0, is the principal ellipse in
the plane of the greatest and least axes. If we conceive a
point on this ellipse to move towards an umbilicus* it will
thus describe one of its own lines of curvature; the second*
perpendicular to this* must, by the law of continuity* gra-
dually collapse as the moving point approaches the umbi-
dit
licus* the tangent —of the collapsing curve at its vertex*
still remaining oo . When the umbilicus is actually reached,
the curve closes, and becomes confounded with the line
of curvature described by the point; the tangent at the
vertex of this closed curve is still however perpendicular
to the axis into which it has merged. But these particu-
lars will perhaps become more intelligible from an in-
spection of the lines of curvature themselves.
To project these on the plane of xy, it will be conve-
nient to put the general equation [37] of the projections
under the form
dy
Dupin remarks, in reference to the foregoing expression for that as
it assumes the form jj nothing can be inferred from it; and, like as in other
parts of his work, affixes to this symbol only the particular value subsequently
determined from it by the application of the calculus. But this view of the form
is an erroneous one. If no more than a particular value were concealed
under that form the fraction would have a vanishing factor common to nume-
rator and denominator, which however is not the case. Numerator and de-
nominator vanish in consequence of two distinct hypotheses or conditions;
and these are such as to render the original equation indeterminate, and there-
dy
fore the values of — innumerable. (See the Differential Calculus, p. 71.)90
MATHEMATICAL DISSERTATIONS·
X*
m2
[39]
by writing for the essentially negative quantity, —-
AO B
ABO
the symbol — m2; and for —r-------- , which· as we have
seen, is positive for one line of each pair, and negative for
the other, the symbol ± n2; the quantities m and n will
then be the semiaxes of the two curves. These quantities
are also otherwise connected with each other by a remark-
able relation — the relation between the quantities for
which we have just substituted them ; and which is seen
at a glance to be expressed by the equation
A
B
1.......[40]
the upper sign having place, as in the preceding equation,
for the elliptic lines of curvature, and the lower for the
hyperbolic.
From this it appears that the semidiameters m, w, of
any one of the projections [39], are respectively equal to
the coordinates m, n of some point in the corresponding
curve [40] ; so that taking the upper sign in these equa-
tions, the coordinates of a point (m, n) in the hyperbola
[40] will exhibit the length of the semidiameters m, n of
one of the elliptic projections [39] ; and taking the lower
sign, the coordinates of a point (m, n) in the ellipse [40]
will be the lengths of the semidiameters of one of the
hyperbolic projections [39].
Hence both series of projections may be constructed,
by help of the auxiliary hyperbola and ellipse, [40], as
follows:LINES OF CURVATURE OF THE ELLIPSOID. 91
On the semiaxes OA, OB of the principal ellipse, coin-
ciding with the plane of projection, set off the distances
0« = A = a
a2 — b2
a2 — b2
0/3=V'B = 6v/-f—2
and with these as semidiameters construct the auxiliary
hyperbola aPQ' whose real vertices shall be upon A'A,
and the auxiliary ellipse aP"/3.
Take any point F in the hyperbola ; transfer its coor-
dinates PE', PD' to OD1, OE', and with these as semi-
diameters construct an ellipse E'MD'; this will be the
projection of one of the lines of curvature. In a similar
manner may every one of the elliptic projections be con-
structed.
(74.) For the hyperbolic projections we must transfer
the coordinates, P E", P D" of each point P" in the aux-
iliary ellipse, to OD", OE"; and with these as semidia-
meters construct an hyperbola D"M, whose real vertices
are upon A'A; this will be one of the hyperbolic pro-
jections and others may be determined in like manner.92
MATHEMATICAL· DISSERTATIONS.
The real diameter 20a or aa', common to the auxiliary
ellipse and hyperbola, is the distance between the umbili-
cal points, projected on the plane which receives the seve-
ral lines of curvature (70); and it is plainly this line aa'
into which the flattening ellipses ultimately merge when
P' approaches towards, and at length reaches the vertex a
of the auxiliary hyperbola.
In like manner as the point P", moving along the aux-
iliary ellipse, approaches towards the same vertex, the
imaginary axis OE" of the hyperbolic projection continu-
ally diminishes to zero, and the real axis OD" continually
increases up to its limit Oa ; hence, when this is reached,
each hyperbolic branch closes, and the curve then merges
into the two straight lines aA, a'A'. The lines of curva-
ture therefore thus circulate as it were round the umbilical
points, always turning their concavities towards them.
On the surface these lines are of course all curves return-
ing into themselves, as the surface is everywhere limited;
the hyperbolic arcs therefore into which one series of them
is projected, are limited by this circumstance within the
bounds of the principal diametral plane which receivesLINES OF CURVATURE OF THE ELLIPSOID. 93
these projections; so that the auxiliary hyperbola is not
required beyond the point QThe curves of double cur-
vature, which furnish the hyperbolic projections, after
passing round the umbilicus of which a is the projection,
widen, embrace the vertex A of the surface, then contract-
ing pass round the other umbilicus, having the same pro-
jection, and return into themselves.
If we take D' any point in A a, then D'P' and AA' will
be the projections of the tangents, or directions, of the
lines of curvature through the point which D' represents.
These directions are at right angles, and they obviously
continue so during the entire progress of D' up to a ; for
D' continues to be the vertex of the collapsing ellipse even
up to a, at which point the ellipse actually closes and
becomes confounded with the line aa'.
If our moving point were taken between these limits,
as at D"; then in like manner the directions D"P", AA',
continue at right angles during the progress of D" up to
a; at which point the hyperbola closes and becomes con-
founded with the lines aA, a'A'.
Hence it may be inferred that at the umbilical point
the two lines of curvature, ordinarily elliptic and hyper-
bolic, become, not confounded with each other, for strictly
speaking they are still distinct, but, confounded with the
unique line of curvature through the point; one line of
curvature with one portion of this unique line, and the
other with another portion; the remaining portion of the
entire line is that into which the other branch of the ulti-
mate hyperbola degenerates.
(75.) It is obvious that if an ellipsoid be cut by a plane
through the mean axis BB', and parallel to the tangent
plane at an umbilicus, the section must be a circle; if
this section incline towards the greatest principal section
there will arise an ellipse having the mean axis of the94
MATHEMATICAL· DISSERTATIONS.
ellipsoid for its minor diameter; but if it incline in the
other direction, towards the least principal section, the
ellipse furnished will have the same mean axis for its ma-
jor diameter; the ellipses thus change their character when
the revolving plane, through the mean axis, passes through
an umbilicus. As the tangent plane parallel to this re-
volving plane in any position always gives an indicatrix
similar to the section made by that plane, it follows that if
a point on the unique line of curvature, and between the
two umbilici a, a', move towards one of these points, the
sections of greatest curvature will, for every position, be
the principal section containing the greatest and least
axes of the ellipsoid; and consequently the section perpen-
dicular to this will always be the least. But when the
moving point passes an umbilicus, the least section will,
on the contrary, be that of the greatest and least axes of
the surface, and that perpendicular to it will be the
greatest.
On these opposite sides of the umbilicus therefore the
character of the surface changes; although at any other
two points on a line of curvature, and symmetrically posited
relatively to the umbilicus (and taken in its neighbour-
hood or not), the surface will exhibit equal or symmetrical
curvatures, as the lines of curvature show.
If the minor diameter of the ellipsoid—and which is the
minor diameter of the ellipse on which the umbilici are
situated—be conceived to lengthen till it become equal to
the mean diameter, the umbilici near A will approach
towards and unite in that vertex, as is plain from the ex-
pression for Oa ; in like manner the other pair of umbilici
will approach towards and meet in the vertex A': the
surface will then become an ellipsoid of revolution; and
the curvature about the united umbilici being uniform in
all directions, lines of curvature will proceed from them in
all directions.LINES OP CURVATURE OP THE ELLIFSOID.
95
(76.) By differentiating the general equation,*
Axy ^ + (Bx2- A/ - AB)^- Bxÿ = 0 ... .[41]
which at the umbilicus a, where y = 0 and x = \/A> has
zero coefficients, we have
Axdf.
A dx3
Aÿë + Bxg-Bÿ = 0,
and this, for the same values of y and x, is no longer in-
determinate ; but is decomposable into the two determi-
nate equations
df
dx1
+ B = 0.
The first of these marks the direction of the unique line
of curvature; the second gives imaginary values for
If the surface be of revolution, about the axis of 0, that
is, if A = B, the differential equation [41], after dividing
by B, remembering that in this case A = 0, becomes
* The student will not fail to bear in remembrance that the differentiation
spoken of here, and in other places, in reference to the differential equation
of lines of curvature, affects only the coefficients of and the term indepen-
ax
dent of it; ~ itself being regarded as constant. These coefficients, being in
doc
dy
the proposed hypothesis all zeros, the values of -j-, if expressed in functions
of them, would necessarily take the form 2 ; that particular value, concealed
under this form, which obeys the law implied in the general expression
whence jj has arisen is, as we know, educed by differentiating numerator
and denominator of that general expression, and introducing the proposed
hypothesis into the results. And it is this that is effected, virtually, by dif-
ferentiating the coefficients at once, as in the text.96
MATHEMATICAL DISSERTATIONS.
Xy^ + %~xy~Q.......t42l’
which is indeterminate for x = 0, y = 0. Hence, differ-
entiating the vanishing coefficients, we have
df
dx3
dj?
dd?
dy
x — y + x — — y = 0.
y	dx *
This also for the same values of x and y is equally inde-
terminate : therefore, differentiating again, we get

which, as the coefficients are exhausted, shows that the
lines of curvature issuing from the point x = 0, y = 0, are
innumerable, and equally close to the osculating sphere
(57). This merely confirms what the general theory has
already established for every surface of revolution.
(77.) The general differential equation [42] of the lines
of curvature of such a surface, when of the second order, is
integrated with great ease; for dividing by xy, it becomes
dx2	v y	x' dx
-1=0,
a quadratic equation, in which the two roots are actually
exhibited within the parenthesis, inasmuch as the product
of these gives the absolute term — 1. Hence
.-.ydy=-xdx,
dy	y " dy	dx
—"» · · — · *
dx	x y	xLINES OP CURVATURE OF THE ELLIPSOID. 97
The integral of the first of these is
if =C — X2 + c, /. y2 + x2 = C.
The integral of the second is
logy = log C, x, .·. y = Cx ar.
Consequently, the projections of the lines of curvature are,
for the first series, all circles described round the origin as
centre; and for the second series straight lines radiating from
this common centre, and, consequently, crossing the circles
at right angles. Through any proposed point (xf, y', z'), only
one circle and one straight line pass : the constant C! for
this point is Cj = the straight line being y = — x.
When the point is the vertex of revolution, x' = o ,y = o;
and the equation is y = ^ x, the lines from that point being
innumerable.
(78.) If the plane on which the lines of curvature of the
ellipsoid, instead of being that of the greatest and mean
axes, were the plane of the least and mean axes, we might
make all our preceding reasonings and deductions apply,
not by changing the plane of xy, which has hitherto been
employed, for that of yz; but, preserving the plane of xy,
as that of projection, by changing the relative magnitudes
of the semiaxes a, 6, c; that is, by supposing
c > b > a,
under which conditions A and B will be positive as before,
and the general equation [33] of the lines of curvature, thus
preserving the same form, will conduct to like results as to
the elliptic and hyperbolic curves into which these lines of
curvature are projected.
798
MATHEMATICAL DISSERTATIONS.
But if the plane of projection be that containing the
greatest and least axes; that is, still making the plane of
projection that of xy, if the conditions be
a > c > b;
• then, since B no longer continues positive, the general dif-
ferential equation becomes modified, and may therefore
refer to curves of a different kind. We shall proceed then
to investigate the nature of the projections when the plane
which receives them is that in which a and b are situated,
these being considered as coincident with the axes of x
and y, and as conforming to the above conditions.
Put, as in the former case,
at___ A . w _ b\a2-b2) _ n .
A	a2—c2	“ A> B ~ c2—b2 ~
then, as before, the integral equation will be
y* = C**+C'........................[43],
the only difference between this and that formerly ob-
tained consisting in the equation of condition which con-
nects the constants C, C'; which equation obviously differs
from the former in having — B' in place of B. Making
this substitution then, we have in the case before us
c,_ A'B'C
A'C - B'
and consequently the general integral equation of the
lines of curvature is
A'B'C
y2 = Cx2 +
A'C -B'
[44]LINES OF CURVATURE OF THE ELLIPSOID. 99
C being the arbitrary constant determinable for any par-
ticular pair of these lines by means of the coordinates of
the proposed point whence they are to issue. But
wherever this proposed point be situated it may be proved
that C must invariably be negative, and consequently that
C' must always be positive. For taking any point z')
the value of C in terms of x'y yf and the constants A', B'
is, as at (67)
„ _ Ay2 + BY2 — A'B' ± s/ { (Ay2 + BY2 — A'B')2— 4A'BY2y2\
2 AY2
where the rational part of the numerator is obviously
greater in absolute value than the irrational quantity con-
nected with it. Now this rational part is negative for
every pair of values of x'y yf; that is to say, we must al-
ways have the condition
Ay2 + BV2 < A'B'
£
B'
<1.
For whatever point on the ellipsoid be projected on the
plane of xy yy that is, the plane of a, 6, the x of it can never
exceed ay nor the y of it b; and it is of course impossible
that these extreme values can simultaneously exist; never-
theless, if we substitute even them for x' and y\ in this
last expression, still the result will be only a unit; for
l·)2 a2 _ c2 — hi2 a2 — c2 
B7 + A' - a2 - A2 + a2 — 62 - ’
hence in every possible case the result is less than a unit.
Consequently, as the value of C is always negative, and100
MATHEMATICAL DISSERTATIONS.
that of the other constant in [43] always positive* it fol-
lows that the projections of the lines of curvature are all
ellipses.
If we put in the general equation [44]
A'B'C	a A'B'
atr*	n > atd/	w
it will take the form
the semiaxes m, n of each ellipse being in virtue of the
substitutions just made related to one another by the
condition
5 + ^ = 1....[45],
so that they are always determined by the coordinates of
a point (m, n) in the auxiliary ellipse [45] constructed on
the semidiameters OX = \/A! and OY = y'B'.
(79.) By referring to the values which A' and B' repre-
sent* we see that these semiaxes exceed those of the prin-LINES OF CURVATURE OF THE ELLIPSOID· 101
cipal section of the surface upon which the lines of curva-
ture are to be projected. There is one point (J in the
auxiliary ellipse of which the coordinates are equal to the
semiaxes of the principal section referred to ; for putting
in the equation [45] a for m there is found to result b for
n; and hence this principal section is one of the lines of
curvature. But as the abscissa of every point P in the
auxiliary ellipse between Q, and X is longer than that of
Q, it follows that the ellipses determined by the coordi-
nates of these points will extend towards X beyond the
limits of the surface, or of the principal section which ter-
minates the projections; and in like manner, since the
ordinate of every point P', between Q, and Y, exceeds that
of Q, the ellipses determined by such points will extend
in the direction OY beyond the bounds of the surface.
With the exception therefore of the principal section
ACA' the lines of curvature will have for their projections
only portions of ellipses. One series of these projections,
determined by that part of the auxiliary ellipse which is
between X and Q, will want the segments towards X and
X'; the other series crossing these at right angles, and
determined by that part of the auxiliary ellipse which is
between Q, and Y, will want the segments towards Y and
Y'. The initial ellipse in the first series, furnished by the
point X in the auxiliary ellipse, is obviously the straight
line XX'; and the ultimate ellipse in the second series,
furnished by the point Y in the auxiliary ellipse is the
straight line Y Y'; so that the three principal sections of
the ellipsoid are lines of curvature.
(80.) It is of importance to notice that the general
equation of these lines of curvature has been derived from
the integral of the differential equation [36], which was
substituted for the equation [35] because the factor
^2 + 5- could not be zero without contradicting the hy-
dx A102
MATHEMATICAL DISSERTATIONS.
pothesis as to the signs of A and B, these quantities being
necessarily positive. But in the present case we have
no such warranty for rejecting the corresponding factor,
which, because A = A', and B = — B', is ^	, fur-
nishing the additional solution
dy
dx
This, therefore, is another integral of the differential
equation of the second order [35], and by which that dif-
ferential equation is evidently satisfied. It contains no
arbitrary constant, nor is it implied in the general integral
involving such a constant, before deduced. It is thus the
singular solution of the equation referred to.
dtt
By substituting this value of ~ in the original differ-
ential equation of the first order [33] we get the new in-
tegral equation.
A y + BV* ± 2 xy v'A'B' = A'B',
involving no arbitrary constant, and distinct from the
general integral [37]. This, therefore, is the singular
solution of the proposed differential equation.
This singular solution is, we know, (see Integ. Calculus,
p. 237,) wrhen interpreted geometrically, the line which
touches or embraces all the curves comprised in the ge-
neral integral It is in the present case decomposable
into the four straight lines represented by the equations
y1/A,±*v'B,= + v'A'B',
which are readily seen to be the four supplemental chordsLINES OF CURVATURE OF THE ELLIPSOID. 103
connecting the four principal vertices of the auxiliary
ellipse.
These four chords therefore touch all the elliptic pro-
jections of the lines of curvature. We already know that
the two lines of curvature through an umbilicus degene-
rate into two opposite arcs of the principal ellipse AC A'C';
and as the supplemental chords just determined touch all
the lines of curvature, it follow’s that the two from each
umbilicus must be touched at their junction, that is, at
the umbilicus itself. Hence the points a, a2, a', of19 wdiere
the supplemental chords touch the principal ellipse are the
four umbilici of the surface. Or we may infer that these
points are the umbilici from considering that as the sup-
plemental chords are the loci of the consecutive inter-
sections of the varying ellipses, they must touch the prin-
cipal ellipse at the points in which the intersections of
a neighbouring ellipse with it settle, when this latter, by
varying, comes to coincide with the former. These con-
secutive intersections are evidently the umbilici; because
the two curves are distinct only so long as they cross each
other at points which are not umbilici; the umbilici alone
are the points at which, when the intersections merge
into them, the curves become confounded, or, as it is
usually expressed, consecutive.104
LINES OF CURVATURE OF THE HYPERBOLOID.
(81.) The projections of the lines of curvature of the
other central surfaces of the second order may be very
easily effected from the foregoing examination of the
ellipsoid, as we have already intimated. Thus in the double-
sheet hyperboloid we shall only have to change the signs
of a2 and b% in the equation [31], and wre shall thus get
for the semiaxes of the auxiliary hyperbola and ellipse,
the values
\/A = a\/
a2 — b2
a2 + c2
y'B = b \/
a2 — b2
b2+c2
and for the equations of these curves, m> n, being as at
(73), put for the .r, y of any point in either, we should
have
B
The real axis of the auxiliary hyperbola will in this case
coincide with axis of y, or with the principal diameter 2b
of the surface. These auxiliary curves being constructed,
the elliptic and hyperbolic projections which, as in the
ellipsoid, will still be analytically represented by the
equations [39] may be traced as before: they will present
to the eye an appearance exactly similar to those already
exhibited for the ellipsoid.LINES OP CURVATURE OF THE HYPERBOLOID. 105
The hyperboloid of a single sheet will be represented by
our original equation [30], by changing the sign of c2; so
that the semiaxes of the auxiliary hyperbola and ellipse
will be expressed as before, and the equations of the curves
themselves will correspond with those already exhibited
for the ellipsoid. The projections will present an appear-
ance similar to those for the ellipsoid.
(82.) The double-sheet hyperboloid has, like the ellipsoid,
four umbilical points, two on each sheet, round which the
projections of the lines of curvature circulate, presenting
always their concavities towards those points.* And
although, as just remarked, the projections in the single-
sheet hyperboloid present the same general appearance, yet
the elliptic sections here never collapse into a straight line
and become confounded with the major diameter, since these
sections cannot contract beyond the throat of the surface,
within which no projections can be situated. At the
vertices of the major diameter of this limiting ellipse the
hyperbolic projections close and become straight lines,
having the limiting ellipse just mentioned for the second
line of curvature of each.
* The four umbilici being determined exactly in the same manner as those
of the ellipsoid are found at page 85, it is unnecessary to repeat the in-
vestigation.ro6
LINES OF CURVATURE OF THE PARABOLOID.
(83.) Let the equation of the paraboloid be
x2 y*
P> + -p —
P and F having the same or different signs according as
the paraboloid is elliptic or hyperbolic. By differentiating
we have
dz	x dz	y
d^~P=¥" d~y = q~¥’
d2z	1	cfz _______
dx2	r P" dxdy	S 9
d?z	1
= P'
These values, substituted in the general differential equation
[22], give for the lines of curvature of the paraboloid pro-
jected upon the tangent plane at its vertex, or upon any
plane parallel to this, the equation
xv	f ______y*_ J______M dl _	_ o ·
P2P' dx2	+ \PP/a	P2F	^ P	P'J dx PP2
or,
/p x _y _ P(P—p;)i jy _ p n
dx2	\P' y x	xy j dx P'LINES OF CURVATURE OF THE PARABOLOID. 107
This equation, by putting
A = F(P-P'), B =P(P-P'),
becomes the very same as that marked [33], the only dif-
ference being in the numerical values of A, B. It may
therefore be written
df	rB x	y
dx2	[Ay	x	xyjdx A
(84.) Let the hyperboloid be elliptic; that is, let P and
F be both positive, and let P > P'; then, as in the ellip-
soid, A and B are both necessarily positive, and conse-
quently all the conclusions follow from this equation that
were before obtained from it for the ellipsoid. We infer,
therefore, that the lines of curvature of the elliptic parabo-
loid, when projected upon the tangent plane at the vertex,
furnish, as in the ellipsoid, a series of ellipses crossed at
right angles by a series of hyperbolas, the concavities of
both series being turned towards the two umbilical points.
The auxiliary hyperbola and ellipse which serve to project
these curves have for their semiaxes the values
v'A^v'PXP-P')
1/ B = v"P(P-F),
and the projections themselves are exhibited in the diagram
at page 91. The preceding values of ±n/A express, as in
the ellipsoid, the distances of the origin, measured along
the axis of x, from the projected umbilici·
(85.) Next, let the paraboloid be hyperbolic, then one of
the coefficients in its equation must be negative: let P be108
MATHEMATICAL DISSERTATIONS.
negative* then B is positive and A negative* and the in-
tegral of the preceding differential equation will differ from
that marked [37] in respect to signs: it will be
y2 = Cx2 +
ABC
B—AC
[46];
and it is plain* whatever point (x'* yr) be taken to deter-
mine the constant C* that this constant can never be nega-
tive* inasmuch as such a negative value would render y'
imaginary.
If the equation [46] be reduced to a quadratic in C*
and then solved* the values of C for any proposed point*
although* as just seen both positive* are nevertheless al-
ways such as to render the corresponding values of
B—AC* the one positive and the other negative. For
referring to the expression for C at page 83, and changing
the sign of A to adapt it to the case before us* it is easily
seen that
2x'2(AC — B) =At/'8 — Bx'2 + AB ^/{ (A/2 + Bi'2 -f- AB)2— 4ABa'V2}
= A/a- Ba'2 -f AB HFVUAy2 — Ba'2 + AB)2 + 4ABV2},LINES OF CURVATURE OF THE PARABOLOID. 109
which is evidently positive or negative according as the
irrational part is positive or negative. Hence AC—B is
positive for one of the values of C, and negative for the
other.
It follows, therefore, that the projections of the lines of
curvature upon the plane of xy9 or upon any plane perpen-
dicular to the axis, are all hyperbolas, of which the equa-
tions are
© "	9	—— A »
nr mr
the auxiliary hyperbolas by which these are constructed
having for equations
ws m2 _ ,
b"a=+l
These denote a pair of conjugate hyperbolas, the absolute
lengths of the axes of which are, seeing that P takes the
sign minus,
V A = V'FCP + P7)
^ B === VPCP + P7):
These auxiliary hyperbolas are similar to the hyperbolic
sections of the proposed surface, made by planes parallel
to that of xy. For putting any constant value for z in the
equation of that surface, we find that the axes of the
section at that distance from the plane of xy9 are to each
other as >^/F to \/P9 and this is the ratio of the axes 2\/A,
2x/B.110
MATHEMATICAL DISSERTATIONS.
