GENERAL METJIODS OF PHYSICAL INVESTIGATION. TAhE objeAt of all Physical Investigationlis to determine the effects of certain natural forces, such as gravity, cohesion, heats light and electricity. For this purpose we subject various bodies to the action of these forces, and note under what circumstances the desired effect is produced; this is called an experiment. Investigations may be of several kinds. First, we may simply wish to know whether a certain effect can be produced, and if so, what are the necessary conditions. To take a familiar example, we find that water when heated boils, and that this result is attained whether the heat is caused by burning coal, wood or gas, or by concentrating the sun's rays; also whether the water is contained in a vessel of metal or glass, and finally that the same effect may be produced with almost all other liquids. Such work is called Qualitative, since no measurements are needed, but only to determine the quality or kind of conditions necessary for its fulfilment. Secondly, we may wish to know the magnitude of the force required, or the temperature necessary to-produce ebullition. This we should find to be about 1000 C. or 212~ F., but varying slightly with the nature of the vessel and the pressure of the air. Thirdly, we often find two quantities so related that any change in one produces a corresponding change in the other, and we may wish to find the law by which we can compute the second, having given any value of the first. Thus by changing the pressure to which the water is subjected, we may alter the temperature of boiling, and to determine the law by which these two quantities are connected, hundreds -of experiments have been made by physi 2 ERRORS. cists in all parts of the world. The last two classes of experiments are called Quantitative, since accurate measurements must be made of the quantity or magnitude of the forces involved. Most of the following experimnents are of this nature, since they require more skill in their performance, and we can test with more certainty how accurately they have been done. Having obtained a number of measurements, we next proceed to discuss them by the aid of the mathematical principles described below, and finally to draw our conclusions from them. It is by this method that the whole science of Physics has been built up step by step. Errors. In comparing a number of measurements of the same quantity, we always find that they differ slightly fronm one another, however carefully they may be made, owing to the imperfection of all human instruments, and of our own senses. These deviations or errors must not be confounded with mistakes, or observations where a number is recorded incorrectly, or the experiment ilnproperly performed; such results must be entirely rejected, and not taken into consideration in drawing our conclusions. If we knew the true value, and subtracted it from each of our measurements, the differences would be the errors, and these may be divided into two kinds. We have first, constant errors, such as a wrong length of our scale, incorrect rate of our clock, or natural tendency of the observer to always estimate certain quantities too great, and others too small. When we change our variables these errors often alter also, but generally according to some definite law. When they alternately increase and diminish the result at regular intervals they are called periodic errors. If we know their magnitude they do no harm, since we can allow for them, and thus obtain a value as accurate as if they did not exist. The second class of errors are those which are due to looseness of the joints of our instruments, impossibility of reading very small distances by the eye, &c., which sometimes render the result too large, sometimes too small. They are called accidental errors, and are unavoidable; they must be carefully distinguished from the mistakes referred to above. Analytical and Graphical iMethods. There are two ways of discussing the results of our experiments mathematically. By the first, or Analytical Method, we represent each quantity by a letter, ANALYTICAL METHOD. 3 and then by means of algebraic methods and the calculus draw our conclusions. By the Graphical Method quantities are represented by lines or distances, and are then treated geometrically. The former method is the most accurate, and would generally be the best, were it not for the accidental errors, and were all physical laws represented by simple equations. The Graphical Method has, however, the advantage of quickness, and of enabling us to see at a glance the accuracy of our results. ANALYTICAL METHOD. 2lean. Suppose we have a number of observations, Al, A2, As, A4, &c., differing from one another only by the accidental errors, and we wish to find what value A is most likely to be correct. If A was the true value, A - A, A2 - A, &c., would be the errors of each observation, and it is proved by the Theory of Probabilities that the most probable value of A is that which makes the sum of the squares of the errors a minimum. Also that this property is possessed by the arithmetical mean. Hence, when we have n such observations, we take A = (Al + A + A3 + &c.). n, or divide their sum by n. Thus the mean of 32, 33, 31, 30, 34, is 160 5 -- 32. It is often more convenient to subtacl't some even number fiom all the observations, and add it to the mean of the remainder; thus, to find the mean of 1582, 1581, 1583, 1581, 1582, subtract 1580 fiom each, and we have the remainders 2, 1, 3, 1, 2. Their mean is 9. 