NOTE.
The expression for R, the radius of curvature of any
normal section of a curve surface, deduced at page 49, is
obtained on the hypothesis that the plane of xy touches
the surface at the point to which R refers. By help of
the equation of the projected indicatrix, given at page 56,
a more general expression for R may be easily derived;
that is, an expression independent of the position of the
rectangular coordinate planes.
The equation referred to is
r (x — cCf + 2s (# — a) (y — b) + t(y — bf = 2h
where k, and the several factors within the parenthesis,
are taken in the limit. This k however is not the same as
the h which enters into our fundamental expression for the
radius of curvature, at page 45,—that being on the nor-
mal, and this on the axis of z. The cosine of the angle y,
between the two, is (Differential Calculus, p. 206,)
1
cosy =	„	1 '■
\/\ + p2 + q
so that the h in the foregoing equation must be multiplied
by this in order to produce the h hitherto employed in the
expression for R. Calling therefore that semidiameter of
the indicatrix, through which the normal section passes, k>
we have by the usual formula (page 45),
R== ______________JL__________________—
r (x — a)2 -r 2s (x — a) (y — b) + t(y — b)2
s/( i +"? + q*)
______________kWC +p2 + ?2)____________
“r(ar — c/)2 + 2s(x — a)(y—b) + Hy — bfSIGN OF RADIUS OF CURVATURE.	Ill
Now it is obvious that (x — a) and (y — b) are the pro-
jections on the axes of x and y of each semidiameter k; so
that putting a and /3 for the angles at which k is inclined
to the axes of x and y, respectively, the preceding ex-
pression for R may be changed into
*y(i+if+ y»)
rk2 cos2a + 2sk2 cosa cosg + tk2 cos2j3
\/(l +P* + 92)
rcos2a -f- 2scosa eosj3	tcos2ff
Just as in the case of plane curves, so here, the radical
which enters into the expression for R has been the source
of some perplexity and confusion in the doctrine of sur-
faces. Instead of viewing this radical in its proper light,
as explained in the first Dissertation, and rejecting the
double sign of it as a superfluous appendage, introduced
into the analysis by giving a wider interpretation to our
symbols than the restrictions of the problem warrant,
analysts have sought to connect this double sign with the
direction of curvature, and have recommended one or other
of the two to be preserved according to circumstances.*
Leroy + states that if we agree to take this radical
always positively, the formula will indicate at once both
the amount and the direction of the curvature of each
normal section. But this convention is opposed to the
pre-established theory of the sign in plane curves, where
the general expression is preceded by a minus ; and it is a
plane curve in the case before us. The proper sign pre-
fixed to R should therefore be minus, and not plus.
* See Dupin. Développements, p. 144.
t Analyse Appliquée à la Géométrie, p. 265.112
MATHEMATICAL DISSERTATIONS.
This sign however has noreference to the radical; but
to the hypothetical conditions implied in the general in-
vestigation of R, as already explained. In our reasonings
in the present dissertation we have taken account only of
the absolute value of R, for the purpose of deducing gene-
ral properties of the surface. At the outset (Art. 31) the
algebraic sign of R was suppressed; and its absolute length
only made to enter our investigations. If this sign be
a2
restored the expression [7] at page 45 will be R = — ~
and, consequently, that deduced above is
r _____________j/(l + p2 4- q2)______
r cos2a -f- cosa cosfl + t cos2j3
If the plane of xy were a tangent at the point to which
R refers, and moreover, if the normal section passed
through the axis of x, then it is plain that the numerator
of this expression for R would be reduced to 1; and the
denominator to r, since in the case supposed cosa =1,
and cosj3 = 0; so that we should have
T5 	1_	1
R	r	¿Fz’
dx2
as at page 28. If, as Leroy proposes, the sign were to
be taken plus, there would then be a discrepancy between
two expressions which ought to be identical.
We take this opportunity of remarking, in reference to
the statement at p. 73, that when the two quantities under
the radical are not, as at p. 83, both essentially positive,
they will, when equated to zero, represent a curve; and the
particular values, adverted to in the text, will then belong
to the intersections of the curve, of which this is the pro-
jection, with, the line of spherical curvature.DISSERTATION III.
ON THE INCOMMENSURABILITY OF THE CIRCUMFER-
ENCE OF A CIRCLE TO ITS DIAMETER $ AND ON
THE SUMMATION OF CERTAIN CLASSES OF INFINITE
SERIES DEPENDENT ON THE NUMERICAL VALUE OF
THE SEMICIRCLE.
8DISSERTATION III.
ON THE INCOMMENSURABILITY OF THE CIRCUMFER-
ENCE OF A CIRCLE ; AND ON THE SUMMATION OF
CERTAIN CLASSES^OF INFINITE SERIES.
(86.) Many attempts have been made by modern
analysts to demonstrate the incommensurability of the
number which expresses the ratio of the circumference of
a circle to its diameter; and thus to put an end to all
hope of effecting the numerical rectification of the circum-
ference, or the numerical quadrature of the inclosed circle,
in finite terms.
Of these attempts we believe but two, at most, have
been regarded as completely successful. Of these, one
was published by Lambert, in the Memoirs of Berlin for
1761, and the other by Legendre, in Note iv. at the end
of his Geometry.	The demonstration of Lambert has
not met with very general acceptance among mathema-
ticians ; and since the publication of Legendre’s method
of proof, which has entirely superseded the former, it is
but very seldom referred to by the more recent writers on
the circle.
The demonstration of Legendre, though no doubt
satisfactory, is nevertheless long and intricate ; resting
upon several subordinate analytical theories which greatly
retard and complicate the proof; so that a short and easy
demonstration, dependent solely upon the well-known116
MATHEMATICAL· DISSERTATIONS.
fundamental principles of analysis, is still a desideratum
in the complete discussion of a problem which has occu-
pied the special attention of mankind ever since the origin
of geometry.
Whether the attempt here made to supply such a de-
monstration fully accomplishes the object desired, it must
be left with analysts to determine. On the score either of
length or of intricacy, the method of proof here proposed
is certainly free from objection; it is as short and as ele-
mentary as can well be expected. Should it be found
upon examination to be as little chargeable with the
graver fault of logical inaccuracy, it may perhaps, as far
as it goes, bear a comparison with the former demon-
strations.
That of Legendre however certainly goes farther; as
it proves the incommensurability, not only of the semi-
circumference 7r, but also of its square 7r2. There is in-
deed every reason to conclude, short of absolute proof,
that not only the square, but the cube, and all higher
powers to infinity, are equally incommensurable. Dr.
Barrow, in his Mathematical Lectures, has advanced
some arguments in support of this opinion; but they
have not the character of demonstration. The conjectures
of Legendre go even beyond this, implying the probability
that the number 7r is not included among algebraical irra-
tional quantities; in other words, that it cannot be the
root of any algebraical equation having a finite number of
terms with rational coefficients. But, he adds, a rigorous
demonstration of this seems very difficult to find.
The short article on Infinite Series, which follows the
demonstration in question, is designed to show the com-
petency of ordinary algebra to effect, and with great ease,
the summation of some extensive classes of infinite series
which would at first sight seem necessarily to require the
aid of definite integrals.
The algebraic principle employed in these summationsCIRCUMFERENCE OF A CIRCLE.
117
is one of extreme simplicity; being no other indeed than
the principle of common subtraction. We have brought
only two or three classes of series under its operation in
the present short specimen; but it is easy to see that
other classes might have been submitted to the same
simple process of summation with equal success.
ANALYTICAL PROOF OF THE INCOMMENSURABILITY
OF THE CIRCUMFERENCE OF A CIRCLE.
(87.) The theorem of De Moivre, when expressed so
as to comprehend fractional values of the exponent, may
be written thus,
(cosa + V — 1 sina)A = (cosAa +l/— 1 sinAa) 1A.
But when integral values only of the exponent are to be
admitted, then the formula will be complete without the
factor 1A. In the present investigation the theorem will
be employed only in this restricted sense; and common
multiplication alone is sufficient to establish its truth. It
obviously furnishes the following four identities, namely,
(cosa + V — 1 sina)A = cosAa +	1 sinAa
= cosaA + V — 1 sinaA = (cosA -f t/—1 sinA)%
where A and a are both integers.
Now if any multiple of the circumference 2w, as118
MATHEMATICAL DISSERTATIONS.
k. 27r, could be a whole number, then putting a = 2kw,
we should have, from the first and last of these identities,
the equation
1A = (cos A + V— 1 sinA)2*71’,
or, substituting for cosA, sinA their developments,
a i A2	· a a As ,
cosA = 1 — — +.........., sinA = A—+----------------
the equation becomes
1A = (1 + l/—"i.A — 2 A2 + ... .)”,r
= {1 +(-1/31-2 A + ....) A }s*,r
At
= 1“*· +	_ 2 A +....) A + &C.,
At
that is, since l2*7’’ is necessarily 1 by hypothesis, as
also 1A,
l=l+2*jr(\/^i—JA+ . . )A+	V (s/—\ _ i A+ . .)’A»+ ..
consequently,
___	1	o hir i
2kir V — l . A ·+· 2kir (— --------) A2 +----= 0 ;
or finally,
2kir \/^l . A - 2 (¿?r)2A2 + .... =0 ....[1]
a series ascending according to the powers of A.CIRCUMFERENCE OF A CIRCLE.
119
Now, by the hypothesis, the roots of this equation,
that is to say, the values of A, being all integral, we must
have
A = 0, = di 1, = ± 2,..........,
for the infinite number of the roots of the equation [1].
But these same values of A, to the exclusion of all
others, are given by the equation
P sinAir = 0,
or, developing the first member in powers of A, the same
values are the roots of the equation.
™ a P?r3 AS
PttA —	■ A3 +
Atr5
1.2.3	' 1.2.3.4.5
or, which is the same thing, of the equation
A5 - .... =0,
PttA+OA2-
Ptt3
1.2.3
a3+oa4+
Ptt5
1.2.3.4.5
A5—. .=0.. [2]
Equations [1] and [2] must therefore be identical, each
being produced from the same factors
A (A ± 1) (A ±2) (A ±3).
or
A (A2- 1) (A2 — 22) (A2 — 32)
>·
,..[3]
Consequently,
2 {kirf = 0,
4
.·. fnr = 0 ;120
MATHEMATICAL DISSERTATIONS.
so that the only admissible value of k is k = 0 ; therefore
no multiple of the circumference can ever be a whole num-
ber. In fact, the product arising from [3] must evidently
always have the coefficients of the even powers of A zero;
so that we may infer, without reference to the develop-
ment [2], that the coefficient of A2 in [1] must be 0.
SUMMATION OF CERTAIN CLASSES OF INFINITE SERIES.
(88.) Since every fraction of the form
_______________1________________
n{n +p) (w + 2p)... (n + mp)'
.[A]
is equal to — the difference between one of the form
mp
_____________1______________
n{n + p)----[n + (w —	1) p]
and another of the form
______________1______________
{n + p) in 4- 2p).... (w + mp)
[B]
[C]
it follows that any series of fractions, having [A] for its
general term, will be equal to the difference between
two series of equal extent, and of which the general term
of the one is [B], and the general term of the other [C].SUMMATION OP INFINITE SERIES»
121
This simple and obvious principle is sufficient to guide
us to the summation of a great variety of series; all those
indeed of which [A] is the general type of the several
terms, as we have already explained at length elsewhere.*
It is our intention in the present short paper to show
that the principle may be extended to classes of series of
much greater difficulty ; those, namely, wherein the seve-
ral denominators are formed not merely of the simple
factors above, but of the squares of these, or of a combi-
nation of both kinds.
We have just observed, and the truth of the statement
is immediately seen, that
A = ¿(B-C).
mp
consequently, squaring each side of this equality,
A2=^<B-C>‘
1
my
(B’ + c2)-JLbc.
mp
Hence any series of which A2 is the general term will be
equal to —5-5 the sum of two series whose general terms
2
are B2 and C2 respectively minus —¿-2 times a third series of
which the general term is BC; these series being all of
equal extent, and commencing each at the same values of
nandp.
Now it is obvious that the two series whose general
terms are B2 and C2, if extended to infinity, will together
be equal to twice the series whose general term is B2
minus the leading term in the same series ; therefore, sup-
Elements of Algebra, chap. viii.122
MATHEMATICAL DISSERTATIONS.
posing n = 1 and calling the series whose terms are of the
form B2, S', and that whose terms are of the form A2, S
the foregoing expression for A2 will give for S the follow-
ing value, namely,
S = my {2S' " ls( 1 + pf (1+ 2pf... [ 1 + (m — 1 )p]*} ’ ’
2 ...
minus —the infinite senes whose terms are of the form
my
n(n + p)2(n 4- 2 pf.... [n + (m — l)p]2(w 4- wp)’ * *	^ *
But from the original relation which connects together A,
B, and C, it is obvious that this fraction D' is equal to
_LJ___________________l_____________________
mp\n{n +p)2(w + 2p)2....[w + (m — l)p]2
(n 4- pf{n 4- 2pf----[n 4- (m — l)p]2(» 4- rnp))
so that the entire series whose terms are generally ex-
pressed by D', will be equal to — the difference between
two series whose corresponding terms are generally ex-
pressed by the two fractions within the braces.
It will be convenient however to regard the latter of
these series, that is, the subtractive series, as originating at
the term immediately preceding that at which it actually
commences, and to correct the error after performing the
subtraction, by adding to the remainder the leading term
previously introduced. This leading term, n being equal
to 1, is
1
12(1 4-p)2(l 4-2p)2-[1 +(m-l)ri.SUMMATION OF INFINITE SERIES.	123
Now the result of the subtraction spoken of is readily seen
to be
— (m — l)pS',
and consequently, by introducing the above correction,
the true difference between the two series will be ex-
pressed by
- {(« - l)j& - J,(1 +i,)’(l + 2p)s....[l+(ro-l);>]}
It follows therefore that the value of the series S must be
2
equal to the expression marked [D] together with ^
the expression just deduced; that is,
s = {2Si — is (i +pf (!+2/>)s—[i+(»»— i)rf} +
or, which is the same thing,
Am — 2 r.	3«p—2/> + 2
S =---^—S' —------Ztt x
P
my

12( 1 +p)s (1 + )s----[1 + (“—1 > ¿Is
By this general formula any infinite series of the form
S =
l*(l+P)a(1+2/')
+
(l+p)s(1+2p)2(1+3^)'
f-- +&C.J
124
MATHEMATICAL DISSERTATIONS.
may be readily summed, provided only that we previously
know the summation of the series S', whose terms differ
from those of S by the absence of the final factor in the
denominator of each.
Knowing therefore the value of the series
S' =
1
(1+P)i +
1
(1+2pf
+ &c.........[c],
we may, by successive applications of the formula [a],
proceed through all the intermediate summations up to
the sum of [6]. And in those cases in which the value of
[c] is unknown, and its approximate summation required
we may arrive at such approximation by first summing
a few terms of the more rapid series [6] ; and then by
means of the formula [a] gradually descending till we
arrive at the proposed form S' = [c].
If in the formula [«] we assume p = 1, we shall have
S
1 f 2 (2m—1) g/
m2\ m
l2.22..,

And if p = 2
3m—1
l2. 32... .(2m—l)2.2
}-----[II].
To render these formulas available for the summation of
infinite series of the forms
S =
1
l2.22.32... .(1 +m)2
+ 22.32 „42___(2+»i)2 +-----
and
S 12.32.52... .(1 +2ot)2 + 32.52.72___________(3+2m)2 + ‘ ‘ ' ’ !SUMMATION OF INFINITE SERIES.
125
we must previously discover the sums of
J_ 1_ . 1
** J2 “f* .>2	32 ^......
and
1 1 1
S — T8 + 3a + 58 + '"
by some independent method. Analysts have been much
occupied with the class of series to which these two forms
belong. Euler especially has devoted considerable space
and labour to their investigation; and it is to his researches
principally that the enquiring student is recommended for
ample information on this curious and interesting subject *
None of the series belonging to the class referred to, where
the denominators which succeed one another are powers of
the same degree of an arithmetical series of numbers, are
summable in finite numbers, although each sum may be
expressed in a finite form; for it is a remarkable circum-
stance, that all these series are so connected with the cir-
cumference of a circle, that the numerical expression for
this circumference, to radius unity, is invariably involved
in the sum of the series. The sum of the first of the
7T2	7T2
series S' is found to be — ; that of the second — ; the cir-
o	o
cumference to radius 1 being as usual represented by 2?r.
These sums are very easily obtained as follows: Since
the values of A which satisfy the condition
sinA** = 0,
are all given by the series
A = 0, = ± 1, = ± 2, &c.,
Analysin Infinitorum, voi. i.126
MATHEMATICAL DISSERTATIONS.
it follows that the roots of the equation,
7T A —
1.2.3
A3 +
7Tb
1.2.3.4.5
-A5 — &c. = 0,
whose first member is only the development of sin Air, are
all represented by the same numbers. Dividing this deve-
lopment by A — 0, or A, the depressed equation, contain-
ing the roots
±: 1, di 2, ± 3, &c.,
is
1 .2.3
A2 +
7T5
1.2.3.4.5
A4 — &c. = 0.
If we divide this by Aît2", the resulting equation
( 1 Y* 7T2	( 1	«-1	7T4	( 1 .
VA*J 1.2.3 VA2) + 1.2.3.4.5 VAa.
— &c. = 0,
will have its n roots the squares of the reciprocals of the
roots of the preceding equation; that is, provided n be
infinite. These roots are therefore
1	1	1 o
___	__ ____ Sic
p> 22’	32»
And as the coefficient of the second term, taken with
changed sign, is equal to the sum of the roots of every
equation, it follows that
1_ 1_
p + 22
+
~ -f &e.
tt_2
"6*SUMMATION OF INFINITE SERIES.
127
Again, the values of A in the expression
7T
cosA - = 0
2
are, exclusively,
A = ± 1, = ± 3, ±5, &c.
so that these numbers are the roots of the equation
1 _ A2 4-
1 1.2 A +
A**
1.2.3.4
A4 — &c. = 0 ;
consequently, dividing by A2”, the roots of the equation
¿)'-o (¿>
+ &c. = 0 ;
where n is infinite, are as follows, namely,
L L L &c
l2’	32’	52’
therefore equating, as before, the sum of the roots with the
coefficient of the second term, taken with the sign changed,
we have
F + F + F + &c·=
7r
~S
and thus the fundamental series [c], in the cases of p = 1
and p = 2, is known.128
MATHEMATICAL DISSERTATIONS
Referring now to the formula [I], we have, when
_	7T2	11	1
m= ,S=——3 =	+ 22 3, + gT^2 + c.
„ _ TTS 39	1	1	1
fn = 2, S = —------------- = Sa n'â 4“ na"8'~7a 4“ ÏÏ2”7Fc2 4“ &C.
4	16	12.22.32	22.32.42	32.42.52
+ &c.
c__éü;2	_______2_____.	1
m --- 3, O -	-- lîni OÎ ^4*
54	216	12.22.32.42	22.32.42.52
and so on as far as we please.
In like manner from the formula [II], we have, when
. a 7T2 1	1	1 1
_1· s=r6_5=	12.32	32.52 + 52.72
-2.S-3'· Ì-	1	1 H	+
’ 256 9	12.32.52	32.52.72
H- &c.
4- &c.
m = 3, S =
5tt2 43
1
4-Ô*
1
46 0 8	4 0 50	12.32.52.72	32.52.72.92
4- &c.
and so on to any extent.
(89.) By means of the same general relation
A =— (B - C)
wp
a variety of other series may be very expeditiously summed.
Let, for example, the form of A be
_____________1____________
nh(n + p)(n + 2p) . .. (n + mp)SUMMATION OF INFINITE SERIES.
129
then, by the general relation just referred to, we may sub-
stitute for it the expression
-i—
mp i.n*(n
1
mp in* (n + p) (n + 2p) .. . [n + (to — l)p]
1
n*”1 (n + p) (n + 2p)... .(n -j- mp)
}
Now by the same relation we may, for the first term
within the braces, substitute the following, namely,
_i_x____________________i_______________
(to— l)p 1.71* (n + p) (n + 2p)-[n + (to — 2)p]
nr
_________________1______________
1 (n + p)(n + 2p)... [n + (m --
And for the first of these we may make a like substi-
tution ; so that, by proceeding in this manner, we shall at
length arrive at
2p \n\n+p)
n‘
1
1 (»+/>) (»+2p)
}
and finally at
IXI________L_i
p\nh nh~l{n+p)]
whence a rule may be easily deduced for the summation of
an infinite series of the proposed form, provided certain
subordinate series can be summed.
9130
MATHEMATICAL DISSERTATIONS.
As an illustration of this, suppose n = 2, and put for
abridgment
I I * J_______________u &c. = s
n* + (n+pf^ (n + 2pf
__!_ ._________l___+__________l------- + &c.=s,
n(n+p)	(n+p)(rc + 2p)	(w+2p)(w+3p)
_____I_______ j_________Î--------N + &c. =B S2
n(?i + p)(n + 2p)	(n +p)(n + 2p)(n + 3p)
and generally,
________\____________j________\-----—- + &c.=Sm,
n(n +p).... (n + mp)	(n +p)---[n + (m+ 1 ) p J
then we have the following rule for obtaining the sum of
the infinite series
______________1__________
n\n +p) (n + 2p)... (w -f mp)
+
_________________1________________
(nH-p)2(w + 2p)... [» + (m+ l)p]
+ &c.
From 5 take and divide the result by p ; from the quo-
tient take S2 and divide the result by 2p ; from this second
quotient take S3 and divide the result by 3p, and so on till
the divisor becomes mp. This divisor will furnish a
quotient, which will be the sum of the proposed series.
As to the values of the subtractive quantities Sj, S2, S3,
&c. they are at once obtained from the fundamental re-
lation with which we set out. ThusSUMMATION OP INFINITE SERIES.
131
Si
1
S	1	= Si
* 2jtm(n+p) 2(n+p)
s ___________1_____ ^
3	3pn(n+p)(n + 2p)	3 (n+2p)
s __ (w-QS-i
l)p]*
As an example, let it be required to find the sum of the in-
finite series
1 1 1
l*.2.3.4 + 2*.3.4.5 + 3*.4.5
6 + &C>
in which we have
w = 1, p = 1, m = 3.
The values of S2, S3, are in this case
S1== 1,
Consequently, arranging these in a row, and prefixing the
subtractive sign, the operation by the rule will be as fol-
lows, namely,
- 1
1
4
7T2
6
18
ir* 5	ir2 49
12 “ 8	36_2Ï6“Sum
7r2	5
6" ~4
7T2	49
12” 72*132
MATHEMATICAL DISSERTATIONS.
If the series to be summed be
1 1
I2.3.5.7 T 3\5.7.9
+ &c.
in which p = 2, the process will be this, namely,
1		1		1	
■ 2		12		90	
7r2	7T2	1	7T2	1	7T2
8	16“	‘4	64”"	12	384
1	7T2	1	7T2	17	
2	Ï6"	“3	64”	" 180'	
17
= Sum
By imitating the steps of the preceding investigation, it
would be easy to deduce a rule for the summation of the
infinite series
1
n(n +p)---(n -f mpf	{n+p)---[n -f (m +1 ) pf
+ &c.
This rule would differ from the preceding one in the
following particulars, namely, instead of S2, S3, S3, &c.
we should have to employ
Si + -,
Sa +
1
n{n+pf
S3 +
1
8 1 w(w+jpXw + 2jd)2
1
Sm +
»(w+^)···· [» + (m— l)p]*9
and these, instead of being subtracted, as before, from theSUMMATION OF INFINITE SERIES.
133
several quantities placed under them, must themselves be
diminished by those quantities.
It is plain that when n and p are each unity, as in the
first of the foregoing examples, the values which here sup-
ply the place of S2, S2, S3, &c., will be the doubles of these
latter quantities.
As an example let the series
1 1
T72T37P + 2737475* + &c·
be proposed.
In this case
» = 1, p— 1, m= 3,
and 2 Sx = 2, 2 S2 = -, 2 S3 = -;
consequently, the operation is as follows:
1
9
7T2	3	31	7T2
12	4	108 ~ 36_ Sum
0 7T	irs 3	31	t1
“ (f	~6~~2	36 _ 12
By comparing the several steps of this process with those
in the work of the first example, we are led to conclude
that the aggregate of two series like this last and the one
referred to will always be accurately expressed in a finite
fraction, provided the factors in the denominators be even
in number, and that the difference of the two series will
also be a finite fraction if the number of these factors be
odd. In all other cases the sum and difference will in-
volve 7r2. Thus the difference between the two series134
MATHEMATICAL· DISSERTATIONS.
Is. 2.3 + 2s. 3.4 + 3a.4.5 + &C'	12	8’
and
1	1	1	„ ar* 3
1.2.32^2.3.42^ 3.4.52^	12	4
is so that, actually subtracting and dividing by 2, we
have
1 1 1 _ 1
la.2.3a + 2a.3.4a+3a.4.5a + C,— 16'
Therefore this last series is the square of the series
1	1	1 _i
1.2.3 + 2.3.4+ 3.4.5~4*
It thus appears that every infinite series of the form
1 1
la. 2.3 .... (m — 1) wa + 2a. 3.4 .... m (m + l)a + &C’
is accurately summable whenever the number of factors in
each denominator is odd. And it is easy to see that the
same is true of every infinite series of the form
l2.3.5 ... (m — 2) m2 32.5.7 .... m (m + 2)2 ^
But when in either case the number of factors is even, then
the sum will always involve tt2.*
♦ For other methods of summing infinite series, whose forms are the same
as those considered in the former part of this article, the student is referred to
two very ingenious papers in the Appendix to the Ladies’ Diary for 1836 and
183T : the first by Mr, Woolhouse, and the second by Mr. Rutherford of
the Royal Military Academy.DISSERTATION IV,
ON SOME PRACTICAL IMPROVEMENTS IN THE THEOREM
OF STURM, WITH NUMERICAL ILLUSTRATIONS.DISSERTATION IV.
ON THE THEOREM OF STURM.*
(90.) The Theorem of Sturm, a detailed account of
which we have elsewhere given, is remarkable in two re-
spects. First, because of the unerring certainty with
which it accomplishes an object of paramount importance
in algebraical calculation; and which, by the former tenta-
tive methods, was often of very doubtful and difficult
attainment; and secondly, because of the simple and ele-
mentary character of the operations involved in its prac-
tical application, these operations being none other than
those employed in the ordinary method of finding the
greatest common measure of two algebraical expressions.
The two expressions which, in the theorem of Sturm,
are thus operated upon, are the original polynomial and
its first derived function or differential coefficient. Of
these a common measure actually exists only when the
proposed equation has equal roots; so that when, as
* It will be presumed, that the reader of this dissertation will already have
made himself acquainted with the nature and objects of Sturm’s theorem.
This theorem was first made known to the British student in the Theory and
Solution of Equations, by the author of the present work.138
MATHEMATICAL DISSERTATIONS.
usually happens, such is not the case, the operation will
not terminate till we arrive at a constant remainder; that
is, at a remainder purely numerical.
Whenever, therefore, the proposed equation is of an ele-
vated order, the process will not only be long, but in con-
sequence of the magnitude of the numerical coefficients, it
may also become arithmetically laborious. But ^whatever
method be proposed for effecting the objects which
Sturm’s theorem so successfully attains, we may always
expect the labour to increase with these same circum-
stances, namely, the degree of the equation and the mag-
nitude of the coefficients. And indeed there is not an
operation in common arithmetic, however simple in itself,
that might not become painfully laborious from the mag-
nitude of the numbers submitted to it.
(91.) In estimating the value of Sturm’s theorem, in
comparison with other methods, we ought not however to
confine our attention merely to the amount of numerical
labour involved in each: it must be remembered, as pecu-
liarly characteristic of Sturm’s process, that every step of
this labour is effective, and does really advance us, and by
a direct path, towards the end in view. In other methods,
wdiich, except the intolerably laborious one of Lagrange,
are all more or less tentative and conjectural, we must work
in doubt and uncertainty till some decided indication pre-
sents itself, either that we are on the right track, or that
our work has been thus far fruitless, and must be renewed
in another direction ; and a very discouraging circumstance
is, that we can seldom foresee at the outset to how re-
mote a point in our progress this indication may be
deferred.
The practical transformations involved in the methods
of Bud an and Fourier, are, when viewed individually,
comparatively easy, much easier in general than the stepsTHE THEOREM OF STURM.
139
of Sturm’s theorem. But the great drawback to the use-
fulness of these methods is, that the information sought
from them, except, indeed, in particular cases, is not of
that precise and determinate character which is requisite
to enable us to pronounce with confidence upon the
constitution of the equation, which it is their object to
analyze.
By successive applications of the various criteria pro-
posed by these authors, we can no doubt reach, in most
cases, a degree of probability as to the nature of the roots
of an equation, amounting almost to absolute certainty;
yet, after exhausting all the expedients which they recom-
mend, there may still remain unsatisfied scruples in the
mind of the computer, of sufficient influence to deter him
from drawing a positive conclusion without either trying
some independent means of verification, or else entering
upon a supplementary calculation, involving almost the
very same amount of numerical labour as that which, by
the method of Sturm, suffices for the complete solution of
the problem.
The only labour worth mentioning which Sturm’s
theorem requires is that implied in finding the greatest
common measure of two algebraical expressions, the one,
as already stated, being the polynomial which forms the
significant side of the proposed equation, and the other
being its first derived function; and this is the very pro-
cess which, after having executed the operations peculiar
to Fourier’s method, it is sometimes necessary to go
through before we can safely interpret the results to which
those operations may conduct us. Fourier, fully aware
of the imperfection or inadequacy of his method, does
himself recommend this supplementary process to be
applied in all those doubtful cases where, within a given
interval, the roots may be either very nearly equal or else
imaginary; and he uniformly regards this determination of140
MATHEMATICAL DISSERTATIONS.
the common measure as essential to the completion of his
own method, and accordingly incorporates it in his rule
for separating the roots of an equation. (See Analyse des
Equations, p. 124.)
Innumerable equations may no doubt be contrived, so
that the rules of Bud an and Fourier will be found amply
sufficient to make known their constitution without resort-
ing to the process for the common measure. But no one
can affirm, a priori, that, propose whatever equation you
may, the directions given by these authors are sufficient to
lead us unfailingly to the knowledge of the number and
nature of its roots. This arises from the circumstance,
that the marks or indications, which are shown always to
accompany certain relations among the roots, may present
themselves without being thus accompanied: if the re-
lations really exist, there,unquestionably, will be the marks;
but these latter may occur in the absence of such relations:
the theorems offered for our guidance are not convertible;
and they thus have the same incompleteness, and leave on
the mind the same indecision, that is known to characterize
all the older rules for discovering the composition of an
equation.
The method proposed by Sturm is free from all these
perplexities. It completely and satisfactorily solves the
great algebraical problem of determining the number and
nature of the real roots of a numerical equation, and all
that is necessary to its perfection is, that some means be
contrived for conducting the numerical application of it in
such a manner as to render the arithmetical operations the
fewest possible.
The complete attainment of this object we cannot but
regard as a matter of the first importance in the present
state of algebra. It appears to us that every future im-
provement in the solution of numerical equations must
originate in some practical reduction of the arithmeticalTHE THEOREM OF STURM·
141
work implied in Sturm’s process; and that no simpler
principle, of equal generality, for the analysis of a nume-
rical equation, is ever likely to be discovered.
We ground this prediction upon the indisputable fact,
that the principle of Sturm is virtually that which must
always either precede as preparatory, or follow as supple-
mentary, every process for analyzing an equation having
roots either accurately equal, or nearly equal to one
another.
In cases such as these—and, it must be remembered,
they are the cases comprehending all the difficulties of the
subject—the operation for the common measure is known
to be indispensable; and this, as already remarked, is the
very operation which constitutes the whole of Sturm’s
method.
It will be our main purpose in the present dissertation
so to conduct the steps of this operation as to reduce the
work of their successive derivation to the smallest amount,
and then to contrive some safe means of restricting the
numerical remainders thus derived within manageable
bounds.
In the early stages of the process we shall never have
occasion for any such abbreviating contrivances; but as
the operation draws to a close, the numerical results of the
final steps are generally found to increase, in the higher
equations, at a very rapid rate; so that without the aid of
some controlling principle, the practicability of the method
would soon cease, merely on account of the magnitude of
the numbers submitted to it.
Nevertheless, by a judicious arrangement of the ele-
ments of the calculation, much saving both of space and
labour may be effected, without the aid of any abbreviating
expedient whatever; and thus the method may be shown
to be readily applicable to equations that have been af-
firmed to lie barely within its reach; we mean equations
of the fourth and fifth degrees.142	MATHEMATICAL DISSERTATIONS.
To devise such an arrangement of the calculation will
be our first and more immediate object; we shall after-
wards show how to check the advance of the numerical
coefficients in the higher equations* and shall exhibit the
efficacy of the method in the analysis of complete equa-
tions of the sixth and seventh degrees.
(92.) The problem that is now to occupy us therefore is
this* viz. to effect* with the least possible expenditure of
numerical labour* the division of a rational polynomial in x,
of the degree w, by another of the degree n — 1* with the
view of discovering the polynomial of the degree w—2*
which forms the remainder* it being understood that we are
at liberty to multiply the dividend by any positive nume-
rical factor that will preclude the entrance of fractions into
this remainder.*
Let the dividend be
axn + bxn~~l + cx*~2 + dxn~3 +...... [A],
and the divisor
a'x"-1 + b’xn~2 + dxn~3 + d'xn~A +.....[B].
The remainder, of degree n — 2* which it is our object to
find* will obviously present itself after two terms of the
quotient have been obtained. The first of these two terms*
when incorporated with the divisor* destroys the first term
of the dividend* and furnishes a remainder of the degree
n — 1* the leading term in which is in like manner de-
stroyed by means of the second term of the quotient: the
second remainder thence resulting is that sought. But, as
* The method of conducting the operation when the two proposed poly-
nomials differ in degree by two or more units, will be considered afterwards.THE THEOREM OF STURM.
143
this remainder is to be without fractions, it is requisite
that both terms of the quotient be without fractions. That
the first of these terms may have no fraction, it is obviously
merely necessary to multiply the dividend by a'; but then
the first remainder, adverted to above, having for its lead-
ing term (a!b—aU) xn~\ would necessitate the entrance of
a fraction into the second term of the quotient, namely, the
.	. a'b-ab' .
fraction------f--, with which fraction therefore the next
a
remainder would be affected in all its terms.
It is generally necessary therefore to multiply the
dividend, not merely by a', but by a/2, after which it is easy
to see that the quotient will be a'ax + (db—aV). Multi-
plying then each of these terms by the divisor, and writing
down the several partial products with changed signs,
instead of actually subtracting them from the dividend, the
work under this dividend will arrange itself as follows, the
several remainders being really obtained by algebraical
addition:
a'2. axn + a'2 . bxn~l + a!2 . cxn~2 + a!2 . dxn~3 -}-
—a'a. a'xn — a'a . b’xn~l — a’a . cfxn~2 — ala. d'xn~3 —
0	— (a'b—ab^a'x*-1—{alb -—abf)b!xn~2— {alb—ab^c'x”*3—
0	-f	al'x+ b"xn~3 -f-
From this mode of arranging the work it is easy to dis-
cover what functions the coefficients a", b", &c., in the
sought remainder, are of the coefficients in the original
expressions. But, in order that a", b"9 &c., may be de-
rived from the proposed coefficients in the most expeditious
manner, the work should be conducted conformably to the
following type of the operations:144
MATHEMATICAL· DISSERTATIONS.
	a'-b' -d-d' -	 X, a + 6 + c + d +	
Multipliers.	
A = (a'i — abf)	-Aô'-Ac'-Ad'-	
B*= aa!	— Be/ —Bd!—Be'—» ......
C= an	Ce -j- C d -j- Ce + ····.·
	a" + y + c" +	
The complete remainder, to which these coefficients be-
long, is
allxn~2 -h b"xn~z -f c'fxn~4 -f-.
In Sturm’s theorem, the signs of the terms of each
remainder are to be changed before that remainder is em-
ployed in deducing, as above, the next subsequent one; so
that the two polynomials concerned in the immediately
following step of the process are as follow, namely, for the
dividend,
afxn~l + bfxn~2 -{- c'a:”-3 -J- d'xn~* -f.....;
and for the divisor,
—	— bnxn~* — c"xn~4 — dflxn~5 —..............
Agreeably to the foregoing type of the work involved in
each step, this last polynomial is to have all its signs,
after the first, changed; the coefficients only are to be
written down, those of the preceding polynomial, with
their proper signs, are to be placed under them, and then
the operations in the model are to be performed. Uniting,
therefore, the consecutive steps, the following will be a
general type of the entire calculation :THE THEOREM OF STURM,
145
a'-V- c'-d'-..
a, + b c -f- d
A—a!b—ab'
B= ao!
C= an
—Abf~Ad—Adf—..
—B(/—Be?'—Be' —..
Cc -f- Crf -f" Ce -f-..
— a" —	— c" —.. =ls£m«. wiM signs changed.
- a"+5" + </' + ..
d+V -f </ -f ..	
A = - a"b'+ a'b"	A'6" + AV' + · ·
B' = - aV"	B'c" + BV"+..
C' = a"2	CV +CV + ..
— a'"—	— .. = 2d rent, with signs changed.
-a"f + b'"
- a"-6"
A" —a!nbn—a"bm
B"= aV"
C" =	a'"2
A'V" + ..
B'V" + ..
- C"c" - ..
3d rem. with signs changed.
&c. &c.
And, consequently, Sturm’s series of functions are,
X = axn + bxn~l + cxn~2 + dxn~~3 -j- ex ”~4 + ....
X! = a!xn~l -j- b'xn~2 -f c'xn~3 + dfxn~* + ....
X2 = — a"xn~2 — b!,xn~3 — d'xn~4 — ... .
X3 = - a,frx*~* - b"’xn-* - ....
X4 = - d'"xn~4 - ....
&c. &c.
10146
MATHEMATICAL DISSERTATIONS.
It is scarcely necessary to state that the cross, inter-
posed between the first two terms of the dividend and
divisor at the head of each step of the work as exhibited
above, is inserted merely for the purpose of pointing out
the mode of forming the first of the three multipliers used
in that step. These three multipliers are arranged in the
first column of the work, and the completion of this column
is to be the first object in each step. The formation of the
second and third multipliers is so simple and obvious, that,
with respect to them, any such contrivance as that adopted
in the case of the first multiplier, to assist the memory,
would be superfluous.
(93.) There are two precepts concerning these multi-
pliers, and the results furnished by them, which, if at-
tended to by the computer, will often save him some por-
tion of the numerical labour: there will indeed occur,
occasionally, other opportunities for abridging the work;
these we shall point out in connexion with the arithmetical
process itself, whenever they present themselves in our
practical examples. The two precepts, however, to which
we now refer, indicating the more important of the means
by which a reduction of labour may be effected, deserve
to be distinctly stated and remembered,—they are as
follow:
1. The multipliers will sometimes have a common mea-
sure, discoverable at sight: when this happens the common
factor should be suppressed, and the multipliers, thus re-
duced, should be employed instead.
It is easy to see that, when either the first term of the
dividend, or the second term of the divisor in any step, has
a factor in common with the leading term of the divisor,
such reduction may always be effected.
When the second coefficient of the divisor is zero, then
it is obvious that the leading coefficient of the divisor will
be a factor common to the three multipliers: it is thereforeTHE THEOREM OF STURM.
147
to be expunged from them, or to be suppressed rather in
writing these multipliers down.
These simplifications in the side-column of multipliers
being always effected when practicable, the several re-
mainders will thus become reduced in numerical value; and
will therefore be more easily manageable in the subsequent
steps of the process.
But they will not always be reduced, by these means,
to their lowest possible amount; since the several coeffi-
cients, belonging to any remainder, may, even after the
reductions here recommended, still involve a common fac-
tor ; so that with a view to the utmost degree of simplicity,
it becomes further necessary to examine the final result of
each step; that is,
2. At the close of each step examine whether or not
a common factor exist throughout the terms of the result;
if such a factor be found to enter expunge it; and then
employ the more simple form in the next step.
If the numerical operations be carried on in imitation of
the preceding model, and the simplifications recommended
in these precepts be always introduced, when obviously
practicable, the labour attending the application of Sturm’s
theorem will be reduced to nearly the least amount pos-
sible.
(94.) Still, however, the utmost degree of simplicity
will not even yet have been reached; for the first step in
the process may be completed without the aid of the
model, and with superior facility; so that to conduct the
entire operation with the least expenditure of time and
labour, it will be necessary to defer the use of the guiding
model till after the first step of the work has been com-
pleted by the shorter method here adverted to. But we
think it better to postpone this final improvement, as also
certain other abbreviating contrivances, for consideration
hereafter; and to work for the present entirely by the148
MATHEMATICAL DISSERTATIONS.
model, in order that the student may become previously
familiar with the general law by which each step is uni-
formly derived from that which precedes it. Let us now
proceed to examples.
1. Required the number and situations of the real roots
of the equation
a;4 — 3a;3 — 2a;2 -fa: — 3 = 0.
In this example the first dividend is the polynomial
a;4 — 3a;3 — 2a;2 + x — 3,
and the corresponding divisor is the first derived function
of it, namely,
4a;3 — 9a:2 — 4x + 1.
Hence the work, conformably to the general type, is as
follows:
4 + 9 + 4-1
X
1— 3—2 + 1—3
-3
4
16
-27-12+ 3
16- 4
-32+16-48
43 + 0 +45
43—0 -45
X
4—9 - 4+1
-9
4
43
0 + 405
— 180
-172 +43
352—448
or, + 4x8
11-14.THE THEOREM OF STURM.
149
One more step would bring us to the final remainder;
but* as the sign of this final remainder is all that we require
to know respecting it, the closing step of the operation
may always be dispensed with, and the required sign
ascertained more easily in another manner; as we shall
presently show.
(95.) The value of the two precepts previously laid
down is clearly seen in the second of the above steps.
By the first precept the numbers forming the column of
multipliers are considerably reduced by the suppression of
the common factor 43; and by the second precept the
result of that step is still further reduced by expunging
the common factor 32.
Omitting the constant remainder, or the final function,
Sturm’s functions are therefore as follow :
X =z x* — 3x3 — 2a2 + a: — 3
X, = 4r* - 9a2 - 4x + 1
X2 = 43a*2 + 45
X3 = 11a- 14.
Now, although we have deduced the function X3, for the
purpose of illustrating both the general method of deri-
vation, and the precepts for simplification, yet there is no
occasion for this function in determining the nature of the
roots of the proposed equation. The first step really com-
pletes the operation; for it is seen in a moment, that the
roots of the equation 43a2 -f 45 = 0, furnished by this
step, are imaginary; and Sturm proves that as soon as
we arrive at a function of this kind the subsequent steps150
MATHEMATICAL DISSERTATIONS.
of the work are superfluous. (See Theory of Equations,
p. 146, also Note ii. at the end of this Dissertation.)
Hence, employing only the functions
X X, x2
we have, for x = — oc , -f — +
..........x = + ao , -j- -f- -f
therefore, as two variations only are lost between — oo and
+ oo, we infer that there are but two real roots.
To ascertain the situations of these, substitute 0,1,2,3,
&c., for x; and then 0, — 1, —2, —3, &c., for it is plain
from the sign of the last term in x that one of these real
roots will be positive and the other negative. We need
not make these substitutions in all three of the functions,
but merely in the first X; since a change of sign in this is
all that can mark the passage of a single root. The posi-
tive root is thus readily found to lie between 3 and 4;
and the negative root between — 1 and — 2.
In the present example the functions Xs and X4 happen
to be unnecessary; the final one X4, or at least the sign
of it, which is all we can ever require to know about the
final function, might have been easily ascertained from con-
sidering that the remainder, arising from the division of a
polynomial by an expression of the form x ± a, is obtained
by the mere substitution of+ a for x in that polynomial,
(Theory of Equations, Art. 10;) so that in the example
14
before us the remainder is found by putting — for x in
43x2 + 45; and we see at a glance that this remainder
must be plus; hence, X4 = —.	And in this manner
should the sign of the final remainder always be deter-
mined.
This method of obtaining the sign of the final remainder
is also suggested from the known property of Sturm’s
functions, that when any one vanishes, for a particularTHE THEOREM OP STURM.
151
value of x, the immediately preceding and following func-
tions take opposite signs for that same value of x. The
function immediately preceding the last vanishes for
x = qp a; hence the sign of the last function must be
opposite to that which the last but two takes for x = + a.
2. Required the number and situations of the real roots
of the equation
*4 — 2a:3 - 7*2 + 10* + 10 = 0.
Here X! = 2x3 - 3*2 - lx + 5.
2 + 3 +7 -5
X
1—2 -7 + 10 + 10
— 1
2
4
—3 —7 +5
14—10
—28 + 40+40
17-23-45
17 + 23+45
X
2—3—7 + 5
-5
34
289
-115 —225
1530
—2023+1445
608-1220
or, -+ 4
152 —305.
Consequently
X = ** - 2x* - 7x2 + 10* + 10
X, = 2x3 - 3x2 - lx + 5
X2 = 17x2 — 23* — 45
X3 = 152* - 305
x<= +152
MATHEMATICAL DISSERTATIONS.
As the leading terms of these functions are all plus, the
roots of the equation are all real ; their situations have
already been determined in the Theory of Equations,
p. 150.
3. Required the number and situations of the real roots
of the equation
a;4 + z3 - 15a;2 - 19a;- 3 = 0.
The leading functions are
xA + a;3 — 15a;2 — 19a: — 3,
and
4a;3 + 3a:2 - 30a; - 19,
and the work is therefore as follows :
4—3 4-30 + 19
X
1 + 1 -15-19-3
1	—3 +30 + 19
4	120+76
16 —240—304—48
123 + 198+29
123-198-29
X .
4+3 -30-19
-141
164
412x3
27918	+4089
-4756
-151290 -95817
128128 +91728THE THEOREM OP STURM
153
Therefore the functions are
X	= z4 +·*3 - 15x2 — 19a; — 3
XI	= 4a;3 + 3a;2 — 30* — 19
X2 = 123*2 4- 198# + 29
X3 = 128128a: + 91728
.·. X4 = 4-
As the leading signs of these functions are all plus, we
infer that the roots are all real; moreover, as for x = 0
there will be one variation, shown by the signs of the final
terms, there is only one positive root. This is found from
the single function X, without the aid of the others, to lie
between 3 and 4. To find the situations of the negative
roots, it will be convenient to convert them into positive;
that is, to change the signs of the alternate terms in these
functions, writing them thus:
X = X* - *3 _ 15a,2 + l9x _ 3
X, = 4*3 - 3#2 - 30* 4- 19
X2 = 123 a;2- 198* 4-29
X3 = 128128a; -91728
X4= +
X Xj X2 X3 X4
for x = 0 .... —	4-	4r	—	4-	three variations
= 1 .... 4-	—	—	4*	4-	two
= 2 .... —	—	4-	4*	4-	one variation
== 3 .··—	4*	4~	4"	4"	one
= 4....4-4-4-4-4-«o variation.154	MATHEMATICAL DISSERTATIONS#
Hence the negative roots of the proposed equation are
situated between 0 and — 1; — 1 and — 2; and — 3
and — 4.