5= 1.8, which added to 1580 gives 1581.8. Where many numbers are to be added, Webb's Adder may be used with advantage. Probable Error. Having by the method just given, found the most probab-b! value of A, we next wish to know llow much reliance Iwe nmay place on it. If it is just an even clhance that the true value is greater or less than A by E, tlhen Eis called its probable error. To find this quantity, subtract the mean from each of the observed vnalues, and place A, - A = e,, A. - A e,, &c. Now the theory of probabilities shlows that E.617/e,' +- e, + &c., + n, from which we can colllpute E in any special ease. As an example, suppose we have measured the height of the bl)rometer twenty-five times, atnd find the mean 29.526 with a probable error of.001 inches. Tlhen it is an even 4 PROBABLE ERROR. chance that the true reading is more than 29.525, and less than 29.527. Now let us suppose that some other day we make a single reading, and wish to know its probable error. The theory of probabilities shows that the accuracy is proportional to the square root of the number of observations, or that the mean of four, is only twice as accurate as a single reading, the mean of a hundred, ten times as accurate as one. Hence in our example we have 1: /25 =.001:.005, the probable error of a single reading. Substituting in the formula, we have the probable error of a single reading, E' = E X V/n =.67/el2 q+ e" + &c..' /n. It is generally best to compute E' as well as E, and thus learn how much dependence can be placed on a single reading of our instrument. Weights. We have assumed in the above paragraph that all our observations are subject to the same errors, and hence are equally reliable. Frequently various methods are used to obtain the same result, and some being more accurate than others are said to have greater weight. Again, if one was obtained as the mean of two, and the second of three similar observations, their weights would be proportional to these numbers, and the simplest way to allow for the weights of observations is to assume that each is duplicated a number of times proportional to its weight. From this statement it evidently follows that instead of the mean of a series of measurements, we should multiply each by its weight, and divide by the sum of the weights. Calling Al, A, &c., the measurements, and wI, w2, &c., their weights, the best value to use will be A = (Al i + A2 w2 + &c.) ( (ta- + w2 + &c.). We may always compute the weight of a series of n observations, if we know the errors el, e2, &c., using the formula w = n - 2(e,2 + e'2 + e32 + &c.). Substituting this value in the equation for probable error, we deduce E-.477 -' /nw if all the observations have the same weight, or E- =.477 - wi + w2 + &c., if their weights are WZ1 W2, &c. Probable Error of Two or More Variables. Suppose we have a number of observations of several quantities, x, y, z, and know that they are so connected that we shall always have 0 = 1 + ax + by + cz. If the first term of the equation does not equal 1, we may make it so, by dividing each term by it. Call the various values x assumes x', x", x"', those of y, y', y, y"', and those TWO OR MORE VARIABLES. 5 of z, z', z", z"', and so on for any other variables which may enter. If we have more observations than variables, it will not in general be possible to find any values of a, b and c which will satisfy them all, but we shall always find the left hand side of our equation instead of being zero will become some small quantity, e', e", e"', so that we shall have:e' = 1 + ax' + by' + cz', e" = 1 + ax" + by" + cz", e"' 1 + ax"' + by"' + cz"', and so on, one equation corresponding to each observation. These are called equations of condition. Now we wish to know what are the most probable values of a, b and c, that is, those which will make the errors e', e", e"', as small as possible. As before, we must have the sum of the squares of the errors a minimum. We therefore square each equation of condition, and take their sum; differentiate this with regard to a, b and c, successively, and place each differential coefficient equal to zero. These last are called normal equations, and correspond to each of the quantities a, b and c, respectively. The practical rule for obtaining the normal equations is as follows: - Multiply each equation of condition by its value of x (or coefficient of a), take their sum and equate it to zero. Thus x'(1 + ax' + by' + cz') + x"(1 + ax" + by". + cz") + &c. - 0, is the first normal equation. Do the same with regard to y, and each other variable in turn. We thus obtain as many equations as there are quantities a, b and c to be determined. Solving them with regard to these last quantities, and substituting in the original formula 0 = 1 + ax + by + cz, we have the desired equation. As an example, suppose we have the three points, Fig. 1, whose coordinates are x ='1, y' = 1, x" = 2, y" -- 2, x"' = 3, y"' = 4, and we wish to pass a straight line as nearly as possible through them all. We have for our equations of conditions: 0 = 1 + a + b, 0 = 1 + 2a + 2b, 0 = I 1 + 3a + 4b. Applying our rule, we multiply the first equation by 1, the second by 2, and the third by 3, the three values of x, and take their 1 sum, which gives 1 + a + b - 2 +- 4a + 4b + 3 + 9a + 12b- 6 + 14a + 17b = 0O. For our sec- Fig. 1. ond normal equation we multiply by 1, 2 and 4, 6 PEIRCE'S CRITERION. respectively, and obtain in the same way 7 + 17a + 21b 0. Solving, we find a- -1.4, b =.8, and substituting in our original equation 0 = 1 + ax + by, we have 0 = 1 - 1.4x +.8y, or y 1.75x - 1.25. Constructing the line thus found, we obtain MIGN, Fig. 1, which will be seen to agree very well with our original conditions. For a fuller description of the various applications of the Theory of Probabilities to the discussion of observations, the reader is referred to the following works. Methode des Moindres Carrees par Ch. Fr. Gauss, trad. par J. Bertrand, Paris, 1855, Watson's Astronomy, 360, Chauvenet's Astronomy, II, 500, and Todhunter's History of the Theory of Probabilities. A good brief description is given in Davies' and Peck's Math. Dict., 454, 536 and 590, also in Mayer's Lecture Notes on Physics, 29. Peirce's Criterion. It has already been stated that all observations affected by errors not accidental, or mistakes, should be at once rejected. But it is generally difficult to detect them, and hence various Criteria have been suggested to enable us to decide whether to reject an observation which appears to differ consid erably from the rest. One of the best known of these is Peirce's Criterion, which may be defined as follows: -The proposed ob servations should be rejected when the probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection, multiplied by the probability of making so many and no more abnormal observations. Or, to put it in a simpler but less accurate form, reject any observations which increase the probable error, allowing for the chances of making so many and no more erroneous measurements. Without this last clause we might reject all but one, when the probable error by the formula would become.zero. See Gould's Astron. Journ., 1852, II, 161; IV, 81, 137, 145. Another criterion has been proposed by Chauvenet, which, though less accurate than the above, is much more easily applied. It is fully described in Watson's Astronomy, 410.:Diferences. To determine the law by which a change in any quantity A alters a second quantity B, we frequently measure B when A is allowed to alter continually by equal amounts. Thus in the example of the boiling of water, we measure the pressure INTERPOLATION. 7 corresponding to temperatures of 0~, 10~, 20~, 30~, &c. Writing these numbers in a table, by placing the various values of A in the first column, those of B in the second, we form a third A. B. n. D". DJ". column, in which each term is 0o 4.6 found by subtracting the value + 4.6 of B from that preceding it; + 8.2 + 2.3 + 8.2 + 2.a the remainders are called the 20~ 17.4 + 5.9 first differences D'. In the + 14.1 + 3.4 30~ 31.5 + 9.3 same way we obtain the sec- + 23.4 ond differences D", by sub- 400 54.9 tracting each first difference from that which follows it, and so on. Interpolation. One of the most common applications of differences is to determine the value of B for any intermediate value of A. This is done by the formula, B - Bm+ n Dm' + n(nj - I)Dmf + a(n - 1) (n - 2)Dm l'+&c, 1. 12 1. 12.3 in which Bm is the measurement next preceding B; Dm', Dme," D,"', the 1st, 2d, 3d, differences, and n a fraction equal to (AAm)' (Am+l - Am), in which A, A, correspond to B, Bm, and Am+1 is the next term of the series to Am. The use of this A. B. D'. D". D".D"". formula is best shown by an 10 1000 example. Suppose, from the 11 + 331 11 1331 _ qt_ 66 accompanying table, we wish + 397 + 6 to find the value of B corre- 12 1728 + 72 0 sponding to A - 12.5. We 13 2197 469 + 6 have Bm = 1728, Dm' --- 469, + 547 - + 6 ff = 78, ),"'= B 61 = A 14 2744 +- 84 - 0 __Dm: — 78, Dimnt 6, Am + 631 + 6 12, Am + = 13, A = 12.5 15 3375 + 90 and n - (12.5 - 12) (13 16 4096 - 12) =.5. Hence, B =1728 +.5(469) + 5(-25)(78) +.5(-.5) (- 1.5)(6)+ 0 ~. 2 1. 2. 3 B = 1728 + 234.5 - 9.75 +.375 + 0 - 1953.125. In this particular case B is always the cube of A, and it may be 8 INVERSE INTERPOLATION. seen that our formula gives an exact result. The reason is that the 4th, and all following differences, equal zero. Inverse Interpolation. Next suppose that in the above example we desired the value of A for some given value of B, as B'; that is, in the equation, B' = Bm + nDm' + n(n - l)Dm"+ n(n - 1) (n - 2)DmI'+ &c. 1.2 1.2.3 we wish to find n. Evidently it is impossible to determine this exactly, but an approximate value may be found by the method of successive corrections. Neglect all terms after the third, and deduce n from the equation B'- Bm + nm'_ + (n. - 1)DM.2 which is a simple quadratic equation. Substitute this value of n in our first equation, and instead of B' we shall obtain another number a little greater or less, B". Now we have approximately B' = B" + fn'Dm' + n'(n' — 1) - 1. 2, in which n' is a very small fraction; this gives a still more accurate value of _B' when added to n and substituted in our original equation, and by continuing this process we finally deduce n with any required degree of accuracy. It is sometimes more convenient to neglect the third term, and deduce n from the equation B' = _Bm + nDm', which saves solving a quadratic equation, but requires more approximations. The values of n(n - 1). 