4. Required the number and situations of the real roots
of the equation
x5 + 5r* — 10x2 + 5x — 1 = 0.
The function Xx, when disencumbered of the common
factor 5y is in this case
x4 4- O#3 + 3x2 — 4ar + l.
1 +0— 3+ 4—1
X
1+0+ 5-10 + 5—1
0
J
1
-3+ 4 -1
5—10 +5-1
—2 +6 -4+1
—2—6	+4-1
X
1+0	+3-4 + 1
-3
-1
2
18-12 + 3
-4 +1
6 —8 + 2
—20 + 19—5
-20-19+5
X
—2 + 6—4+1
-41	779—205
20	100
200	-800+200
- 79 + 5THE THEOREM OF STURM·
155
Hence the functions are
X = x8 4- 5xs — \ Ox* 4. 5x — 1
Xi =s x4 -f 3x* — 4a? -f 1
X2 = — 2xs -j- 6x2 — 4x 4- 1
/
X3 = - 20x2 4* 19x -5
X4 = - 79x 4- 5
·*. Xa — +
(96.) In this example the last two functions have been
deduced merely for the sake of further illustrating the
mode of derivation. They are not necessary to the solution
of the problem; for since 4(5 x 20) > 19s the roots of
X3 = 0 are imaginary.
Employing therefore only the functions
X Xi x* x5
we have, for x = — oo , — 4- 4- — two variations
...........x = 4- 00 , 4- 4- — — one.
Hence there is but one real root 5 and, since the last term
of X is negative, this root must be positive. A superior
limit to this positive root will be found by dividing the
greatest negative coefficient, 10, by the greatest positive
one, 5, which precedes it, and then adding 1 to the quo-
tient. (Theory of Equations, Proposition IV. Chap. III.)
This superior limit will therefore be 3; so that we need
try only the numbers from 0 to 3, in the single function
X. We thus find that x = 1 is the real root.
5. Required the number and situations of the real roots
of the equation156	MATHEMATICAL DISSERTATIONS.
a* + 3a·4 + 2a;3 — 3a;2 — 2a; — 2=0.
The function Xt is
5a4 4- 12a;3 4- 6x2 — 6a; — 2
5—12 —6 + 64-2
X
1+3	+2 —3 —2—2 ;
1 -	■	36-18 + 18+ 6 ■	30 + 30 + 10 50-75-50-50
	16 + 63+22+44
	16-63-22-44
	X
	5 + 12 +6 —6—2
123	7749+2706 + 5412
80	-1760-3520
256	1536-1536 —512
—7525+2350—4900
or, -~52
-301+94—196.
The operation terminates here, as in the preceding exam-
ple, because 4 (301 x 196) > 942. The functions, as far as
necessary, are therefore
X = x5 + 3a;4 + 2a;3 — 3a;2 — 2a; —2
Xj = 5x4 + I2ar* + 6a;2 — 6a; — 2
X2 = 16a;3 + 63a;2 + 22a; + 44
X3 = -301a;2 + 94a;-196
These give, for a; = — oo,-----1-----two variations
a; = + oo, + + 4-------one variation.THE THEOREM OF STURM.
157
Hence there is but one real root, which must be positive,
because the last term of X is negative. Moreover, a su-
perior limit to this root, determined as in last example,
is 2 ; so that no number beyond 2 need be substituted in
X. Putting x == 0 in X, the result is minus ; putting
x = 1, the result is still minus ; we infer therefore that the
root is between 1 and 2.
(97.) If we had had occasion to extend our computation
up to X4 in this example, the preparation for the additional
step would have been
—301—94+ 196
X
16 + 63+22+44
—20467
-4816
3012
But as 94, 196, 22, and 44 have the common factor 2, we
should have suppressed that factor ; and, preserving the
side column of multipliers as it is, should have employed
the two horizontal rows, as if they had been written thus,
— 301 - 47 + 98
16 + 63+11+22
and then have proceeded as usual.
Whenever at the commencement of any step the num-
bers in the first horizontal row, beginning with the second
of those numbers, each involve a factor common also to
the numbers in the second row, beginning with the third,
as in the instance before us, the common factor may always
be neglected, and thus the subsequent \Vork shortened.
We shall give an example in which this simplification
may be introduced.158
MATHEMATICAL DISSERTATIONS.
6. Required the number and situations of the real roots
of the equation
x4 - 14a:3 4- 66a:2 - 123* 4- 90 = 0.
In this example the reduction referred to above may be
applied to the first step of the work, the numbers being
divisible by 3. We shall therefore commence the step as
usual; and place the results of this division over the
original numbers as follows :
14	44	41
44-42—1324-123
X	22	41 30
	1 —14 4-66—1234-90
—7	—98 4-308—287
2	—88 4-82
8	176^3284-240
	10 —62+47
104-62 —47
	4—424-132—123
-43	-26664-2021
10	- 470
52	3300-3075
	- 1644-1054
	or, 4-2
	— 82 4-527
Consequently the functions are
X = a:4 — Ux* 4- 66a:2 — 123a: 4- 90
Xj = 4a:3 - 42a:3 4- 132a: - 123
X2 = 10a;2 — 62a; 4- 47
X3 = — 82 a: 4-527
X4 = -THE THEOREM OF STURM.
159
As there is one variation in the signs of the leading
terms we infer that the equation has a pair of imaginary
roots. Also since there is one variation for x = oc, and
three for x = 0, we conclude that the two real roots are
both positive. By substituting 0, 1,2..........for x we find
that X vanishes for x = 5, and for x = 6; these therefore
are the two real roots of the equation.
(98.) The following equation is one of some celebrity.
It is that to which Dr. Wallis was led in his solution of
“ Colonel Tituses Problem ” One of the roots of this
equation is determined by Mr. Horner’s method, as far as
fifteen places of decimals, in an historical dissertation on
the problem of Involution, published in the Companion
to the Almanack for 1839;* and the same root is com-
puted to seventeen places of decimals, by the Newtonian
• Written, we suspect, by Professor De Morgan, who, in reference to the
equation mentioned, says,“ This is the equation resulting from the celebrated
problem proposed by Dr. Pell to Colonel Silas Titus, and by him to Dr.
Wallis, who seems to have considered it as a sort of challenge from Pell,
and has discussed it in a most masterly manner. A great deal may be found
on the subject of this very equation in the Scriptores Logarithmici, and other
publications, of Baron Maseres.’7
To this we may add that the problem itself is to find x, y, and s, from the
three equations,
ra -j- t/s = 16, y*-|" x* = IT, ia -J- ary = 18.
It has been solved by many algebraists ; as Wallis, Halley, Ivory, and
F rend ; but one of the easiest solutions that the author of this work has met
with was given in 1825, by Mr. John Whitley, of Bradford, a geometer of
great ability, in a periodical work published at Liverpool, under the title
of The Apollonius, and conducted by another excellent geometrician, Mr.
J. H. Swale.
Professor Davies too has discussed this celebrated problem at length,
in his recently published Key to Hutton's Course of Mathematics, a work
replete with curious and useful information, and forming a very valuable ac-
quisition to the library of the mathematical student.160
MATHEMATICAL DISSERTATIONS.
method of approximation, by Mr. Lockhart.* The
equation is this :
7.	a;4 — 80a;3 + 1998a;2 - 14937a; + 5000 = 0,
and the numerical operation for the discovery of the num-
ber and situations of its roots is as follows :
4+240 -3996 +14937
X
1—80	+1998 -14937 + 5000
—20 -4800+79920-298740
1 -3996+14937
4	7992-59748 +20000
804-35109+278740
804 + 35109-278740
X
4-240 +3996 -14937
-4377
268
804 x 67
-153672093 +1220044980
—74702320
215256528 -804626316
13117885 —415418664.
Consequently the functions are
X = a;4 — 80a;3 + 1998a;2-14937a; + 5000
X1 = 4a:3 - 240a:2 + 3996a; - 14937
X2 = 804a:2 — 35109a: + 278740
X3 = 13117885a; - 415418664
X4 = +
* We cannot refer to this venerable mathematician without expressing our
admiration of the ardour and perseverance that, at the age of eighty years,
supports him through the long and laborious calculations in which he is en-
gaged. Our readers will be both gratified and instructed by the perusal of the
following works recently published by that gentleman:
1.	“ Resolution of Equations” quarto, Oxford, 1837.
2.	“ Resolution of two Equations” <fec. 1839.
3.	“ Extension of the Celebrated Theorem of C. Sturm,” 1839.THE THEOREM OF STURM.	161
Hence the roots are all real ; they are moreover all posi-
tive, since for x = 0 there are four variations. The situa-
tions of these roots are found thus :
X X! x2 x3 x4
for x = 0...........+	—	+	—	+	four variations
= 1.........— — + — three
= 10........-	+	+	-	+
= 20........+ + — — + two
= 30......... +-----------+
= 40..........+	-l·	+	+	+	no variation.
We conclude therefore that there is one root between
0 and 1, one between 10 and 20, and two between 30 and
40; that is, the leading integers of the roots are zero, ten,
and thirty, the latter number belonging to a pair of roots.
(99.) The w ork of the foregoing example is well adapted
to show the amount of numerical labour usually involved
in the application of Sturm’s theorem to a complete equa-
tion of the fourth degree, whose coefficients are rather
high numbers. Means will be furnished hereafter for
greatly reducing this labour, which is, to a considerable
extent, superfluous. But, as before stated, w^e think it de-
sirable to postpone all abbreviating contrivances for the
present, in order that ample illustration of the general
mode of derivation may first be exhibited to the student.
8. Required the number and situations of the real roots
of the equation
x6 — 36r* + 72a:2 - 37x + 72 = 0
The leading polynomials here are
x6 + O*4 - 36x3 + 72*2 - 37x + 72,
and
5x* + 0xs — 108a?2 + 144* — 37,
11162
MATHEMATICAL DISSERTATIONS.
and the work is as follows:
5 + 0+108-144 +37
y;
1+0 -36 +7*2	-37+72
0
1	108-144 +37
5—180 + 360—185 + 360
72-216 + 148-360
or, -+4
18-54 +37-90
18+54	—37	+90
X
5 + 0	—108 +144—37
15
5
18
810 —555 + 1350
— 185 +450
-1944 + 2592 —666
1319—2487 —684
1319+2487 +684
X
18—54	+37—90
- 490x54*
13I9x 18
13192
—65806020 —18098640
16239528
64371157-156578490
— 14804665 + 174677130
or, +-5
— 2960933 +34935426
* As the number 2487 is divisible by 3, and 3 times 18 make 54, the leading
multiplier in the side column may be considered as formed from
1319+829
x
54-54
that is, from
(829 —1319)54 = —490 X 54.THE THEOREM OF STURM.	163
consequently, Sturm’s functions are
X = a:5 - 36a:3 + 72a:2 — 37a: + 72
X1= 5a:4—108a:2 + 144a:—37
X2= 18a:3— 54a:2 + 37a: —90
X3 = 1319a:2 - 2487a: - 684
X4 = — 2960933a: + 34935426
X5=-
Substituting — x> and + oo successively for x in these
functions, we have the series
x = — oo,.... — + — + -f — ....four variations
a: = + oo , .... + -f + + — — .... one variation.
Hence the equation has only three real roots.
To determine the situations of these roots, we have
for x = 0 .... -f	—	—	—	-f	—
= l....-j-	+	—	—	+	—
= 2 .... -h	—	—	—	-f	—
= 3.... —	—	-+■	+	+	—
= 4 ... .-+ + + + —
= 5 .... -f
As there are two more variations for ;r=0 than for x = -f oo164
MATHEMATICAL DISSERTATIONS.
we infer that there are two positive roots; the third root
therefore must be negative, because the sign of the last
term of X is positive. The variations show that these
positive roots are the one between 2 and 3, and the other
between 4 and 5. By substituting the negative numbers
— 1, —■ 2, — 3, &c. in the single function X, or the posi-
tive numbers 1 , 2, 3, &c. after changing the alternate signs
in that function, we find that the remaining root lies be-
tween — 6 and — 7.
In order to diminish the number of substitutions, and to
discover at once the immediately superior limit to the
sought root, which, when taken positively, is the greatest
root of the equation
x5 — 36x3 — 7 2 xa — 37x — 72 = 0,
we have only to seek the smallest value of x, that will
satisfy the inequality
x5 > 36x3 + 72xa + 37x + 72,
or
1 >
36	72	72
X2 X3 X4 x6 ’
This value we immediately see must be 7,* so that the
greatest root must lie between 6 and 7, and therefore the
root of the original equation between — 6 and — 7.
The foregoing example is given by BouRDON,fwho has
* However, 6 is more immediately seen to be a close inferior limit, so that
we might commence our substitutions with 6.
f Algebre, [>. 582, 1837.THE THEOREM OP STURM,
165
deduced incorrect values for X3 and X0 and has moreover
assigned a wrong place to one of the positive roots·
(100.) We shall now show the application of the theorem
to literal equations of the fourth and fifth degrees.
9. Required Sturm’s functions for the equation
x1 + ax2 + bx + c = 0
4 + 0—2a—6
X
l+0+a+6+c
0
1 -2 a-6
4	4a + 46+4c
—2a—35—4c
—2 a + 36+4c
X
4 + 0 + 2 a + 6
36
-2a
962—126c
—8ac
2a3 + a26
A + B
A-B
x
—2a—36—4c
2aB—36A
—2aB2+36AB
A2
—4c A2
4c A2— 36AB+2aB2.*
* This final result, as before remarked, is the same as that which would
g
arise from putting — ^-for x in the result preceding that whence this ex-
pression for x is deduced; that is, in — 2ax2 — ‘Sbx — 4c, and then changing
the signs.166
MATHEMATICAL DISSERTATIONS.
Hence the functions are
X = x4 -f ax3 + 6x + c
= 4x3 + 2ax -j- b
X,= - 2ax3 - Ux - 4c
X3 = (Sac — 2a3 — 9A2) x — b (a2 + 12c)
= Ax + B
X4 = 4cA* — 36AB + 2aB*.
If X4 = 0, the equation will have a pair of equal roots,
namely, the roots of the equation
(Ax -f B)* = 0,
B
each root being =-----t-.
The equation containing the other two roots is therefore
x „	2 B A2
(Â7ÏÏÏ?=0’ or^-2ro: + -c = 0.
If X3 = 0, then there will be two pair of equal roots in the
proposed, and the equationTHE THEOREM OF STURM.
167
2ax2 4- 3bx 4- 4c = 0
will furnish one of each pair, unless it should so happen
that this latter equation has itself equal roots ; that is, un-
less 962 = 32ac, in which case the proposed equation will
have three equal roots, of which the equation X2 = 0 con-
tains two. These two being together equal to the coeffi-
cient of x in X2, when the sign of that coefficient is changed,
and the division by 2a, the coefficient x2 performed, it fol-
lows that the expression for each of the three equal roots
3 b
is x = — — ; and as in the proposed equation the sum of
all the roots, with changed signs, is 0, it follows that the
fourth root is —.
(101.) Given the equation
10.	x5 + ax3 + bx2 + cx + d = 0
to determine Sturm’s functions.
The two leading polynomials are in this case
xb + Ox4 4- ax3 + bx2 4- cx 4- d,
and
5x4 4- Ox3 4- 3ax2 4- 2bx 4- c,
and the work of derivation is as follows:168
MATHEMATICAL DISSERTATIONS.
5 + 0—3a—26—c
X
1 —|— 0 —|— ct —|— ^ c -f- c?
0
1 —3a—2 b—c
5	5a-\-5b + 5c+5d
— 2a—3Ô—4c—5d
—2a 36 -j- 4c 5d
x
5 -J- 0 -J- 3a -f" 25 -f· c
15b
-10a
4a
4562 + 605c+756rf
—40ac—50ad
12a3+8a25 + 4a2c
A + B + C
A-B-C
X
—2a—35—4c—5d
2aB-35A
—2a A
A
—	(2aB—35A)B-(2aB—35A)C
2aAC
—	4c A2—5dA2
D + E
D-E
X
A + B + C
BD-AE
D2
-(BD-AE) E
CD2
F.THE THEOREM OP STURM.
169
Hence Sturm^s functions are as follow:
X = x5 + axz + bx2 -f cx + d
= 5#4 -f 3 ax2 + 2&r + c
X2 = — 2 ax3 — 3 bx2 — 4 cx — 5d
X3 = Ax2 + Bar + C
X4 = J)x -f E
Xs=F;
where A, B, C, D, E, F, are the following functions of the
coefficients, namely,
A = 40ac — 12a3 — 45b2
B = 50 ad — 8a26 — 605c
C = — 4a2c — 15bd
D = 4cA2 + (2aB - 35A) B - 2aAC
E = 5dA2 + (2aB - 35A) C
F = (BD - AE) E - CD2*.
• If by means of these forms we can discover, without actual calculation,
the signs of A and D, and are able also, from an estimate of the values of the
preceding coefficients, to predict the sign of F, we shall obviously be in pos-
session of information sufficient to enable us to find at once the number of
real roots in the equation. The character of these roots, as to positive and
negative, will also be ascertained from a comparison of the signs of C, E, and F,
in connexion with the sign of d. But for a very complete and detailed exa-
mination of all these circumstances, the student is referred to an interesting
paper by Professor Gill, of New York, published in the eighth number of
The Mathematical Miscellany, an American periodical very ably conducted,
and well worthy of attention from students of the modern analysis. The work
is published in London by Groombridge, Paternoster Row, at intervals of six
months.
An abridgment of Mr. Gill's paper, here adverted to, may be seen in the
Ladies’ Diary for 1839.
The examination referred to will place in a striking light the superiority of170
MATHEMATICAL DISSERTATIONS.
(102.) In applying these general formulas to any parti-
cular example, there will be no occasion to execute the
actual computation of F, as we have already remarked,
since the sign of F, which is all we require to know about
it, is easily inferrible from X4 and X3. It is easy to see,
however, that there will be but little saving of actual labour
effected by employing these forms in preference to going
through the process which leads to them, at least as re-
spects the more complicated functions X3 and X4. The
functions preceding these, may indeed sometimes be ob-
tained more readily by the formulas, or rather by these
somewhat modified. Thus, for the purpose of computa-
tion, the following will usually be found to be the more
convenient method of writing the expressions for A, B,
and C, each of the former expressions for these quantities
being divided by a2.
A = 40 -	—	12a	— 45 -	.	-
a	a	a
B = 50-	-8b	-60-	.	-
a	a	a
c =	_ 4c-75- . -.
a a
Sturm’s method over that of Lagrange, who was fully aware however of the
labour involved in his own process ; and accordingly remarks with great can-
dour, after discussing the general equation of thefourth degree, ** On pourrait
de même trouver les conditions qui rendent les racines des équations du
cinquième degré toutes réelles, ou en partie réelles et en partie imaginaires ;
5.4
mais comme, dans ce cas, l’équation des différences monterait au degré -~-= 10,
le calcul deviendrait extrêmement prolixe et embarrassant.” Traité de la
Résolution des Equations Numériques, p. 43,1826.
We need scarcely add to this quotation, that, notwithstanding the recent
improvements in this subject, the profound and elegant work of Lagrange
will richly repay the analytical student for a careful perusal.—See Note at
the end.THE THEOREM OF STURM.
171
(103.) As an application of these forms, let us take the
equation of the fifth degree before given, namely,
a:5 - 36a:8 + 72a;3 - 37a: + 72 = 0.
Then, from the general expression for X2, we have im-
mediately
X2 = 72a;3 - 216a;3 + 148a; - 360
18a;3 — 54a:3 + 37a: — 90.
Also, since
d
a
we have, from the general expressions above,
A =	41^ + 432-J 80 =	293£
B = — 100 — 576 + 123£ = — 552f
C =
148 - 300 = - 152
Consequently,
x3 = 293^ a:2 - 552£ a: - 152 ;
or, without fractions,
X3= 1319a;3 - 2487a; - 684.
A glance at the general expressions for D and E will suf-
fice to showT, that the very same labour is involved in their172
MATHEMATICAL DISSERTATIONS.
computation, as is necessary to complete the step for X4 by
the regular process: we should therefore recommend X4 to
be derived in this latter way.
(104). We should find, by proceeding in a similar man-
ner with a general equation of the sixth degree, wanting
its second term; that is, with the equation
11. x6 -f- ax4 -}- bx3 + cx2 + dx -h c = 0
that
X2 = — 2 ax* — 3 bx* — 4 cx2 — 5dx — 6e,
and that the coefficients A, B, C, D, of the next function
X3 = Ax3 -j- Bar2 + Cx + D
are as follow:
A = 24ac — 8a3 - 2762
B = 30 ad — 6a2b — 366c
C = 3 6ae — 4a2c — 456c?
D = — 2 a2d — 546c
or somewhat more commodiously,
	c	r 6	h
A = 4.6	- — 8a —	3.9-	. -
	a	a	a
	d	_ 6	c
B = 5.6	- -66-	4.9-	
	a	a	a
	e	. 6	d
C = 6.6	- — 4c —	o.9 -	
	a	a	a
D =	-2d —	6.9- . a	e aTHE THEOREM OF STURM.
173
And we might, with the same ease, obtain the functions
to the same extent for any higher equation. There would
indeed be but little difficulty in extending the series fur-
ther, so as to complete the set. But the algebraical ex-
pressions for the coefficients in the subsequent functions
would offer no facilities for the actual computation of them,
beyond those of which we might avail ourselves in the
derivation of them by the regular process ; so that it would
be useless to exhibit them in their algebraic form.
(105.) It must be acknowledged, however, that it is
always in the derivation of these subsequent functions that
the irksomeness of the numerical labour is most felt,
because of the rapid rate at which the coefficients generally
increase as we approach towards the final step of the
operation.
It is true that these large numbers may be sometimes
reduced by the suppression of common factors ; but it is
not always easy to discover these, even when they exist,
except indeed they are of such small amount as to render
the division by them, in the case of these very large num-
bers, of little or no importance.
It is plain, therefore, that any considerable reduction of
the work here, provided it could always be made without
shaking our confidence in the numerical results, would
constitute a capital improvement in Sturm’s Theorem.
But if the numerical coefficients which enter these
advanced functions are always required to be exhibited in
full, with the cumbersome redundancy of figures which
they so frequently present, then, we confess, that we have
but little expectation of any reduction worth notice ever
being made in the work by which these large numbers are
obtained. The case would be different if the labour com-
plained of consisted in the intricacy or complication of the
means by which the several results are arrived at : we
might then reasonably hope that, as in other practical en-174
MATHEMATICAL DISSERTATIONS.
quines, time and application would eventually bring to
light simpler and shorter methods of attaining the same
end. But here there is no complication of means; each
step of the work may almost be said to involve but the
fewest possible operations, and these of the simplest pos-
sible character. The difficulty felt in practice is attri-
butable solely to the magnitude of the numbers with which
we have to deal; and we cannot conceive of any reduction
of this difficulty being practicable, that has not for its main
object the contracting and confining within moderate
bounds these overgrown coefficients.
(106.) We are disposed to think that the general mode
of conducting the numerical work of Sturm’s theorem, as
exhibited in the preceding pages of this dissertation, re-
duces the operation to the utmost degree of practical con-
venience. To give it perfection there only remains to be
united with it, in its closing steps, some abbreviating con-
trivance, of easy application, that shall at once diminish
the actual labour, and restrain the resulting numbers within
moderate limits.
As to any means being hereafter discovered for lessening
the labour of the common multiplication of large numbers,
thus increasing the facilities at present furnished by our
tables of logarithms and of squares and products, we have,
as already remarked, little or no hope.* We think such a
reduction of labour to be altogether impracticable, by any
purely numerical or algebraical contrivance whatever. It
may possibly be accomplished by instrumental or mecha-
nical methods; and we should suppose that the several
steps of Sturm’s theorem, in their unabbreviated form,
however large the numbers involved in them, might be
* Barlow’s Tables will be found of much assistance in the practical
operations of Sturm’s Theorem. The work has been recently reprinted by
the Society for the Diffusion of Useful Knowledge.THE THEOREM OF STURM.
175
worked out, with comparative ease, by the aid of the
admirable calculating engine of Mr. Babbage.
But even admitting the necessity of undertaking this
labour by the common arithmetical process, which neces-
sity, however, as we shall shortly show, does not exist, yet
the amount of work in Sturm’s theorem has, we think,
been somewhat overrated in the comparisons that have
been made between it and other modes of operation.
When the labour is made the ground of rejection of this
method in favour of any other at present known, it is
surely overlooked, that be this labour whatever it may, yet
it is indisputable that Sturm’s is not only the simplest,
but is also the shortest unfailing method that has ever been
proposed. All the work which Sturm’s method requires
is essentially comprehended in the method of Lagrange,
in conjunction with the additional labour peculiar to his
process. The same is true of that of Fourier or Budan.
And no infallible rule for finding the number and situations
of the real roots of a numerical equation, chosen at random,
can be adduced, in the present state of analysis, that shall
not imply the process for finding the common measure of
the proposed polynomial and its first derived function.*
What entitles Sturm’s method to the precedence of them
all is, that although it also requires the finding of this
common measure, yet it requires nothing else: this is the
single and unique operation that comprises the whole of
his process, and it is certainly an operation of a very ele-
mentary character.
(107.) Thus the labour of Sturm’s method, even when
viewed in the most unfavorable light; that is, when all
• “ Avant d'entreprendre la résolution d'une équation par quelque méthode
que ce soit, il est tou jours nécessaire de s'assurer si elle a des racines égales.”
Lagrange, Résolution des Equations Numériques, p. 115, ed. 1826.
It is almost superfluous to add that the search after equal roots neces-
sarily involves the operation for the common measure adverted to in the text.176
MATHEMATICAL DISSERTATIONS.
abbreviations of it are precluded, is, after all, a labour so
entirely free from ambiguity and intricacy, a labour of so
ordinary a kind, that the merest tyro is competent to per-
form it. Every successive step is characterized by the
same simplicity and undeviating uniformity, each being
the model or pattern of the next in succession, so that the
work can be carried on with little or no mental exertion^
and can be broken off at any step, and resumed again at
pleasure without endangering the accuracy of the operation.
(108.) In certain cases the transformed equations of
Budan will, no doubt, effectually make known the num-
ber and nature of the roots, with, comparatively, a trifling
amount of labour; such will happen when these roots are
all real, and moderately wide apart. It is possible, in some
practical enquiries, that the problem which conducts to
our equation may furnish extraneous information as to the
nature of the roots, or may afford strong grounds for con-
jecturing that the roots can none of them be imaginary,
nor yet nearly equal. Under such circumstances it may be
prudent to apply the tests of Budan. But when, as
usually happens, we are in total ignorance of the consti-
tution of our equation, nothing can effectually supply the
place of Sturm’s infallible method.
Nevertheless, either from prejudice, or from an imperfect
apprehension of the comparative merits of the two rules,
that of Fourier is sometimes represented as superior to
the method of Sturm. It would be easy to show, from
the examples selected by Fourier himself for the purpose
of illustrating his own rule, that in cases of any delicacy
Sturm’s process surpasses his, as well in ease and sim-
plicity, as in certainty. In confirmation of this we shall
take, from Fourier’s book, an equation of this kind, and
one in which every figure employed in the work is actually
exhibited in the several steps, there being no bye ope-
rations whatever. By a comparison of the two pro-THE THEOREM OF STURM.
177
cesses it will be seen that that here given is by far the
simpler.
12. Required the number and situations of the real roots
of the equation
a?4 — 4x3 — 3x + 23 = 0*
4+12 + 0 + 3
1 —4 + 0 — 3 + 23
—	12-0—3
[	0 + 3 0 — 12 + 92
12 +9—89	
	12—9 +89 X 4 —12 —0—:
-15	135-1335
4	356
12	0 -36
-491 + 1371
* Fourier, Analyse des Equations, p. 137.
As this equation, when put in a complete form, is
x4_4a,3 + Oi2 — 3a? + 23 = 0,
it is evident at once, from the criterion of De Gua, that two of its roots are
imaginary, (see Theory of Equations, p. 61.) To remove all doubt as to the
nature of the other roots, Fourier recommends, in addition to his own pecu-
liar method of examination, that we should go through the very operation
which constitutes the whole of Sturm’s process: the operation, namely,
of seeking the common measure of the proposed function and its de-
rivative.
12178
MATHEMATICAL DISSERTATIONS.
Writing now the functions of Sturm we have
X = x4 — 4x3 — 3x + 23
X, = 4x* — \2x2— 3
X2= 12*2 + 9ar —89
X3 = — 491a* + 1371
.\X4=-
from which we see that there are two real and positive
roots, inasmuch as two variations are lost between x = 0
and x = oo . Moreover, since the greatest negative coeffi-
cient of the proposed equation increased by unity is 5, it
follows that 5 is a superior limit to these roots. But since
for such limit
x4 + 23 > 4#3 -f- 3æ
or
,	23	4	3
1 H--4	>----1--3
it is easily seen that 4 is also a superior limit. The roots
are thus readily found to lie between 2 and 3, and between
3 and 4.
But a more striking example of the great inferiority of
Fourier’s mode of proceeding, in any but the simplest
cases, is furnished by the equation
13.	x6 + x4 + a?3 — 2xa + 2x — 1=0
the analysis of which, as detailed in the work before re-
ferred to, (Anal. des Eq. p. 146,) is very laborious; re-
quiring, in conjunction with several trial-calculations, noTHE THEOREM OP STURM.
179
fewer than three operations for the common measure· The
whole process* by the theorem of Sturm* is as follows:
5—4—3-f4—2
X
1 + 1 +1 -2+2 — 1
1	—4 —3 +4—2
5 —15 + 20 —10
25	25—50 + 50—25
—6 + 33—44 + 27
—6—33 + 44—27
X
5 + 4 + 3 —4 + 2
-63
-10
12
2079—2772+1700
—440 +270
36 — 48	+24
-1675 + 2550-1724
Now, since it is immediately seen that
4 x 1675 x 1724 > 2550%
the operation terminates at this step.* Moreover* as two
* The abrupt conclusion of the work, as soon as we arrive at a function,
the roots of which are known to be imaginary, is an important feature in
Sturm’s method, and is often the means of saving a considerable portion
of the numerical labour. There is nothing analogous to it in the method of
Fourier.
The criterion of imaginary roots is of very easy application in equations of
the second degree. It would be easy too to investigate as unfailing a criterion
for those of the fourth degree ; and, if such criterion were of convenient appli-
cation, it would be worth while to do so. But the complicated character of
the test is such as to render it quite useless in connexion with Sturm’s
theorem. We have given it however in Note I. at the end, where it is shown
that one of the conditions furnished by Sturm, for the reality of the roots, is
unnecessarily complicated; and, what is a little remariv:*ble, the tests of
Lagrange, generally of such laborious application, suggest, in this case, the
proper simplification of Sturm’s criteria.180
MATHEMATICAL DISSERTATIONS.
variations are furnished by the leading signs for x = — oc ,
and only one for x = + oo, we infer that there is but one
real root.
From the sign of the last term in the proposed equation,
we conclude that the real root must be positive, and it is
immediately seen to lie between 0 and 1.
If an example be taken in which there are not only
imaginary roots, but also pairs of roots very close together,
the superiority of Sturm’s method will be still more con-
spicuous. Fourier, however, has but sparingly intro-
duced into his work examples of this kind.
(109.) We shall now endeavour to obviate the objection
advanced against Sturm’s method on the ground of the
intolerable labour of executing the closing steps of the
work in equations beyond the fourth degree, and of the
great extent to which the coefficients in the resulting
functions reach.
We propose to put an effectual check upon this ten-
dency of the successive coefficients to increase to an un-
manageable extent; this check is suggested by the follow-
ing considerations:
There are three multipliers connected with every step
of the work; these, in our arrangement of the process, oc-
cupy the side column in each step; and we have seen how
to reduce their magnitude whenever they involve a factor
common to all three. But to the second and third there is
always a common factor; and one too of considerable
magnitude in the advanced steps. It is plain that this
factor may always be suppressed, provided the first multi-
plier be divided by it, and the quotient be put to supply
the place of this multiplier. Even taking into account the
trouble of performing this division, as indeed we ought,
there will still be a considerable saving of labour, when the
second and third multipliers are very large, by this simple
modification; but as we wish to reduce the results of theseTHE THEOREM OF STURM.
181
steps, as well as the labour of executing them, we shall
perform the several multiplications required, not at full
length, but according to the contracted method, so com-
monly resorted to in multiplication of decimals, when we
desire accuracy in our results only to a certain proposed
extent of leading figures. By introducing this means of
abbreviation into the work of Sturm’s steps, we may re-
strict the several resulting coefficients to any number of
leading figures we please, which shall be all correct as far
as they extend, and the labour will thus be saved, as well
as the inconvenience avoided, of introducing into the
closing steps of the work a long array of redundant figures
A glance at the last step of the operation at page 162.
will be enough to show that the amount of labour expended
on it, and the extent of figures obtained in the result, far
exceed the actual demands of the case ; two or three only
of the leading figures — 296.......,349........, would have
been amply sufficient ; and to obtain these, by the means
described above, would have cost us comparatively but
very little trouble.
(110.) But, instead of returning upon this step for the
purpose of illustrating the importance of the abbreviations
here recommended, it will be more instructive to take an
example of much greater difficulty ; and we therefore se-
lect the following complete equation of the fifth degree.
14. Required the number and situations of the real roots
of the equation
a:5 + 7x* _ 144a:3 + 61 lor8 — 928® + 362 = 0.
The leading functions here are
X = a:5 + 7a:4 — 144a:3 + 611a:2 - 928a: + 362,
and
Xx = 5xi + 28a:3 - 432a·2 + 1222a - 928.182
MATHEMATICAL DISSERTATIONS.
and the entire work by the general method, abbreviated
after the first step, in accordance with the foregoing di-
rections, is as follows :
7
5
25
5—28	+432	-1222	+928
X
1+7	-144	+611	-928	+	362
— 196 +3024 —8554 +6496
2160 —6110 +4640
-3600+15275-23200 +9050
1636-12189+27114-15546
1636+12189—27114+15546
v
5 + 28	—432	+1222—928
65-2524
5
1636
795362—1769254 + 1014414
—135570	+77730
-706752 +1999192-1518208
46960 —307668 +503794
46960 + 307668—503794
X
1636—12189 + 27114—15546
-1470-4
1636
46960
—4524 + 7408
-8242
12733—7300
33 -108.
It may be proper to remark, for the better elucidation of
the foregoing work, that the leading multiplier 65*2524,
in the second step, is extended only to a sufficient number
of decimals to secure accuracy in the integral parts of the
several products resulting from it. These products are all
obtained by contracted multiplication, the decimal part
being rejected.
In the next following step of the work it is proposed toTHE THEOREM OF STURM.
183
secure accuracy only as far as the first four ox jive figures
in the several products; in other words, jive figures are
uniformly rejected from the right, in each product, or
rather these products, wanting the five terminating figures,
are obtained by contracted multiplication. As in the pre-
ceding step so here, the leading multiplier — 14704 con-
tains none of those decimals whose influence would be
confined only to the figures thus rejected.
We may mention, in reference to the manner in which
these leading multipliers are obtained, that the operations
implied in
and in
1636+12189
X
5 + 28
46960+307668
X
1636-12189
are not actually performed, and the division in the one
case by 1636, and in the other case by 46960, afterwards
executed; because this course of proceeding w ould involve
unnecessary trouble.
But the first of these multipliers is obtained by dividing
12189 by 1636, then multiplying the quotient 7*45048 by
5, and adding 28 to the product. In like manner, to ob-
tain the other multiplier, 307668 is divided by 46960, the
quotient 6*5517 is then multiplied by 1636, and — 12189
added to the product.
(111.) The above example is by no means a favorable
one for our abbreviated method; as three roots of the pro-
posed equation lie pretty closely together. We have a
sufficiently plain intimation of this in the circumstance
that the final function turns out to be comparatively near
to zero; and, in addition to this, the results of the second184
MATHEMATICAL DISSERTATIONS.
step are such that, considering the magnitudes of the num-
bers, four times the product of the extreme terms, differs
but little from the square of the mean term: the less this
difference is, the more nearly of course do the roots of the
quadratic, of which these numbers are the coefficients,
approach to equality.
It is quite clear, however, that the final function, just
adverted to, and the sign of which we infer as usual from
108
substituting for x in the preceding quadratic function,
is of finite magnitude ; and, with change of sign, is plus.
If in any case this final function, or indeed any preceding
function, should be actually zero, most likely there will be
equal roots; yet in case the roots, instead of being rigo-
rously equal, should have a minute difference, attention
should be paid to the accuracy of the terminating right-
hand figure of each of the coefficients of the function at
which our work terminates.
In the example above, although as just remarked one
of some delicacy, it is of no moment whether or not the
33 and the 108 err by a unit in the last figures 3 and 8.
Nevertheless, in the multiplications which contribute to
produce these numbers, it is as well to notice, that the 33
is in reality 33*3 ; and the two numbers over the 108 are,
in strictness, 7300*4 and 7407*8; so that 108 is a unit in
excess. The correct value is therefore 107.
We have just observed that when our final result is
zero, most likely there will be equal roots. We cannot
affirm that in such a case there are necessarily equal roots,
because our result, for aught we know to the contrary,
may be zero only within the limits to which the coefficients
last deduced are restricted; and thus the roots may be
equal only as far as this prescribed extent. But for a
figure or two, we may always in such a case safely infer the
equality of a pair of roots. The situations of the roots, in
such critical instances, will therefore always be clearlyTHE THEOREM OF STURM.
185
pointed out ; the leading figure of each/ and generally two
or three of the leading figures, being made known by
equating our terminating function, as for instance, the
function of the first degree above, to zero.
It appears then we can never be at a loss, in consequence
of our abridged mode of working, as to the character of
the roots of the proposed equation ; although, when roots
have several leading figures the same in each, we may not
be able to say whether they differ at a remote figure or not
without additional examination.* But we shall revert to
this presently.
(112.) We shall now proceed to determine the character
and situations of the roots of the preceding equation. The
functions just determined are
X	= xb 4- 7x4 — 144a;3 + 611a:2 — 928a: + 362
XI	= 5a;4 + 28a;3 — 432a;2 + 1222a; - 928
X2= 1636a;3- 12189a;2 + 27114a;- 15546
X3 = 46960a:2 - 307668a: + 503794
X4 = 33a: —107
X5= +
As the signs of the leading terms of these functions are all
plus, we infer that the roots are all real. And as the signs
of the Jinal terms present four variations, we conclude
* It will be proved in Note II. at the end, that when the last, or constant
function is actually zero, then the preceding function, or that of the first de-
gree in x, when superfluous factors are expunged, must have its leading co-
efficient a factor of that in the.proposed equation ; and the remaining number
also a factor of the final number in the proposed equation. It will always be
easy to ascertain whether or not such is the case.186
MATHEMATICAL DISSERTATIONS.
that there are four positive roots.* We shall seek the
places of these*
for x = 0.........+	—	—	4-	—	4-	four variations
= 1.......— — 4- + — + three „
-2........-	+	+	+	-	+
= 3.........-	+	+	+	-	+
= 4........4-	+	+	4-	+	-|-	no variation.
Hence there is one positive root between 0 and 1* and
there are three between 3 and 4.
The negative root is found, from the single polynomial
X, to lie between 10 and 20; that is, its first figure is 1 in
the tens’ place.
The simplifications introduced into the foregoing work,
by means of the abbreviations made use of in the second
and third steps, are very considerable; as will be soon felt
by any one who will take the trouble of going through the
several multiplications in their unabbreviated form.
Mr. Lockhart, in the publication on Sturm’s Theorem,
• The number of positive and negative roots in any equation may always
be readily ascertained from an inspection merely of the leading and final signs
in Sturm’s functions, without making any actual substitutions in them. The
positive roots will always be equal in number to the excess of the variations in
tbe final signs above the variations in the leading signs.
This rule for the number of positive roots will not require any modification
even when the functions X, Xt, X2, &c. are fewer in number than n -}- I in
an equation of the wth degree. But as regards the imaginary roots there will
always be as many pairs of them as there are functions wanting to make up
the ordinary number n 4-1, besides the pairs indicated by the variations in the
leading signs of the incomplete series. We shall have occasion shortly to
notice a modification necessary to be introduced into the numerical operation
of Sturm’s theorem whenever these incomplete series occur.THE THEOREM OF STURM.
187
before referred to* has executed this labour, availing him-
self of only the ordinary abridgments; and he finds the
complete functions to be as follow:
X = a:9 + 7a;4 — 144r* -f 611a;2 — 928a; + 362
X, = 5a?4 + 28a?3 — 432x2 + 1222x — 928
X2 = 1636a;8 - 12189a;2 + 27114a; - 15546
Xs= 3073059a;2-20133742* + 32968246
X4 = 20969919426169a; - 67576648427974
X»= +
The three roots lying between 3 and 4 are
3*213....... 3-229_____ 3-414......
It may be interesting to notice, as corroborative of the
accuracy of both Mr. Lockhart’s calculations and mine,
that if the number 209699......be divided by 33*3, the
quotient 6297...., taken as a divisor of 675766......,
gives exactly 107.
(113.) We have already adverted to the circumstance
that the remainder X9 is comparatively a small number;
for as we may readily foresee the two or three leading
figures of it become zero. Now if in consequence of our
having rejected an unwarrantable number of figures from
the right of the numbers in X3, we had deprived ourselves
of the means of getting any significant result for X5, in
consequence of the remainder, as far as our numbers could
influence it being zero, then, as remarked above, we should
have concluded that there are at least two roots, each equal
to the value of x in X* = 0, to as many figures as this
value of x can be depended upon as accurate consistently
with the result X5 = 0.188
MATHEMATICAL DISSERTATIONS.
To determine whether or not these roots are equal
throughout, would require one or other of these three
methods of proceeding :
1.	We might correct the latter steps of the work, com-
mencing at where the abbreviations are first introduced,
and thus remove all doubt.
2.	We might take the leading figure or figures furnished
by the simple equation X4 = 0, and extend these by
Mr. Horner’s method of approximation, till we should
arrive at that figure at which the roots separate, or till suf-
ficient evidence is afforded that this point of separation is
indefinitely postponed, and that therefore the roots are
equal.*
3.	Or finally, and which is generally the shortest course
to our object, we might see if the two numbers in X4 have
the property mentioned in the foot-note at page 185. If
they have not, the roots are unequal. If they have, then
divide X by the square of X4, the common factors being
expunged, and the roots under examination are equal or
* Cauchy has determined a fractional expression, involving only the first
coefficient of the proposed equation, and the one which is greatest in abso-
lute value, which he proves to be less than the smallest difference between
any two real and unequal roots. If, therefore, two or more roots, as the ap-
proximation proceeds, keep together up to a figure as small in the numerical
scale as this fraction, we may infer that they must necessarily be equal, with-
out extending the approximation farther. But unless the original coefficients
are small, and the degree of the equation not beyond the fifth, the extreme
smallness of this limit will render it we fear of but little value.
The expression referred to is
1_______
A{2(fc+ 1)}*»-1
where A is the coefficient of the first term, k the greatest coefficient in
absolute value, the coefficients being supposed whole numbers, and
m = ^ n (n— 1), where n is the degree of the equation. (See Journal de
Mathématiques, May 1838.) In connexion with this subject the student may
also consult Note II. at the end of the present work, wherein it is shown that
no difficulty can ever arise from the entrance of equal or nearly equal roots
into the equation.THE THEOREM OF STURM.
189
not according as the remainder of this division is zero
or not.
We have here been supposing an extreme case. It is
scarcely necessary to state that in practice the better plan
will always be to secure at least four or five places of
figures even for our last step; or so many that should the
final or constant function become zero, we may immedi-
ately conclude the values of the roots indicated to three or
four places of figures. It will be a very rare occurrence
indeed for unequal roots to keep together throughout this
extent, and then to differ at a more remote figure; yet, as
the thing is not absolutely impossible, it is requisite that
we know how to proceed should it be required to find, in
addition to the nature and number of the roots and the
leading figures of each, whether two or more remarkably
close roots are really separable or not. But we shall not
enter at greater length into these critical cases at present;
we propose to resume the consideration of them in Note II.
at the end, at which place it will be shown that there can
never be the slightest perplexity attendant upon our me-
thod on account of the presence of equal roots, or of roots
differing ever so little from each other.
(114.) We shall now give an example of the application
of our method to an equation of the 6th degree, selecting
for the purpose one which is adduced in a recent publica-
tion as evidence of the impracticability of Sturm^s Theo-
rem in the higher equations.* The equation is this,
15.	a:6 + xb — a:4 — x3 + x2 —x +1=0
w’hence
Xj = 6x5 + 5x4 — 4a:3 — 3a:2 + 2x — 1,
* Du Théorème de M. Sturm, et de ses Applications Numériques. Par
M. E. Midy, Professeur au Collège de Nantes. Paris, 1836.190
MATHEMATICAL DISSERTATIONS.
and the operation is as follows :
6-5	+4 +3-2+1
X
1 + 1 -1 -1 +1 -1+1
1	_5 +4 +3 —2 +1
6	24 + 18-12+ 6
36 -36—36 + 36-36 + 36
17 + 14-27 + 32—37
17 —14	+27	-32 +37
X
6 + 5	— 4	— 3	+2-1
1	—14 +27 -32 +37
102	2754-3264 + 3774
3 72	— 1156 —867 +578-289
	— 1584 +4104 —4320 +252
or, -+4x9
—44 +114 —120 + 7
—44 —114 +120	—7
X
17 + 14	-27 +32-37
-1277	145578-153240 +8939
22x17	-44880 +2618
44x22	-26136 +30976—35816
	-74562 +119646 + 26877
-74562-119646-26877
X
—44+114	—120 + 7
-21*69
22
37281
2595
-591
-4474
+ 583
+261
2470	-844
or, -+2
1235	-422THE THEOREM OF STURM.
191
Consequently the functions are as follow:
X = x8 4- x6 — a:4 — xz 4- x2 — x -f 1
Xi*= 6a:5 4- 5a:4 — 4a:3 — 3a:2 4- 2x — 1
X2 = 17a:4 4- 14a:3 — ‘27a:2 4- 32a; — 37
X3 = — 44a:3 4- 114a:2 — 120a: + 7
X4 = — 74562a2 4- 119646a: 4- 26877
X5 = 1235a: —422
A Xe = -
X Xj x2 x3 x4 x5 x6
For x = — oo....4- — 4- — — — — three variations,
= 4-00------4- 4- 4- — — 4- — three „
Hence the roots are all imaginary. And this we might
have inferred at once without writing down the functions*
because there are always as many pairs of imaginary roots
as there are variations in the leading signs. (Theory of
Equations, p. 144.)
(115.) The following equation of the seventh degree is
from Fourier (page 111).
16. Required the number and situations of the real roots
of the equation
x7 — 2a:5 — 3a:3 4- 4a:2 — 5x 4- 6 = 0.