1. 2, n(n - 1) (n- 2)' 1. 2. 3, &c., may be more readily obtained from Interpolation tables than by computation. A good explanation of this subject is given in the Assurance ]Magazine, XI, 61, XI, 301, and XII, 136, by Woolhouse. When the terms are not equidistant the method of interpolation by differences cannot be applied. In this case, if we wish to find values of B corresponding to known values of A, we assume the equation, B = a + bA + cA2 + dA3 + &c., and see what values of a, b, c, &c., will best satisfy these equations. If we have a great many corresponding values of A and -B, the method of least squares should be applied. In general, however, it is much more convenient to solve this problem by the Graphical Method de NUMERICAL COMPUTATION. 9 scribed below. See C(auchy's Calculus, I, 513, and an article in the Connaissance des Temps, for 1852, by Villarceau. Nurumerical C(omputation. Where much arithmetical work is necessary to reduce a series of observations, a great saving of time is effected by making the computation in a systematic form. In general, measurements of the same quantity should be written in a column, one below the other, instead of on the same line, and plenty of room should always be allowed on the paper. When the same computations must be made for several values of one of the variables, instead of completing one before beginning the next, it is better to carry all on together, as in the following example. Suppose, as in the experiment of the Universal Joint, we wish to compute the values of b in the formula, tan b = cos A tan a, in which A = 45~, and a in turn 5~, 10, 150~, &c. Construct a table thus: a 50 100 150 200 250 300 log tan a 8.94195 9.24632 9.42805 9.56107 9.66867 9.76144 log cos A 9.84948 9.84948 9.84948 9.84948 9.84948 9.84948 log tan b 8.79143 9.09580 9.27753 9.41055 9.41815 9.61092 b 30 32' 70 6' 10~ 44' 14~ 26' 14~ 41' 15' In the first lille write the various values of a, in the second the corresponding values of its log tan, and so on throughout the computation. An error is purposely committed in the above table to show how easily it may be detected. It will be noticed that the values of b increase pretty regularly, except that when a = 25~, and that this is but little greater than that corresponding to a = 20~. Following the column up we find that the same is the case for log tan b but not for log tan a, hence the error is between the two. In fact, in the addition of the logarithms we took 6 and 4 equal to 10, and omitted to carry the 1; log tan b then really equals 9.51815, and b = 18~ 15'. If the error is not found at once this value of b should be recomputed. Besides these advantages, this method is much quicker and less laborious. When we have to multiply, or divide by, the same number A a great many times, it is often shorter to obtain at once 1A, 2A, 3A, 4A, &c., and use these numbers instead of making the multiplication each time. This is useful in reducing metres to inches, &c. There are many 10 SIGNIFICANT FIGURES. other arithmetical devices, but their consideration would lead us too far from our subject. Significant Figures. One of the most common mistakes in reducing observations is to retain more decimal places than the experiment warrants. For instance, suppose we are measuring a distance with a scale of millimetres, and dividing them into tenths by the eye, we find it 32.7 mm. Now to reduce it to inches we have 1 metre = 39.37 in., hence 32.7 mm. = 1.287399. But it is absurd to retain the last three figures, since in our original measurement, as we only read to tenths of a millimetre, we are always liable to an error of one half this amount, or.002 of an inch. Then we merely know that our distance lies between 1.2894 and 1.2854 inches, showing that even the thousandths are doubtful. It is worse than useless to retain more figures, since they might mislead a reader by making him think greater accuracy of measurement had been attained. If we are sure that our errors do not exceed one per cent. of the quantity measured, we say that we have two significant figures, if one tenth of a per cent. three, if one hundredth, four. Thus in the example given above, if we are sure the distance is nearer 32.7 than 32.8 or 32.6, we have three significant figures, and it would be the same if the number was 327,000, or.00327. In general, count the figures, after cutting off the zeros at either end, unless they are obtained by the measurement; thus 300,000 has three significant figures if we know that it is more correct than 301,000 or 299,000. In reducing results we should never retain but one more significant figure than has been obtained in the first measurement, and must remember that the last of these figures is sometimes liable to an error of several units. Successive Approximations. This method is also known as that of trial and error. It consists in assuming an approximate value of the magnitude to be constructed, measuring the error, correcting by this amount as nearly as we can, measuring again, and so on, until the error is too small to do any harm. As an example, suppose we wish to cut a plate of brass so that its weight shall be precisely 100 grammes. We first cut a piece somewhat too large, weigh it and measure its area. If its thickness and density were perfectly uniform we could at once, by the rule of GRAPHICAL METHOD. 