Here the leading functions are
x7 4- Oa:6 — 2a:5 4- Oa:4 — 3a;3 4- 4a;2 — 5a: 4-6
and
7x* 4- Oa;5 — 10a:4 4- Oa;3 — 9a;2 4- 8a: — 5,
* In the work of M. Midy, before referred to, the entire series of functions
after this are erroneous; but the character of the roots is nevertheless cor-
rectly determined.192
MATHEMATICAL DISSERTATIONS.
and the following is the numerical operation,
7+0 + 10 +0	+9	—8 + 5
X
1+0— 2 +0	-3	+4 -5 +6
0
1	10 +0 +9	—8	+5
7 —14 +0—21 +28 —35 + 42
4+ 0+12 —20 +30—42
or, -+ 2
2	+0 +6 -10 +15-21
2-0	-6 +10 -15 + 21
X
7 —0 —10	—0	—9 +8— 5
0
7 —42+70 — 105+ 147
2 —20 —0 —18 +16 — 10
62—70+ 123 —163 + 10
35
31
312
62 + 70 -123 +163 -10
X
2 + 0	+6	—1,0	+15—21
2450—4305 +5705	—350
—3813 + 5053	—310
5766-9610 + 14415-20181
—4403 + 8862 — 19810+ 20531
—4403—8862 +19810—20531
X
62—70
+ 123	—163 + 10
-54-81672
-62
4403
485786 -1085919+1125442
— 1228220+1272922
541569 —576793	+44030
200865 +489790 — 1169472
200865—489790 +1169472
-4403+8862	-19810+20531
19615
-4403
200865
-96072 +229392
-51492
—39791	+41240
187355 -270632THE THEOREM OF STURM.
193
Hence the functions are
X = x1 — 2x5 — 3xs 4- 4x2— 5x 4- 6
X, = 7x* - 10x4 — 9x2 + Sx - 5
X2 = 2x5 + 6x3 - lOx* 4- 15x -21
X3 = 62x4 - 7Ox3 + 123x2 - 163x + 10
X4 = - 4403xs + 8862X2 - 1981 Ox + 20531
X5 = 200865x2 + 489790x — 1169472
Xfl = 187355x — 270632
A X7= +
As there are two variations in the signs of the leading
terms, we infer that the equation has two pairs of ima-
ginary roots, but not a greater number : there are there-
fore three real roots ; and since the variations in the final
signs exceed those in the leading signs by two, it follows
that two of the roots are positive.
Since the greatest negative coefficient in the proposed
equation is 5, it follows (Theory of Equations, p. 38,) that
6 is a superior limit to the positive roots ; so that our sub-
stitutions may be confined to numbers below 6.
For x = 0.... 4- — — 4- 4- — — 4- four variations
= 1----------4- —--------4---------4- four	„
= 2....4" 4* 4· 4· — 4” 4· 4* two	,,
Consequently the two positive roots are both situated be-
tween 1 and 2.
13194
MATHEMATICAL DISSERTATIONS.
Fourier has not succeeded in discovering the charac-
ter of the roots of the foregoing equation. He seems to
imply that Jive of them are real instead of three only.*
As to the negative root of the equation, we easily find,
by changing the alternate signs, and writing it thus,
a?7 — 2a?5 — 3a?3 — 4a?2 — 5a? — 6 = 0,
that x = 1, and x = 2, give opposite signs; the negative
root is therefore between — 1 and — 2.
(116.) The equation just analyzed, though of the seventh
degree, is not complete in all its terms. It is easy to see
that this circumstance operates favorably in restraining the
magnitude of the numerical results, especially those of them
which are more immediately dependent on the second term
of the proposed equation. We shall now take a complete
equation of the seventh degree; and one, moreover, in
which the leading term has a coefficient greater than
unity.
17. Let the equation
4a?7 — 6a?fl — lx5 + 8a?4 + 7a?3 — 23a·2 - 22a? — 5 = 0
be proposed.
Then the functions X, Xx are
X = 4a?7 — 6x6 — 7a?B -f 8a?4 -f 7a?8 — 23a?2 — 22a? — 5
X! = 28a?6 — 36a?5 - 35a?4 + 32a?8 + 21a2 - 46a? — 22 ;
and the following is the numerical operation, the abbre-
viations being first introduced at the commencement of the
third step of the work.
* Analyse des Equations, pages 111 and 153.THE THEOREM OF STURM·
195
28 + 36 + 35 —32 —21	+46 +22
4-6 -7	+8	+7	—23 —22—5
-3
14
98
—108—105 +96	+63 —138 —66
490—448—294 +644 +308
—686 + 784+686—2254—2156—490
304-231—488 +1547 +1986 + 556
304+231 + 488 — 1547 — 1986—556
X
28—36	—35	+32	+21 —46—22
— 1119
2128
76 x 304
-258489 —546072+ 1731093 + 2222334 + 622164
1038464—3292016—4226208—1183168
—808640 +739328 +485184—1062784—508288
28665 + 3098760 + 2009931	+23618—113876
-33059*2937
304
28665
28665—3098760 —2009931	—23618 + 113876
304—231—488 + 1547 + 1986+556
102442817 + 66446899 +780794—3764660
—611019	-7180 +34618
-13989	+44345 +56929 +15938
-101817809-66409734 -872341+3748722
-101817809 + 66409734 +872341 -3748722
28665 + 3098760 +2009931+23618-113876
—3080074’—20454689 —268687 +1154634
—28665?	-2501 + 10746
101817809	20464677 +240473 —1159460
—7487	+17468	+4826
-7487 — 17468 -4826
X
— 101817S—664097— 8723 + 37487
3040	-5310
1018	-491
7	-6
-1467
+26
5807 · +1441
or-i- 11
528 + 131.196
MATHEMATICAL DISSERTATIONS.
This last function, containing x only in the first degree,
gives, when equated to zero,
131
*	528’
If this be substituted for x in the preceding function
of the second degree, namely, in
- 7487xs + 17468a: + 4826,
the result of the first two terms will be — 4795; hence the
whole result will be -f- 31; consequently, the final function
takes the sign minus.
We shall have no occasion to write down the several
functions to which the foregoing sets of coefficients belong;
for it is at once seen, from the signs of the leading terms
in those functions, that the equation has only one real root,
and the place of this is readily determined from the original
function X.
The last term in this function being minus, the root
sought is of course plus. It is found, in the usual way, to
lie between 2 and 3.
(117.) It will have been observed that, in all the pre-
ceding examples, the derived functions descend regularly,
in degree, by unity. No instance has occurred in which
two consecutive functions differ in degree by two or more
units. Such instances, indeed, are by no means common;
for an equation of the rath degree usually furnishes » + 1
functions, X, X15 X2, &c. Nevertheless it sometimes hap-
pens that the functions are fewer than n + 1 in number,
owing to an interruption taking place in the regularity
with which the degrees of these functions ordinarily dimi-THE THEOREM OF STURM.
197
nish. In every such case a corresponding interruption to
the regularity of our uniform process of deriving these
functions will occur; so that, commencing at the point
where the break happens, the step next in advance must
be executed in a manner somewhat out of harmony with
the prevailing mode of procedure.
This circumstance, however, will not at all add to the
complication or difficulty of the process ; on the contrary,
the oftener these interruptions take place, the shorter and
easier will our work become ; but it is very important to
keep in remembrance the necessary modification which
our method is to undergo at these stages.
What this modification is, is soon discovered from in-
specting the general formulas at the commencement of this
dissertation. Those formulas were constructed on the
express condition that the several remainders (see page 142)
descend regularly in degree by unity, when the proper
powers of x are inserted ; and we have now to ascertain,
when two consecutive remainders differ by two units, how
we are to derive from them the third remainder.
If
ax” + bxH~l + cxn~2 + ...	[A]
be the first remainder, and
arx'*~2 -f- b'xn~~3 -f c'x*~A + ....	[B]
the second, the third must be of the form
b'"x”-3 + c"'*"-4 + . . . .
to obtain which will evidently require three terms in the198
MATHEMATICAL· DISSERTATIONS.
quotient of [A] by [B] instead of two, as in the case con-
sidered at page 143, the third term being chosen so as to
cause the third term in the equation [C] at page 143 to
vanish, in addition to the two preceding terms. This third
term, as far at least as the coefficient is concerned, is
a"
obviously or, to avoid fractions, if we first multiply [C]
by a', the third term of the quotient will be a"; and, there-
fore, the remainder sought arises thus:
aV'#"-2 -f afbffxn~3 -f a! d'xn~4 -f ....
af,a/xn~2 + af,b'xn~3 + a"dxn~* -f- ....
0 -f bf"xn~3 -f d"xn~*	....
Consequently, in the case supposed, this is the supple-
mentary operation which must be introduced to complete
the step. Hence we must proceed in the usual way, that
is, as at page 144, to the remainder
a" + A" + c" + ....
we may then bring down, under this, the preceding
remainder
a! + b' + d +______
and, multiplying b" + d1 -h.....by a', and V + d -f-......
by a", we may arrange the products under one another, and
subtract the lower set from the upper.
This appears to us to be the most convenient way of
arranging the additional work, and that which best unites
with the general process. Instead of actually subtracting,
however, we shall, as in the other parts of the work,THE THEOREM OF STURM.
199
convert the operation into one of addition by changing
the signs.
(118.) An example will sufficiently illustrate the nume-
rical operation.
Let the two consecutive functions or remainders be
4x5 — 3a:4 + 0x% + 2x2 — x + 1
and
2x3 + Oa?* — x — 4.
Then executing the work of an ordinary step first, and
afterwards supplying the necessary addition, the whole
will stand thus:
2+0 +1+ 4
X
4-3 +0+2 -1+1
0	-3-	-12	
4 + 16			
0	1 +4	-2	+2
-4	— 17 + 14		—2
2	+0	+i	+4
—34+28 —4
0 -4-16
—34+24—20
The result —4—17 + 14 — 2, which closes the ordinary
step, has all its signs changed; also the terms + 0+1+4
have had the signs changed: but the multiplication of
these by the — 4, the sign of this having been changed,200
MATHEMATICAL DISSERTATIONS.
restores the signs of the product 0 — 4 — 16 to what it
would have been had none of these changes been made.
It appears, therefore, that of the two rows of terms
—	34 -f 28 —4
0 -4-16
the first has all its signs the opposite of those really be-
longing to it, and the second has its proper signs, which
become, in effect, changed by adding the lower row instead
of subtracting it, and thus our final result
—	34 + ‘24 - 20
is the true remainder with changed signs; and it is with
such change that, in Sturm’s theorem, we wish the re-
mainders to present themselves. The final result therefore
under the prescribed arrangement harmonizes with all the
other results of the process.*
As the multipliers — 4 and 2 have the common factor 2,
this factor may evidently be suppressed; and thus, like as
with the other multipliers, the operation simplified. The
final result, when thus reduced, will therefore be
—	17 -f 12 — 10
To render the method still plainer we shall give another
example.
* It is easy to see that the entire operation of Sturm’s theorem might be
conducted according to this plan, every successive function being obtained at
two steps instead of at one. The intermediate steps, in this mode of proceed-
ing, would furnish the intermediate or auxiliary remainders ; hut the method
employed throughout this dissertation will, in general be found to require both
less labour and less space.THE THEOREM OP STURM.
201
Suppose the two consecutive functions or remainders
to be
2æ4 — 6a:3 + 3æ2 + x — 2
and
— 3a:2 + 2a; — 4
—3—2 +4
2—6 +3	+1—2
-28	+56	
-24		
27	+9 -	-18
25	-65	+ 18
-3	—2	+4
	130	—54
	—50+100	
	80	+46
	or	-r-2
	40	+23
(119.) If the proposed functions differ in degree by three
units instead of two, then four terms in the result [C],
p. 143, must be rendered zero; two are made zero by the
usual step; so that we shall only have to employ the result
of this step, in conjunction with the original divisor, and
to proceed with them in the ordinary way, in order to ren-
der two more terms zero; that is, in order to arrive at the
desired remainder. We shall, however, as before, employ202
MATHEMATICAL DISSERTATIONS.
addition instead of subtraction, that this remainder may
appear with changed signs. The following example will
sufficiently illustrate the method of proceeding.
Let the two functions be
2x6 — 3x5 + 2x4 +xs — 3x2 — 3x +4
and
4x3 — 6xz + æ + 1
44-6 —1 -1
X
2—3 4-2 +1 —3—34-4
0
1	-1 -i
2	4 4-2 —6 —64-8
3	—1 4-6 4-6—8
4	4-6 -1 -1
X
—3 — 1 4-6 4-6 -8
-11
-6
8
-66 + II4-II
6 +6
48+48—64
-12+65—53
Hence the remainder, with changed signs, is
- 12a:2 + 65x - 53
It is scarcely necessary to remark, that if our final re-THE THEOREM OF STURM.
203
mainder is to be still further postponed, till it reach one
degree lower, as will happen when the functions differ in
degree, by four units, then we are to continue the work
as in the former examples, and so on. Whenever the
sought remainder is deduced, a thick line, or rule,
should be drawn under it, to mark distinctly the close of a
step.*
(120.) We shall now give an example in which Sturm’s
functions do not regularly descend in degree by unity, and
in the working of which the modification just explained
will therefore be necessary. This example, like the last, is
from Fourier.
18. Required the number and situations of the real roots
of the equation
*6 — \2x5 + 60x* + 123x2 + 4567x — 89012 = 0
* The above example, as well as the one first given, is from the Penny
Cyclopaedia, article Involution. The writer of that article, we believe
Mr. De Morgan, gives a method of deducing the remainder, which is some-
what shorter than that which we have recommended, and recommended
chiefly because of its more readily uniting with our general numerical process.
Mr. De Morgan’s method of conducting and arranging the work may perhaps
be preferred by some, and we therefore refer to his paper, as also to remark
that the operations in his method, in that of M. Midy, and in the steps em-
ployed in the present Dissertation, are all suggested by one and the same
general principle, and that a principle of extreme obviousness and simplicity.
It is solely in the mode of applying this principle to actual practice, that is, in
the mere numerical arrangement of the work, that the labour involved in it is
to be economized, and reduced to its least possible amount.204
MATHEMATICAL DISSERTATIONS.
6+60-240	+0 -246 -4567
X
1-12 +60	+0	+123	+4567-89012
—2 —120+480	+0	+492	+9134
1 -240	+0-246- 4567
6	360	+0+738+27402-534072
0-480-492-23327 +524938
-480+492 +23327-524938
6-60	— 240	-0 — 246-4567
441	2169724-10287207		— 231497658		
-40	-9330804-20997520				
3200	768000 4-0		4-787200	4-14614400	
	*—51892	-31284727	4-230710458	—	14614400
	—480	4- 492	4- 23327	—	524938
		150167	- 1107410	4-	70149
		—255	— 12105	4-	272401
		149912	— 1119515	4-	342550
149912+1119515—342550
X
—480—492—23327 +524938
-4076-8
—480
1499
—4564 -f 1397
—164
— 3	+80
4731—1477
* The two numbers in this side column, and which are employed as multi-
pliers for the completion of the step before us, agreeably to the directions
already given at (118), have a common factor, 4. Expunging this, therefore,
we might have employed as multipliers, the numbers 12973, and 120. We
have preserved the original numbers in order to render the method of proceed-
ing the more plain ; but in practice it will always be advisable to substitute for
the numbers themselves those which arise from expunging the common
factor.THE THEOREM OF STURM.
205
Consequently, Sturm’s functions are as follow:
X =	— 12x5 + 60a;4 + 123a:2 + 4567a; — 89012
Xx = 6a;5 — 60a:4 + 240a;3 + 246a: + 4567
X3 = — 480a:3 — 492a;3 — 23327x + 524938
X3 = 149912xa — 1119515a: + 342550
X4 = 4731a: — 1477
/. X5= +
We conclude at once, and without even writing down the
functions, that the equation has four imaginary roots; one
pair being indicated by the absence of the function of the
fourth degree, and another pair by the leading sign of X2.
There are thus only two real roots; and since, moreover,
the variations in the final signs exceed those in the leading
signs by owe, it follows that one of these two roots is posi-
tive, and the other negative. Their situations therefore
may be readily determined from the single function X, the
others being useless for this purpose.
To find a superior limit to the positive root, we have the
inequality
1 +
60
123	4567	12	89012
X4	X5 ^ x X6
which is satisfied for x = 7, and for every higher number,
but not for x = 6; hence the positive root is between
6 and 7.206
MATHEMATICAL DISSERTATIONS.
For the negative root, change the alternate signs of X,
taking account of the term Or*, writing it thus,
x6 + 12a;5 + 60a;4 + 123a;2 — 4567a; — 89012,
and it will soon be found that the consecutive numbers,
which give opposite signs to this function, are x = 5
and x = 6. Consequently, the negative root lies between
— 5 and — 6.
(121.) Enough has now been done to put the student
fully in possession of what appears to us to be the best
method of conducting the numerical operations in the
practical application of Sturm’s Theorem ; and, at the
same time, to enable him to institute a fair comparison
between this method and the methods of Budan and
Fourier. To render such a comparison the easier, we
have selected all our more difficult examples from the work
of Fourier· Some of these, however, are not fully ana-
lyzed by Fourier. The examination, for instance, of the
equation of the seventh degree, which we have considered
at p. 191, is left by Fourier incomplete; and his results
are, in consequence, presented in an imperfect, if not in-
deed in an erroneous form. We are not aware whether or
not Sturm’s theorem has ever till now been applied to
equations of the sixth and seventh degrees; but the fact
here established of its ready application to such equations,
and equations too whose coefficients are by no means re-
markably small, will serve to moderate the objections
which have been made to the theorem on the ground of
its general impracticability in equations of a degree higher
than the fourth.
By conducting the operation as at first proposed by
Sturm, that is, by going through the ordinary routine of
seeking the common measure of two algebraic functions,THE THEOREM OP STURM.
207
the work would, no doubt, become both fatiguing and
cumbersome in equations of the fifth and sixth degrees.
But by employing only detached coefficients, as in
Horner’s method of solution, suppressing the useless quo-
tients, precluding fractions, and rejecting the redundant
figures in the closing steps, or rather providing against
their entrance—by combining all these facilities, and dis-
posing of the whole work so as to effect the greatest saving
in numerical labour, we have seen that an equation of even
the seventh degree may be completely analyzed in reason-
able space, and with but a moderate amount of arithmeti-
cal calculation.
(122.) Although the theorem of Sturm is thus reduced
to a practicable form, and is, on account of its infallibility,
invaluable in the analysis of numerical equations, yet the
rules of Fourier, or rather the transformations of Budan,
are not without their use. In equations of a high order it
will frequently be found that these transformations, which
may always be effected with comparatively little trouble,
will furnish us with useful preliminary information as to
the partial construction of the equation; and in favorable
cases the entire constitution of it will thus be made known.
Any foresight into the nature of the roots, before actually
applying Sturm’s theorem, is of advantage, either in the
wray of checking our numerical results, or of enabling us
to anticipate or to dispense with, the closing steps of the
work. As, for instance, if a preliminary examination of an
equation by the rules and tests given by Budan, shows
us that there is but one doubtful interval, and we after-
wards, in the work of deriving Sturm’s functions, obtain
a result with its leading sign minus, we may conclude at
once that the equation has two, and only two, imaginary
roots.
If we had submitted the equation at p. 203 to these
tests, we should have found two doubtful intervals; or, at208
MATHEMATICAL DISSERTATIONS.
least, one interval doubtful, and one indicating a pair of
imaginary roots ; and, as the result of the very first step in
Sturm’s process shows us, that there are at least ¿wo pairs
of imaginary roots, we might have stopped the work at that
step, and have affirmed with confidence, that the two un-
ambiguous roots, made known by Budan’s transforma-
tions, are all that exist in the equation.
Many instructive examples of the value and efficiency
of Budan’s rules, in certain cases, are given in Mr.
Lockhart’s Resolution of Equations, before referred
to ; and besides this latter work, the student may consult
with advantage, on the subject of these rules and criteria,
the second edition of Budan’s own treatise,* a paper by
M. Vincent, on the Solution of Numerical Equations in
the Journal de Mathématiques, for October 1836', Professor
Davies’s Key to Hutton9$ Mathematics, and the fourth
chapter of the Theory of Equations, by the author of the
present work.
(123.) We have now to notice a short and easy method
of performing the first step in Sturm’s process ; that is,
of obtaining the function X2.
Abridgment in this early stage of the work is, we must
acknowledge, of comparatively but little importance in
equations of elevated degrees ; still in those of the third
and fourth degrees, which, after all, occur more frequently
than those of the higher orders, the saving of trouble
effected by this simplification of the general method, al-
though extending only to the first step, does really bear a
perceptible proportion to the entire work. In such equa-
tions therefore, at least, we think the improvement to bq
noticed should always be introduced ; and even in equa-
tions of more advanced degrees, whose coefficients are
* Nouvelle Méthode pour la Résolution des Equations Numériques.
Paris, 1822.THE THEOREM OF STURM.
209
large numbers, this more brief and rapid method of per-
forming the first step is well worth employing.
The improvement we have in view is derived from the
general process, in the following manner:
(124.) Let
X = axn -f bxn~l + cxn~2 + dxn~3 -f ....
then
Xj = naxn~l -f (w —■ 1) bxn~2 + (n — 2) cxn~~3 + ....
and, proceeding according to the rule already explained,
X2 is determined as follows:
	na—(n—1 ) b — xA a-\-b +	(n—2)c c	+	(n—3)d d	
b	-(n-l)b2 -	(n—2 )bc	—	(n—3)bd	
na	—n(n—2 )ac —	n(n—3 )ad	—	n(n—4 )ae	
n2a	n2ac +	n2ad	+	n2ae	+
{(w —l)62—2nac! +{(n— 2)bc— 3nad\ -H(n —-3)6d—^nae) +..
Consequently
Xs=(«-1)62	£n-2+(w—2)6c	xn'	~3 + (n—3 )bd
—2 nac	— 3 nad		— 4 nae
And from this general form is immediately derived the
following mode of computing the coefficients of X2 in any
particular case, namely,
Arrange the coefficients of X! in a row, with their proper
signs. Underneath these write the coefficients of X, com-
mencing with the second, after having multiplied them in
order by the numbers 1, 2, 3, 4, &c., taking care however
14210
MATHEMATICAL DISSERTATIONS·
to put down all the results of these multiplications, except
the first, with changed signs. We shall thus have two
rows of figures of equal extent.
Cut off the leading term in each row for multipliers,
and combine them with the other figures, as at page 199 ;
that is, the under multiplier with the upper row, and the
upper multiplier with the under row ; and the step will be
completed by adding the results.
Note. If the second term of X be zero, that is, if
6=0, in the preceding general expression for X2, the
coefficients in that expression will be reduced simply to
X2 = — 2 nac — 3 nad — Anae — ....
or, still more simply, to
— 2 c* — 3d — Ae —______
so that to obtain the coefficients in this case we shall
merely have to multiply the coefficients of X, commencing
with the third, (or the second that actually appears) by
— 2, —3, —4, &c. respectively.
It is scarcely necessary to add that, like as in the gene-
ral process, if the multipliers have a common factor, it is
to be expunged.
And moreover, if the numbers in the lower row, that is,
the coefficients, after the first, in the proposed equation, are
found to have a common factor when the multiplications
by 1, 2, 3 have been performed, that factor may be neg-
lected.
(125.) We shall apply these rules to the examples given
in the Theory of Equations ; as the work for the common
measure is not there exhibited at length, except in one
instance.THE THEOREM OF STURM.
211
1. Let the proposed equation be
x3 — 5x2 +8* — 1=0
Xj = 3x2 — lOx + 8,
and by the rule
3 — 10 +8
—5 — 16 +3
50 —40
—48 +9
2—31
.·. X2 = 2* — 31
2. Let the equation be
*3 + 11 ¿c* — 102* +181=0
.·. Xj = 3x2 + 22* — 102,
and proceeding by the rule
3	-f22—102
11 + 204 —543
242 — 1122
612 — 1629
854—2751
.·. X2 = 854*—2751.
(126.) From these two examples the student cannot fail
to recognize the extreme simplicity to which everything
connected with the complete solution of a cubic equation212
MATHEMATICAL DISSERTATIONS.
is now reduced. The means by which X2, and thence the
sign of the final function X3 is obtained, are so easy, and
the computations so very trifling, that one would scarcely
be prepared to expect that such simple operations actually
comprehend all the work essentially necessary in seeking
the common measure of X and X2.
In this improved method indeed all analogy to the ori-
ginal course of proceeding seems to have disappeared;
and it might be taken for an entirely independent rule for
separating the roots of cubic equations. In conjunction
with Mr. Horner’s process of continuous approximation
to the roots, nothing farther can surely be desired in the
way of simplifying the solution of equations of the third
degree; although but a few years ago the complete ana-
lysis of this class of equations was numbered among the
difficulties of algebra.
3. Let the equation be
*3 — lx + 7 = 0,
which is an equation of some celebrity on account of the
difficulty attending the separation of the roots by the older
methods.*
Here,
Xj = 3x2 — 7,
and by the Note
X2= 14*-21,
or
2x — 3
.·. X. = +,
and the roots may now be separated with the greatest ease.
* See Lagrange. Traité de la Résolution des Equations Numériques,
pp.33 and 237.	1826.THE THEOREM OF STURM.
213
4. As another example of this kind, let the equation be
8a?3 — 6a? — 1 = 0.
As before, the functions may be written at once from
mere inspection, without any numerical work ; they are
X = 8r* - 6a? - 1
Xj = 4a?2 - 1
Xa = 12a? + 3
or
4a? + 1
X3 — +.
As a last application of our method to cubic equations,
we shall take the following, which has been selected by
Mr. Lockhart on account of its difficulty; and which
that gentleman has analysed at length, by a combination of
the methods of Newton and Budan, in the work before
referred to.*
5. X = 12a?3 — 120a?2 + 326a? — 127 = 0
Xx = 36a:2 - 240a? + 326.
Consequently, by the rule,
Resolution of Equations*, p. 26. Oxford, 1837.214
MATHEMATICAL DISSERTATIONS.
3*
36	—240 + 326
— 10
— 120	—652 + 381
2400 — 3260
-rl956 -f 1143
444—2117
X2 == 444a? — 2117
/. X3 =
Hence the proposed equation has two imaginary roots.
And in this manner may the character of the roots of a
cubic equation always be ascertained with very little
trouble. We shall proceed to biquadratic equations.
6. Let the equation be
X = x'4 — 2a:3 — 7a*2 4- 10a: -f 10 = 0
/. Xj = 2a:3 — 3a2 —lx 4- 5,
and by the rule
2 —3 —7 -f5
—2 4-14—30—40
6 + 14—10
28—60—80
34—46—90
* As 36 and 120 are both divisible by 12, leaving for quotients 3 and 10,
we have employed these for multipliers instead of the original numbers; and
have placed them in small characters over the latter. But in practice it will
be better to expunge these, putting the smaller numbers in their place.THE THEOREM OF STURM.
215
As the multipliers here have a common factor 2, we ought
to have expunged it, and to have employed the multi-
pliers 1 and — 1. When properly reduced therefore {he
operation is as follows:
1
— 1
—3 —7 +5
+ 14—30—40
3 +7 —5
14—30—40
17—23—45
X2= 17a:2 — 23# — 45.
We might now, commencing with this function and the
preceding one Xi, make the determination of X3, the first
step of the general rule; and this indeed is the only
step requisite. To separate the roots of a biquadratic
equation therefore can never be a work of much labour or
difficulty. But this will be still more distinctly shown in
Note II. at the end.
As an application of the precept in the Note to an equa-
tion of the fourth degree, let us take
7.	X =2x4 —13x2 + 10a:—19
X, = 4a:3 — 13a: + 5
and by the Note
X2 = 26a:2 —30a: + 76
or
13a:2 — 15a; + 38
and since in this latter function
4 x 13 x 38 > 152
the operation terminates here.216
MATHEMATICAL· DISSERTATIONS.
Asa final example we shall now show the amount of
abridgment effected by our rule in the first step of the
operation when the equation is one of the sixth degree;
taking for that purpose the equation at page 203, namely,
8. a?6 — 12a:5 + 60a;4 + Ox3 + 123a;2 + 4567a; — 89012 = 0
/. X, = 6a;5 — 60a;4 + 240u,3 + Oa;2 + 246a; + 4567,
and by the rule
—60+240	+ 0 +246 +4567
—120 —0	—492—22835+534072
120—480 — 120 —0	— 0 —492 —9134 —492—22835+534072
0—480	—492 —23327 +524938
Consequently
X2 = — 480a;3 — 492a;2 —23327a; + 524938.
And commencing now with this function, and the pre-
ceding one Xi, we may, as at page 204, proceed with the
process by the general method.
(127.) We here terminate what we had to deliver upon
the subject of Sturm’s Theorem; simply adding two
Notes, devoted to some details essentially necessary to
the completion of our purpose in the present Essay, but
which we have thought it desirable to postpone till the
end.
The first of these notes briefly points out a relation be-
tween Sturm’s coefficients and those which appear inTHE THEOREM OF STURM.
217
Lagrange’s equation of the squares of the differences;
and the second enters into an examination of the efficacy
of the abbreviated method in those very delicate cases,
where roots, either accurately equal or differing by only a
minute quantity, exist in the equation.
The latter of these cases, namely, that in which several
roots differ but very little from one another, is that which
invariably proves a source of perplexity, if not of insur-
mountable difficulty, whenever the analysis of the equation
is attempted by any method except that furnished by
Sturm. And it may at first seem questionable whether
even that method, when modified and curtailed as we have
recommended, may not allow some particulars to elude
detection, which the unabridged, but more laborious pro-
cess, would fully develope. It is of consequence there-
fore to prove that our abbreviations may be as safely em-
ployed in these critical cases as in those wherein the roots
have no remarkable proximity to one another. Note II.
is chiefly occupied with establishing this truth.218
Note I. (page 170).
On the Equation of the Squares of the Differences, in
connexion with Sturms Functions.
(128.) There is a very intimate connexion between the
coefficients of Sturm’s functions and those of the equa-
tion of the squares of the differences proposed by Waring
and Lagrange ; both, in fact, resting ultimately on the
same principle, namely, the principle involved in the pro-
cess for seeking the common measure of two algebraic
functions.
In the methods of Waring and Lagrange, however,
the number of conditions, for determining the character of
the roots of an equation, speedily becomes so great and
so complicated in their nature, as to be of little practical
value in the solution of numerical equations. As remarked
in the text, the conditions for determining the character of
the roots of an equation of the fifth degree are ten in
number. They are exhibited at length by Waring, in
the Philosophical Transactions for 1763. This eminent
analyst was unquestionably the first who proposed the
equation of the squares of the differences as a means for
discovering the number of real and imaginary roots in any
proposed equation.
The ten conditions which Waring showed to be im-
plied in this equation of the squares of the differences,
when the proposed is of the fifth degree, are also given by
Lagrange at page 108 of the work referred to; a glance
at them will be sufficient to show that their determination
must involve an enormous amount of numerical labour.THE THEOREM OF STURM.
219
Lagrange, however, with his usual penetration, sus-
pected that some of these conditions might be embodied
among the others ; and there is little doubt, if everything
absolutely unessential about them could be ascertained
and suppressed, that we should find the resulting con-
ditions thus modified to be the same in effect as those im-
plied in Sturm’s functions.
(129.) Lagrange’s conditions for the equation of the
fourth degree,
æ4 -f Br* — Cx -f D = 0,
are as follow,
a = — 8B
b = 22B2 + 8 D
c = - 18B3 + 16BD + 26C2
d = 17B4 + 24B2D - 7.16D2 + 3.16BC2
e = - 4B5 - 2.27C2B2 - 8.27C2D + 3.43BD2 — 2.42B3D
ƒ = 44D3 - 23.42B2D2 + 42.32C2BD + 42B4D - 4C2B3 - 33C4.
If all the quantities a, 6, c, dy e9f are positive, the roots
will all be real ; and they will all be imaginary if the quan-
tity f is positive, and any of the others negative. Now
Lagrange finds that ƒ being positive, all the other quan-
tities will be so too, provided there exist at the same time
B < 0, and B2 — 4D > 0.
These, therefore, in conjunction with the condition f > 0,220
MATHEMATICAL DISSERTATIONS.
are all the conditions essentially necessary to warrant the
inference that the roots are all real.
In order that some one of the quantities a, b, c, d, e
may be negative, and consequently the four roots ima-
ginary, the only essential condition is
B > 0, or B2 — 4D < 0.
In the solution adopted in our general example of the
fourth degree in the text, a == B, 6 = — C, c = D; making
therefore these substitutions in the expression for ƒ, we
have
ƒ = 4V —' 23.4W + 42.3Vac + 4Vc - 45V -336\
and the first pair of conditions above is
a < 0, and a2 — 4c > 0.... [ 1 ]
which must exist, in conjunction with f > 0, if the four
roots are real.
Now, by referring to Sturm's functions, page 166, we
see that in order that the roots may all be real, we must
have the conditions.
X4 = 4cA2 - 36AB + 2aB2 > 0,
a < 0, and 8ac — 2a3 — 962 > 0.... [2]
Computing the first of these expressions, in terms of the
coefficients a, 6, c we find it to be
4Vc3 - 2\42aV + 42.3262a3c + 4Ve - 46V - 3W,THE THEOREM OP STURM.
221
or, dividing by the essentially positive quantity a2,
4V - 23.4W + 42.32ò2ac + 4Vc - 4Va* - 33ò\
which is the very same as Lagrange’s ƒ.
(130.) But the third condition [1], determined by
Lagrange, is undoubtedly simpler than that [2] imme-
diately given by Sturm ; yet the latter is reducible to the
former ; for
— 2a3 + Sac — 9b2
a2 — 4c
— 2a —
9 b2
a2 — 4c *
and it is impossible that a — 4c can be negative ; for as
— 2a3 + 8ac — 9b2 is by hypothesis positive, and a nega-
tive, both terms of the second member of this equation
would be necessarily positive, while the first member of
the equation would be negative.
It follows therefore that when [2] exists [1] must ne-
cessarily exist also ; and conversely.
Hence, that the roots may all be real, we must have the
conditions
a < 0, a2 — 4c > 0, ƒ > 0,
and, that they may all be imaginary, we must have, in
conjunction with f > 0, either
a > 0, or a2 — 4c < 0.
Lagrange’s equation of the differences is an equation
in y, got by eliminating x from the proposed equation
ƒ(*) = <>....[A]222
MATHEMATICAL DISSERTATIONS.
in conjunction with the equation
/'(*) + ƒ" (*) A + /'"(X) jig + .... =0 ... . [B].
(See Theory of Equations, p. 214.)
This elimination is effected by seeking the common
measure of [A] and [B], the process terminating when we
arrive at a remainder in y independent of x: this* it is
obvious, will require just as many steps as would bring us
to a remainder independent of x when for [A] and [B] we
substitute the functions
f(x) and ƒ'(a:)
and it is further obvious that this latter remainder would
be that which Lagrange’s remainder would become for
y = 0. But the remainder spoken of, disregarding sign,
is that which constitutes the final function in Sturm’s
theorem. Consequently, with the exception of the sign,
Lagrange’s equation of the differences, which is only his
final remainder in y equated to zero, will furnish Sturm’s
final function when the terms in y are expunged; that is,
the constant term in the equation of the differences, and
which is also the constant in the equation of the squares
of the differences, is, abstracting from the sign, the
same as the constant or final term in Sturm’s series of
functions.*
Lagrange’s constant term in the equation of the
squares of the differences for the general equation of the
fourth degree before quoted, is ƒ; and we have already
* It is proper to mention that this circumstance is also noticed in a paper
entitled “ On the Existence of a Relation among the Coefficients of the Equa-
tion of the Squares of the Differences of the Roots of an Equation/’ pub-
lished in No. VI. of the Cambridge Mathematical Journal, a work containing
much instructive matter, and one which students of analysis will do well to
consult.THE THEOREM OF STURM.
223
seen, without reference to this general truth, that f is the
same function of the coefficients as X4, when the factor a2,
introduced at the outset, is suppressed.
In like manner the result of every step of the operation
with [A] and [B], that is, the successive remainders will,
except sign, become the very same as the corresponding
remainders of Sturm ; provided that in each of the former
y be made zero.
(131.) No one has ever attempted to exhibit the con-
ditions for ascertaining the number of the roots beyond
the point reached by Waring ; for Lagrange’s admis-
sion, before quoted in the text, that for the fifth degree
even, the computations requisite for this purpose are ex-
cessively laborious, applies as well to the deduction of the
general expressions themselves, as to the numerical calcu-
lations implied in them. But by Sturm’s theorem, carrying
on the work of it agreeably to the type we have proposed,
the deduction of the general expressions for the sixth,
seventh, eighth, or even higher degrees, is a process of
but a very moderate degree of labour or difficulty. If the
equation of the squares of the differences was ever of any
practical value, the general functions of Sturm must be
of incomparably more value.
(132.) It is pretty obvious that the process for obtain-
ing the equation of the squares of the differences may be
conducted in a manner similar to that which has been em-
ployed to deduce Sturm’s functions. As an illustration,
we shall take the example given at page 215 of the Theory
of Equations, remembering that here the remainders are
not to have their signs changed.
In the example referred to we are required to find the
common measure of the functions.
X	= a3 + px -(- q.
XI	= 3a2 -f 3yx + if + p,224
MATHEMATICAL DISSERTATIONS.
or, rather, the final remainder which the operation for the
common measure leaves.
Taking only the coefficients of x and its powers, the
work is as follows :
3—3 y—(y*+p)
X
1+0 +p +q
—y 3if+yijf+p)
i —(y*+p)
3	3p+3q
2(y3+p)+(y3+py+3?)
2(y3 +p)—(y3+py+3q)
X
3 + 3 y + (y3+p)
3 (y3+py—3q) — 3(y3 +py — 3q)(y3+py + 3q)
4 (y2+P)2	4(y2+p)3
4 (y3+pf— 3(y3 +py—3q)(y3 +py + 3 q).
Hence
4iy3+p)3—3(y3+py—3q)(y3+py + 3q)=0
is the final equation in y sought.
In a similar manner may the remainder, independent of
x, be obtained in the general theory of Elimination. (See
Theory of Equations, Part II. Chap. III.)225
Note II. (page 189.)
On Equations containing Equal Roots, or Roots nearly Equal.
(133.) In investigating the theory of numerical equa-
tions, by whatever method, the greatest, and almost the
only difficulty of consequence, with which the analyst has
to contend, is one which is inseparably connected with the
existence of equal, or nearly equal, roots. In the ordinary
methods this difficulty is increased in consequence of those
methods furnishing no ready criterion for distinguishing
such roots from those that may be imaginary; since the
same indications often present themselves, whether the
roots be actually equal, close together but unequal, or
imaginary.
The unabridged process of Sturm leads to perfectly
conclusive information upon all these particulars. And it
is plain that even the results of the abbreviated method can
never leave us in any doubt as to what roots are real and
what imaginary. The precluding of all doubt on this head
we cannot but regard as the perfection of any method for
analyzing a numerical equation. If the number of the
imaginary roots, and the leading figures of all the real
roots, are infallibly discovered in all cases, everything which
calculation can require, such a method, in conjunction with
a sure process for the indefinite development of each root,
is competent to supply.
The abbreviated mode of conducting the work of Sturm’s
theorem is clearly infallible in supplying the leading
figures of all the real roots, be they equal or unequal; and
Mr. Horner’s process of continuous approximation makes
known the subsequent figures of each root to as remote
15226
MATHEMATICAL DISSERTATIONS.
a decimal as we please, when such root is incom-
mensurable.
If from Sturm’s theorem we learn that two or more
roots commence with the same figures, Horner’s further
development will either separate them, if they are se-
parable, or will show, up to any extent of decimals we may
assign, that the roots continue equal.
To approximate to a root to the extent of seven or eight
places of decimals, even in an equation of the seventh or
eighth degree, is certainly not a formidable task; and if
two or more roots keep together throughout this range,
they may, no doubt, be pronounced equal, even supposing
that they really separate at a decimal still more remote.
If, however, we desire to bring the matter to an infal-
lible test, we may do so by restoring the latter steps of
Sturm’s theorem to their unabridged form; when the
sought information will of course be obtained.
But this method of proceeding, if it were absolutely ne-
cessary, would imply an imperfection in the abbreviated
process, from which it is in reality free. We advert to it
simply because it is the most obvious mode of removing
the doubt in question; and because, moreover, as the final
function tends to zero, the equation, in the case supposed,
contains within itself a check to the advance of the figures
in the closing steps, usually of sufficient power to render
extraneous restrictions unnecessary; so that the work of
these steps might be carried on without abbreviation, and
yet without any risk of a continued increase of figures, as
in the case of unequal and widely different roots.
We shall show, however, in the course of this note, that
our abbreviated results are in all cases fully effective; and
that the corrections, here adverted to, are not at all neces-
sary to the completeness of the method.
(134.) As an example of equal roots, we shall give the
following, from the work of M. Midy before quoted:THE THEOREM OF STURM.
227
X = a;6 - 7a;4 + 4a;3 + 7a;2 - 2a; - 2 = 0
Xj = 3x5 — 14a;3 + 6x2 + 7a; — 1
3 + 0 +14 -6 —7	+1
X
1 + 0 —7 +4 +7 —2 -2
14 -6 —7 +1
—21 +12 +21 -6 —6
7	—6 -14	+5	+6
7+6 +14	-5	-6
X
3-0 — J 4	+6	+7	-1
18
21
49
108 + 252 -90—108
294—105—126
— 686+294 + 343 —49
284—441 — 127 + 157
284 + 441 + 127—157
X
7-6	—14 + 5+6
1383
1988
2845
609903 + 175641—217131
252476—312116
- 1129184+403280+483936
266805-266805-266805
or
1-1-1
1+1 +1
X
284—441 — 127 + 157
-157
284
1
— 157 —157
284
-127 +157
0 0228
MATHEMATICAL DISSERTATIONS.
Consequently,
X	= x6 — lx4 -h 4a:3 + 7xa — 2x — 2
XI	= 3a;5 — 14a:3 + 6a;2 + lx — 1
X2 = 7a:4 — 6x3 — 14a:2 -f 5x -f 6
X3 = 284a:3 — 441a:2 — 127a: + 157
X4 = x2 — x — 1
X5 = 0.
The equation proposed has therefore two pairs of equal
roots. They are contained singly in the equation
x2 — x — 1 = 0,
and are therefore easily determined. Dividing the func-
tion X by (a;2 — x — l)2, or
a:4 — 2a:3 + x2 + 2a: + 1,
the quotient is x2 + 2x — 2. Hence the remaining roots
of the proposed equation are those of the quadratic
a:2 -f 2a: — 2 = 0.
In this example a very decided indication of equal roots,
or of a function by which the original polynomial is ac-
tually divisible, is furnished at the close of the third step
of the work. And as our abbreviations never commence
till the numbers in the course of derivation threaten to
become inconveniently large or unmanageable, in a case
like this these abbreviations would never be introduced;THE THEOREM OF STURM.
229
but the work would be allowed to proceed, unabridged, to
the close.
(135.) It will often happen, however, in applying our
method to the higher equations, that, on account of the
increasing magnitude of the numbers, we shall commence
our abbreviations before these indications of equal roots
can discover themselves, these latter occurring at a more
advanced step, and after we have ceased to attend to the
common factors in our several results for the purpose of
reducing their magnitude.
We shall give an example of this, and then show,
finally, how, by disregarding all these indications, and car-
rying on our process up to the close of the work, as in the
examples already given in the text, we may obtain, free
from every ambiguity, a complete analysis of the equation
under examination.
The following equation is quoted by Cagnoli from a
Memoir on Equations by Ruffini, whose work, however,
we have not seen.
Required the analysis of the equation
x6 — 16x5 + 85x4 — 144x3 — 57x2 + 126x + 54 = 0.
We shall work the first step of this equation conformably
to the shorter method explained at p. 209.
The coefficients in X and Xx, those of the latter function
being divided by 2, are
1 _ 16 + 85 — 144 — 57 + 126 + 54
f
3 _ 40 + 170 — 216 - 57 + 63,
and the work is as follows:230
MATHEMATICAL DISSERTATIONS.
3 —40 +170 —216	—57	+63
— 16 —170 +432 +228 —630 —324
640—2720 + 3456 +912—1008
—510 + 1296 +684—1890 —972
130-1424+4140 -978-1980
or +-2
65—712 +2070 —489 —990
65 + 712 -2070 +489 +990
X
3—40	+170 -216 —57+63
-464
195
652
—330368 +960480-226896-459360
—403650 +95355 + 193050
718250-912600-240825 + 266175
15768—143235+274671+193185
or +· 33
584	—5305 +10173	+7155
584+5305	—10173	—7155
X
65-712	+2070	-489—99
-121*5462
65
584
— 644803 +1236489 + 869663
-661245 -465075
1208880 —285576—578160
97168 -485838—291503
97168+485838 +291503
X
584—5305+10173+7155
-2385
584
97] 68
— 11587—6952
1702
9885 + 6952
0 0THE THEOREM OF STURM.
231
From this last result we may conclude that the equation
has equal roots, these roots being contained singly in the
preceding function equated to zero. This preceding func-
tion must therefore be a divisor of the original polynomial;
and consequently the numbers, at present involved in it,
must admit of a very large common divisor. In fact, as the
leading coefficient of the original polynomial is unity, the
leading coefficient of the reduced divisor must be unity
also : hence the three numbers in the function referred to
must be divisible by the first of those three. Actually
dividing them by this first, namely, by 97168, we find for
our divisor the expression
x2 — 5x — 3.
It is true 'that 5 times our divisor do not rigorously give
485838, but 485840, differing from it by two units; nor
do 3 times the same give 291503, but 291504. These
minute differences are of course to be expected, on account
of the rejected figures to the right. But we conclude,
with as much confidence as if these figures were actually
preserved, that the function in question, w hen reduced by
dividing by the leading coefficient, can be no other than
that just obtained.
There may, however, seem to be a bare possibility that
our final result might turn out to be a minute finite quan-
tity, had there been no curtailments; and that therefore,
however near to equality certain roots may be, they may
nevertheless not be rigorously equal; nor, in consequence,
the reduced expression above rigorously a divisor of the
original polynomial.
But every scruple of this kind is removed by the final
step which completes the analysis of our equation; that is,
by the actual division of X by x2 — 5x — 3, or rather by
the square of this.232
MATHEMATICAL DISSERTATIONS.
The result of this divisiofi is
x2 — 6a: 4- 6,
and hence our equation is made up of the three quadratics
a:8 — 5# — 3 = 0
x2 — 5x — 3 = 0
x2 — 6x +6 = 0
and therefore its six roots are
5±V 37	5±^37 0 ^ /Q
----o---->	----o----' d ± V
If our final result had not been accurately zero; and, in
consequence, the roots supposed equal had really a minute
difference, the circumstance would have been unequivocally
made known by the occurrence of a remainder after the
division by the supposed common measure, or complete
divisor employed above.
(136.) It is this that removes all objection to the abbre-
viations which so greatly reduce the labour of Sturm’s
theorem. We need never remain in any perplexity at the
close of the abbreviated process, as to whether, from the
character of our final result, roots absolutely equal are in-
dicated, or roots differing from each other by only a
minute quantity. If the numbers in this final result have
wholly disappeared, or have dwindled down each to a mere
unit, or at most to two or three units, the great probabilityTHE THEOREM OF STURM.
233
is that equal roots exist. If such be the case, the imme-
diately preceding function may be so reduced, by the sup-
pression in it of a common factor, that the extreme coeffi-
cients in it may become divisors of those in the original
polynomial, each of each. If, however, such common
factor does not exist, we conclude at once that the roots
whose character we seek are not equal, but only nearly so;
and this preceding function, just adverted to, will supply
us with the leading figures of those roots, each root en-
tering twice into the original equation.
The number and situations of all the real roots are then
to be determined as follows ; the leading figures, that is,
the situations of the roots of the function last deduced
being known :
Suppose this last function, or that furnishing the first
figures of the nearly equal roots to be Xw; and, as usual,
write the series
X, Xj, X2 .... xm.