11 three, determine the exact amount to be cut off. As, however, it will not do to make it too light, we cut off a somewhat less quantity and weigh again; by a few repetitions of this process we may reduce the error to a very small amount. This method is sometimes the only one available, but it should.not be too generally used, as it encourages guessing at results, and tends to destroy habits of accuracy. GRAPHICAL METHOD. Suppose that we have any two quantities, x and y, so connected that a change in one alters the other. Then we may construct a curve, in which abscissas represent various values of x, and ordinates the corresponding values of y. Thus suppose we know that y is always equal to twice x. Take a piece of paper divided into squares by equidistant vertical and horizontal lines. Select one of each of these lines to start from. The vertical one is called the axis of Y; the other the axis of X;, and their intersection, the origin. Make x = 1, y will equal 2, since it is double x; now construct a point distant 1 space from the origin horizontally, and 2 vertically. Make x = 2, y = 4, and we have a second point; x = -1, gives y = -2, &c., and x - 0, gives y = 0. Connecting these points we get a straight line passing through the origin, as is evident by analytical geometry fiom its equation, y - 2x. Again, let y always equal the square of x, and we have the corresponding values x = O, y 0; x-, y = 1; x = -1, y-=l; x = 2, y = 4; connecting all the points thus found we obtain a parabola with its apex at the origin, and tangent to the axis of X. As another example, suppose we have made a series of experiments on the volumes of a given amount of air corresponding to different pressures. Construct points making horizontal distances volumes, and vertical distances pressures. It will be found that a smooth curve drawn through these points approaches closely to an equilateral hyperbola with the two axes as asymptotes. Now this curve has the equation xy - a, or y = a - x, that is, the volume is inversely proportional to the pressure, which is Mariotte's law. Owing to the accidental errors the points will not all lie on the curve, but some will be above it and others below, and this will be true however many points may be observed. 12 INTERPOLATION. In general, then, after observing any two quantities, A and B, construct points such that their ordinates and abscissas shall be these quantities respectively. Draw a smooth curve as nearly as possible through them, and then see if it coincides with any common curve, or if its form can be defined in any simple way. To acquire practice in using the Graphical Method it is well to construct a number of curves representing familiar phenomena, as the variation in the U. S. debt during the late war, the strength of horses moving at different rates, and the alterations of the thermometer during the day or year. It is by no means necessary that the same scale should be used for vertical, as for horizontal distances, but this should depend on the size of paper, making the curve as large as possible. The greatest accuracy is attained when the latter is about equally inclined to both axes. It is sometimes better when one of the variables is an angle to use polar coordinates. In this case paper must be used with a graduated circle printed on it. The points are constructed by drawing lines from the centre in the direction represented by one variable, and measuring off on them distances equal to the other. For ordinary purposes circles may easily be drawn, and divided with sufficient accuracy by hand. Laying off the radius on the circumference divides it to 600; bisecting these spaces gives 30~, and a second bisection 15~. By trial these angles may be divided into three equal parts, which is generally small enough, as the observations are usually taken at intervals of 5~. Interpolation. All kinds of interpolation are very readily performed by the Graphical Method. After constructing one curve to find the value of y, for any given value of x as x', we have only to draw a line parallel to the axis of Y; at a distance x', and note the ordinate of the point where it meets the curve. Inverse interpolation is performed in the same manner, and this method is equally applicable, whether the observations are at equal intervals or not. As by drawing a smooth curve the accidental errors are in a great measure corrected, this method of interpolation is often more accurate than that by differences. Residual Curves. The principal objection to the Graphical Method, as ordinarily used, is its inaccuracy, as by it we can rarely obtain more thanl three significant figures, although Regnault, by RESIDUAL CURVES. 