Then, since we know between what numbers all the roots
of XOT are situated, we know how often XOT vanishes, in
substituting the numbers 0, 1, 2, ... oo, and the num-
bers 0, — 1, — 2, ... — oo, for ,r.* Consequently, we
always know how many variations are lost (or gained) in
the series
X., X.
+ 1 '
* It will be remembered that when we know the number of roots which any
intermediate function has between a and b, we at the same time know how
many variations are lost in the series, from this function to the end, by the sub-
stitution of a and 6. For these lost variations, by the nature of Sturm’s
theorem, must equal in number the roots spoken of; so that, knowing the
places of these roots, we shall have no occasion for the subsequent functions
themselves.234
MATHEMATICAL DISSERTATIONS.
up to the end; without requiring the actual exhibition of
this series. This follows from the very nature of Sturm’s
theorem. Hence the number and situations of the real
roots are easily determinable without the aid of the func-
tions beyond Xm.
As to the situations of the roots of Xm = 0, supposed to
be previously ascertained, they may always be discovered
with but little trouble. If Xm = 0 be a simple equation, a
mere inspection of it is sufficient; if it be a quadratic, the
increase of labour is very trifling; and the same may be
said in the case of XTO = 0 being a cubic equation, pro-
vided we adopt in the enquiry the short method explained
at p. 209. It would be useless to advert to the case of
Xm = 0 being an equation of the fourth degree; or to oc-
cupy ourselves about supposed equations having eight
roots differing from one another only at an advanced
figure.
Let us suppose, now, that the coefficients in Xm really
have a common factor, and one of sufficient magnitude to
reduce the leading and final numbers in Xm to the small-
ness of factors of the corresponding numbers in X. Divide
X by the square of XTO thus reduced. If the remainder is
zero, Xm = 0 furnishes singly the equal roots of the pro-
posed equation. If the remainder is not zero, the same
equation will, as before, involve singly the roots nearly
equal; and we shall conclude that none absolutely equal
has place in the equation.*
* We all along presume that the common divisor, when such really exists,
has not itself equal roots. When this divisor is of no higher degree than the
second, an inspection of the coefficients in it is sufficient to make known whe-
ther its simple factors are equal or not. But, when the divisor is of a higher
degree than the second, its composition cannot be so readily discovered. In
this case, therefore, we should divide X simply by Xm, and not by its square,
since X is divisible by the square of Xm only when the roots of Xm = 0 are
unequal. The greatest common divisor of X and X! is always made up of theTHE THEOREM OF STURM.
235
(137.) We have now pointed out the method of pro-
ceeding, in order to remove all ambiguity in a case, the
actual occurrence of which in practice we regard as in the
highest degree improbable; the occurrence, namely, of
roots so exceedingly close together, as to render it doubtful
whether our final function, preserved true to four or five
places of figures, indicates, by the actual evanescence of
those figures, equal roots or not.
Let it be remembered that the function preceding this,
true to a still greater number of figures, must in case of
unequal roots imply the coincidence of these roots to the
extent of at least five or six places of figures. We can give
no examples of equations of this kind to illustrate the pre-
cepts just laid down, for we have never met with an equation
containing roots which keep thus far together and then se-
parate at a more advanced figure. Nor do we know in what
manner to frame such an example, at least one of a mode-
rately high degree, except by the introduction of very large
coefficients into the original equation. Nevertheless, the
method here explained provides satisfactorily even for ex-
treme cases of this kind; so that the abbreviated mode of
deducing Sturm’s functions, while it renders his theorem
practicable in its application to the higher equations, pre-
multiple factors of X, each raised to a power a unit lower than that with which
the same factor enters X.
If in the case of roots very nearly equal, that is, in the case where all the
coefficients in the function next to Xm become nearly evanescent, it should so
happen that there are also roots in XOT = 0 so nearly equal as to render it doubt-
ful whether a pair of them may be real or imaginary, we should proceed as if
they were real, and determine, as explained in the text, the variations lost, in
the interval which comprises them, in both the partial series from X to Xm,
and from Xm to the end. If our supposition be correct, it will be verified by
these variations. When Xm is a quadratic, there should be a loss of four or
at least of three variations in the assumed interval, and the same when Xm is a
cubic, otherwise the roots tested are imaginary.236	MATHEMATICAL· DISSERTATIONS.
serves, at the same time, all the accuracy and certainty that
belong to the unabridged process.
(138.) It seems, then, that there remains but one more
point to be established; and which is, that in the case of
equal roots the function preceding the last, when the com-
mon factors are expunged, does really present coefficients
remarkably small; the first and last never exceeding the
first and last in the proposed equation, but always being
factors of them, each of each.
To prove this let the original polynomial be
axn 4- bxn~l + cxn~~2 -f	^	[1],
and let the function in Sturm’s series that divides thi
without remainder be
afxm -f- b'x”1"1 4- * — b! .... [2],
The coefficients in [1] have been deprived of all common
factors, so have those in [2]. This being borne in mind,
let it be supposed that in order to perform the division of
[1] by [2], and at the same time to preserve the quotient
free from fractions, it be necessary to multiply [1] by some
factor P. Let the quotient of the division, freed from
fractions by this preparatory operation, be
a"xp 4- b"x*~l 4-______kn______[3];
this quotient has of course no common factor; otherwise the
multiplier introduced into [1] would contain that factor andTHE THEOREM OF STURM.
237
would therefore be needlessly large. Now, as the operation
terminates without remainder, we have
PI x [3] = P x [1].
The second member of this equation is divisible by the
number P: hence the first member must be divisible by P.*
But neither [2] nor [3] are divisible by any number but
unity. Consequently P = 1.
It follows therefore that [1J is accurately divisible by [2]
without any preparation, giving a quotient free from frac-
tions. Hence a' is either equal to, or else a factor of a;
and kf either equal to or a factor of k.
As in the case of equal roots, entering in pairs, the
original polynomial is divisible, not only by the last of
Sturm’s functions, but also by the square of it, common
factors being suppressed, we see that the extreme numbers
in the original polynomial must be divisible by the squares
of the extreme numbers in the final function, each of each,
if such equal roots really exist. And, more than this, the
extreme numbers in every one of the preceding functions
must be divisible by the corresponding extremes of the
common divisor, although not necessarily by their squares.
It is scarcely necessary to add that when the division of
[1] by [2] leaves a remainder, the more minute this re-
mainder is, the nearer does [2] approach to a measure
(139.) We shall merely make one or two observations
in conclusion, chiefly for the purpose of showing the uni-
versality of a principle of useful application in the process
* If any rational and integral polynomial be divisible by a number P, then
one, at least, of the integral factors whose product give that polynomial must
be divisible by the same number P. (See Bourdon, Algebre, p. 614, edit.
1837.)238
MATHEMATICAL DISSERTATIONS.
for the common measure, but which has hitherto been re-
stricted to particular cases.
It is a condition stipulated for by Sturm, in the inves-
tigation of his theorem, that equal roots be first removed
from the equation. He afterwards shows, however, that
the presence of such roots will not neutralize his method,
although he gives no examples in which they enter.
Nearly all who have discussed and exemplified the theo-
rem, expressly exclude equal roots from their examples.
Now we would remark here that, with this restriction,
much of the labour which these authors bestow on their
examples might be spared. In all cases the derivation
might terminate when the quadratic function is reached ;
for it will be easy to discover, by a glance at the co-
efficients, if the roots of this quadratic are equal, and if so,
to find the common leading figure ; or if the roots are un-
equal, the places of each may be easily found by a few
trials ; after which all the necessary particulars may be
determined without the aid of the subsequent functions,
as explained above.
(140.) Sturm himself has not been inattentive to this
circumstance. For he observes, <c Whenever in seeking
the greatest common divisor of V and V2, we shall arrive
at a polynomial Yn (one of the second degree for example,)
which equated to zero, will give only imaginary values of
r, it will not be necessary to carry on the divisions further,
for this polynomial Yn will constantly be of the same sign
as its first term for all the real values of x, so that it may
be taken for the last of the auxiliary functions V2, V2, &c.
We might, indeed, even stop at a polynomial V„, which
would be destroyed for real values of x9 provided that we
could determine all these values. For representing by
p,q,r,....those of such values which would be comprised
between A and B, after having arranged them in the order of
their magnitude, commencing with the least, and observingTHE THEOREM OF STURM.
239
that Vn preserves the same sign for all values of x com-
prised between A and p, we should find, by the application
of the theorem, how many roots the equation V = 0 has
between A and p—uyu being a very small quantity; in
like manner, Vn having also a constant sign for all values
of x comprised between p and q, we should find how many
roots V=0 has between p + u and q — uy that is, between
p and qy by taking u sufficiently small: we should in a
similar manner ascertain how many roots V=0 has be-
tween q and r, and so on. This is on the supposition that
the equation V=0 has no equal roots, or that a value of x,
which destroys Vn, does not at the same time destroy V.
These cases, in which we may diminish the number of
auxiliary functions, deserve particular attention, for the cal-
culations necessary for the determination of the functions
V2, Y3......are very long, especially when we arrive at the
last functions, in consequence of the magnitude of the nu-
merical coefficients.*
(141.) To this we may add that, without any hypothesis
as to equal roots entering the equation, we may in all cases
proceed upon the above principle, when we have reached
a quadratic function having imaginary roots; and we have
accordingly resorted to it in some of the earlier examples
in this dissertation, without any previous stipulation as to
the nature of the roots. Sturm does not appear to have
perceived the universality of this principle; but it may be
established as follows:
Suppose the quadratic function itself is a divisor of the
proposed polynomial X, then this function, being the only
common divisor of X and Xi, the division of X by it wrill
be a polynomial Xm, containing the remaining roots, which
roots will be all unequal.
Now it is plain, from the nature of the process, that the
Spiller’s translation of Sturm’s Memoire, p. 15.240
MATHEMATICAL DISSERTATIONS.
several functions or remainders derived from X„, as a pri-
mitive, will be respectively equal to the corresponding re-
mainders derived from X, divided each by the quadratic
function common to them all; which function is, by hy-
pothesis, always positive. Hence there must be the same
variations of sign in the functions arising from X as in
those arising from X*,. Consequently the former series of
functions will determine the number and places of the real
roots of X=0, whether the final quadratic, with imaginary
roots, divides X or not.*
That the function of the first degree, which would be
given by continuing the operation a step further, cannot
divide X is obvious; because, if so, it would also divide the
quadratic itself, which would thus have a real root. When
therefore the operation terminates in a quadratic with
imaginary roots, we may conclude that the proposed equa-
tion cannot have equal roots unless they be imaginary
also; and that the determination of the real roots is alto-
gether independent of this circumstance.
(142.) This is a useful truth, not only as respects the
method of Sturm, but in reference to the labour attending
the practice of the common measure in general. It is al-
ways easy to see whether or not the extreme terms of any
quadratic function give a product which exceeds the fourth
part of the square of the mean. If we find that they do
give such a product, then we shall know that it is useless
to extend the work; for if any common measure exist, it
must be the quadratic expression at which we have ar-
rived ; and in order that this may be a common measure,
the first term in it must be a divisor of the first term of
every one of the remainders preceding it, as well as of the
* The same reasoning will evidently apply whatever be the degree of the
function containing only imaginary roots; but the property is of little use
except in the case of functions of the second degree, on account of the diffi-
culty of applying the necessary tests to the expression.THE THEOREM OF STURM.
24]
first terms of the original polynomials; the last term must
in like manner divide the last terms of the same expressions.
The actual division however, by the supposed common di-
visor, is generally attended with but very little trouble; far
less than would be requisite to prolong the operation for
the common measure one step further.
(143.) We might extend the principle just established
to the case in which the quadratic arrived at in Sturm^s
method had real roots, without making any preliminary
condition as to equal roots. For whether or not this qua-
dratic function is itself a divisor of the proposed polyno-
mial may be ascertained as already directed; and, on the
supposition that the roots of the proposed are unequal, we
may discover their number and situations by help of the
places which the roots of this quadratic occupy, as ex-
plained at page 233.
But if this supposition should prove wrong, the equation
really having a pair of equal roots, which would have been
discovered by continuing the work beyond the quadratic,
then we shall find one real root unaccounted for by our
previous examination; so that we shall immediately infer
the existence of an additional root between one of the
pairs of limits which that examination makes known.
The proper pair is found by substituting each pair in the
original polynomial; for if the place of the wanting root
is between limits already comprehending an odd number
of roots, the substitution of these limits will give like signs;
but if its place be among an even number of roots, the
limits will give unlike signs. Its proper situation may
therefore be easily discovered.
(144.) In equations of a very high degree, those of the
ninth or tenth degree for instance, it will generally be
found convenient to make use of this principle $ postponing
our abbreviations to no later a period in the work than may .
16242
MATHEMATICAL DISSERTATIONS.
be necessary to provide against ambiguity in the function
of the second degree; to close the operation at that step,
and then to proceed as just described.
If, for example, we had adopted this course in the equa-
tion of the seventh degree, examined at page 195, and
had made the fourth step of the work the last, we might
have abridged that step more freely; computing it thus :
-1018178 + 664097	+ 8723—37487
X
28665 + 3098760+2009931+23618—113876
-3080074
-28665
1018178
—204547—2687 + 11546
-25 +107
204647 + 2405—11595
—75 +175	+ 49
The quadratic function in this case is therefore
X5 = — 75a;2 + 175a: + 49.
Equating this to zero, the roots are easily found to lie, one
between 0 and — 1, and one between 2 and 3. Hence the
functions
X5, x«> x7,
gain two variations from x — — co to i = +	, seeing
that the leading sign of X5 is minus.
Also the functions
X X1 X2 X, X4 X*
giving for x = — oo, — + — + + —
and for x = + <»:+ + + +------------THE THEOREM OF STURM.
243
lose three variations within the same limits. Consequently,
upon the whole, only one variation is lost between the
limits — oo and -f oo, in the entire series of functions.
X, Xi.... x7,
and therefore the equation has but one real root. The
place of this is found, from the original equation, to lie
between 2 and 3, as before.
The actual determination of the situations of tfye two
roots of X5 = 0 happens to be, in this case, superfluous;
as the equation turns out to have but a single real root,
the place of which is ascertained without any aid from the
derived functions. But when several real roots are found
to exist, then the actual situations of them can be dis-
covered only when we know the particular intervals be-
tween which the changes of sign in our final function
happen; that is to say, only when we know whereabouts
the roots of this final function are situated.
All this however has been sufficiently explained already.
(145.) We remark, finally, that the method of proceeding
just explained will, in most cases, greatly shorten the work,
even when no numerical abbreviations are introduced.
We have already seen, page 214, that a biquadratic
equation may be completely analyzed by employing the
general method of derivation in one step only. From what
has now been said, we see that even this step may be dis-
pensed with, without any danger of committing error from
the presence of equal roots, whether these be real or imagi-
nary. This is a circumstance well worthy of attention; and
it was with especial reference to the facilities thus intro-
duced into the analysis of equations of the fourth degree,
that we characterized the operation for separating the roots
of a biquadratic equation, at page 215, as one of so little
labour or difficulty.244
Mathematical dissertations.
In fact the observations in article (126), setting forth
the extreme simplicity to which the problem of analysing
a cubic equation is now reduced, may, with equal pro-
priety, be extended to equations of the fourth degree. For,
as in cubics so here, the single function JC2> in conjunction
with the two preceding ones, immediately given by the
proposed equation, is all that is indispensably necessary
for the separation of the roots ; and we have alreàdy seen
with what ease the function X2 may always be computed.
On this account the analysis and solution of equations of
the third and fourth degrees might with propriety be de-
tached from the general theory, and incorporated among
the elementary operations of common algebra.
Instead therefore of the Theorem of Stürm being im-
practicable beyond equations of the fourth degree, as some
authors have stated it to be, on account of the labour at-
tending the numerical computations, we may truly affirm
that, under the form in which we have here presented it,
it is not till we have passed this limit that any compu-
tation, worth mentioning, is really indispensable to the
complete analysis of the equation.
Even in equations of the fifth degree, the function of
the second order, at which we see the process of deriva-
tion may be made to terminate, may generally be reached
without any abbreviations being employed ; and yet with
, but a very moderate amount of numerical computation.
In equations of the sixth and of a more advanced
degree, it will usually be found expedient to resort to
the abbreviated method before the quadratic function is
arrived at. The only object of this abbreviated method
being to restrain the numerical results within manageable
bounds, it should of course always be delayed till these
results threaten to become inconveniently large ; and when
it is introduced it must be employed with that discretion
which it is requisite to exercise in all abbreviated methods;
that is to say, our curtailments must be made with suchTHE THEOREM OE STURM.
245
moderation that our abridged forms may still retain all
the important peculiarities of the complete functions them-
selves. The precepts and examples already given will, we
think, furnish guidance sufficient, in this matter for every
case that can occur in actual practice.
Having thus settled the point at which the abbrevia-
tions are to commence, we may then either carry on the
several steps of the work till it terminates of itself, as in
most of the examples given in the text; or else, conform-
ably to the directions in the present note, we may discon-
tinue the process of derivation when the function of the
second degree is obtained, and then complete the analysis
of the equation in the manner which has just been ex-
plained.
SUPPLEMENTARY NOTE ON THE BINOMIAL
THEOREM.
(146.) In the third edition of the Elementary Treatise
on Algebra, recently published, the author has offered an
investigation of the Binomial Theorem, which he conceives
to have some claims to originality. That part of the
proof, however, which establishes the law of the coeffi-
cients, and which commences at page 183, has appeared
before; at least so much of it as follows equation [8].
This he has virtually acknowledged at page x of the
preface to the work itself. Very soon after the publi-
cation of that treatise, the author perceived that the part of
the proof which he had borrowed might be superseded by a
much more simple and obvious method of deduction; and,246	MATHEMATICAL DISSERTATIONS.
as he has here a blank page at his disposal, he feels *5T
duced to occupy it with the improvement which then
gested itself.
From equation [8], page 183, we have
+ 2Q b + 3R b2 + 4S£* + &c.
Also from [6], page 182,
(\+b)m= 1 + mb+Qb2 + Rb3 + S64 + &c.
Hence, multiplying this by m and the former by (1 + i)»
and then equating the right-hand member of each, we
have
m -f m2b -f-	mQb2	-f wiRA3 -f- wiS64 -f- ?&c.
= m + (2Q + m)b + (3R + 2Q )b2+(4S + 3R )63+(5T + 4S)64 + &c.
Consequently
2Q -f- ?n = m2
3R-b2Q=mQ
4S -f*3R=mR
/.Q=-
.·. R =
.*. S =-
m(m— 1)
2
m(m—l)(ra—2)
273
m(m—l)(m—2)(m—3)
'	2.3.4
&c.	&c.	&c.
which establishes the law of the coefficients Q, R, S, &c.
Printed by C. Adlard, Bartholomew ( lose.ANALYTICAL COURSE OF MATHEMATICS,
BY J. R. YOUNG,
PROFESSOR OF MATHEMATICS; ROYAL COLLEGE; BELFAST.
I.
ELEMENTS of GEOMETRY ; containing a New and
Universal Treatise on the Doctrine of Proportion, together with Notes,
in which are pointed out and corrected several important Errors that
have liitherto remained unnoticed in the Writings of Geometers. 8vo, 8s.
AN ELEMENTARY TREATISE on ALGEBRA, Theo-
retical and Practical; with attempts to simplify some of the more
difficult parts of the Science, particularly the Demonstration of the
Binomial Theorem in its most general form ; the Summation of Infinite
Series, &c. The third Edition, 6s.
hi.
A KEY to the above. New Edit, by W. H. Spiller. 6s.
THE GENERAL THEORY and SOLUTION of ALGE-
BRAICAL EQUATIONS, in which are embodied the recent Re-
searches of Budan, Fourier, and Horner; together with a complete
Solution of the important Problem of the Separation of the Real and
Imaginary Roots of an Equation, under all circumstances ; being the
substance of the remarkable paper by M. Sturm, printed among the
Mémoires présentés par divers Savans à l'Académie Royale des Sciences
de l’Institut de France. 12mo, 9s.
ELEMENTS of PLANE Ind SPHERICAL TRIGONO-
METRY, with its Applications to the Principles of Navigation and
Nautical Astronomy, with the necessary Logarithmic and Trigonome-
trical Tables. By J. R. Young. (To which is added, some Original
Researches in Spherical Geometry ; by T. S. Davies, f.r.s.e. f.r.s.a.)
6s. cloth.
VI.
MATHEMATICAL TABLES ; comprehending the Loga-
rithms of all Numbers from 1 to 30,600; also the Natural and Lo-
garithmic Sines and Tangents ; computed to seven places of Decimals,
and arranged on an improved Plan ; with several other Tables, useful in
Navigation and Nautical Astronomy, and in other departments of Prac-
tical Mathematics. 6s. 6d. cloth.
VII.
AN ELEMENTARY TREATISE on the COMPUTA-
TION of LOGARITHMS. Intended as a Supplement to the various
Books on Algebra. A new edition, enlarged. 12mo, 5s.
Vili.
ANALYTICAL GEOMETRY, Part I. on the Theory of
the Conic Sections. Second Edition. 12mo, 6s. 6d.
IX.
ANALYTICAL GEOMETRY, Part II. on the General
Theory of Curves and Surfaces of the Second Order. Second Edition.
12mo, 7s. 6d.Souters List of Mathematical Works.
x.
ELEMENTS of the DIFFERENTIAL CALCULUS;
comprehending the General Theory of Curve Surfaces and of Curves of
Double Curvature. Second Edition. 12mo, 9s. in cloth.
Also an Octavo Edition for the Universities. 12s.
XI.
ELEMENTS of the INTEGRAL CALCULUS ; with its
Applications to Geometry, and to the Summation of Infinite Series,
&c. 9s. in cloth.
XII.
ELEMENTS of MECHANICS; comprehending the Theory
of Equilibrium and of Motion, and the first Principles of Physical
Astronomy, together with a variety of Statical and Dynamical Prob-
lems. Illustrated by numerous Engravings. 10s. 6d. in cloth.
XIII.
A LECTURE ox the STUDY of MATHEMATICS. 2s. 6d.
XIV.
A CATECHISM of ALGEBRA; Part I. Being an easy
Introduction to the first Principles of Analytical Calculation. 9d. sewed,
.or Is. bound.
xv.
A CATECHISM of ALGEBRA; Part II. 9d. sewed, or
Is. bound.
XVI.
EUCLID’s ELEMENTS; the first Six and the Eleventh
and Twelfth Books. Chiefly from the Texts of Simson and Playfair.
With Corrections, and an Improved Fifth Book; also Supplementary
Treatises on Plane Trigonometry, Incommensurable Quantities, the
Composition of Ratios, and the Quadrature of the Circle: together
with Notes and Comments. Designed for the use of Colleges and
Private Students. Second Edition, considerably enlarged, 5s, cloth.
Also lately published,
LACROIX’s ELEMENTS of ALGEBRA; translated from
the French. By W. H. Spiller. 12mo, 7s. 6d.
ON THE SOLUTION of NUMERICAL EQUATIONS.
By C. Sturm. Translated from the Mémoires présentés par divers
Savans à l’Académie Royale des Sciences de l’Institut de France, by'
W. H. Spiller. 4to, sewed, 7s. 6d.
AN ESSAY on MUSICAL INTERVALS, HARMONICS,
Ænd TEMPERAMENT ; in which the most delicate, interesting, and
most useful parts of the Theory are explained, in a manner adapted to
the comprehension of the practical Musician. By W. S. B. Woolhouse.
12mo, 5s.
A BRIEF TREATISE on the USE and CONSTRUC-
TION of a CASE of MATHEMATICAL INSTRUMENTS. By
George Phillips, b.a. Queen’s Coll. Cambridge. New edition. 2s. 6d.
JOHN SOUTER, 131, FLEET STREET, LONDON.