13 using a large plate of copper and a dividing engine to construct his points, attained a higher degree of precision. It will be found, however, that in many of the most carefully conducted researches the fourth figure is doubtful, as for example, in Regnault's measurements of the pressure of steam, and even in Angstr6m's and Van der Willingen's determinations of wavelengths. By the following device the accuracy of the Graphical Method may be increased almost indefinitely. After constructing our points, assume some simple curve passing nearly through them. From its equation compute the value of y for each observed value of x, and construct points whose ordinates shall equal the difference between the point and curve on an enlarged scale, while the abscissas are unchanged. Thus let x', y' be the observed co6rdinates, and y — f(), the assumed curve. Construct a new point, whose coordinates are x' and a [y' —f(x')], in which a equals 5, 10, or 100, according to the enlargement desired. Do the same for all the other points, and a curve drawn through them is called a residual curve. In this way the accidental errors are greatly enlarged, and any peculiarities in the form of the curve rendered much more marked. If the points still fall pretty regularly, we may construct a second residual curve, and thus keep on until the accidental errors have attained such a size that they may be easily observed. To find the value of y corresponding to any given value of x, as xn, we addf(x,) to the ordinate of the corresponding point of the residual curve, first reducing them to the same scale. Most of the singular points of a curve are very readily found by the aid of a residual curve. See an article by the author, Journal of the Franklin Institute, LXI, 272. fiaxima and lMinima. To find the highest point of a curve, use, as an approximation, a straight line parallel to the axis of X; and nearly tangent to the curve. Construct a residual curve, which will show in a marked manner the position of the required point. The same plan is applicable to any other maximum or minimum. Points of Inflexion. Draw a line approximately tangent to the curve at the required point. In the residual curve the change of curvature becomes very marked. 14 ASYMPTOTES. Asymptotes. Asymptotes present especial difficulties to the Graphical Method, as ordinarily used. Suppose our curve asymptotic to the axis of X; construct a new curve with ordinates unchanged, and abscissas the reciprocals of those previously used, that is equal to 1'. x. It will contain between 0 and 1 all the points in the original curve between 1 and oo. It will always pass through the origin, and unless tangent to the axis of X at this point the area included between the curve and its asymptote will be infinite. When this space is finite, it may be measured by constructing another curve with abscissas as before equal to 1.', and ordinates equal to the area included between the curve and axis, as far as the point under consideration. Find where this curve meets the axis of Y, and its ordinate gives the required area. A problem in Diffraction is solved by this device in the Journal of the Franklin Institute, LIX, 264. Curves of Error. This very fruitful application of the Graphical Method is best explained by an example. Suppose we wish to draw a tangent to the curve B'A, Fig. 2, at the point A. Describe J a circle with A as a centre, through which --- pass a series of lines, as AB, AD, AE. /',,/VNow construct C by laying off BC equal to AB', the part of the curve cut off by A ~~ the line. We thus get a curve CD, called the curve of error, intersecting the circle at D, and the line AD is the required tangent. This is evident, since if we made our construction at this point we should have no distance intercepted, or the line AD) touching, but not cutting, the curve. A similar method may be applied to a great variety of problems, such as drawing a tangent parallel to a given line, or through a point outside the curve. Three Variables. The Graphical Method may also be applied where we have three connected variables. If we construct points whose coordinates in space equal these three variables, a surface is generated whose properties show the laws by which they are connected. To represent this surface the device known as contour lines may be used, as in showing the irregularities of the ground in a map. First, generate a surface by constructing points in which CONTOUR LINES. 15 ordinates and abscissas shall correspond to two of the variables, and mark near each in small letters the magnitude of the third variable, which represents its distance from the plane of the paper. If now we pass a series of equidistant planes parallel to the paper, their intersections with the surface will give the required contour lines. To find these intersections, connect each pair of adjacent points by a straight line, and mark on it its intersections with the intervening parallel planes. Thus if two adjacent points have elevations of 28 and 32, we may regard the point of the surface midway between them, as at the height 30, or as lying on the 30 contour line. Construct in this way a number of points at the same height, and draw a smooth curve approximately through them; do the same for other heights, and we thus obtain as many contours as we please. They give an excellent idea of the general form of the surface, and by descriptive geometry it is easy to construct sections passing through the surface in any direction. An easy way to understand the contours on a map is to imagine the country flooded with water, when the contours will represent the shore lines when the water stands at different heights. This method is constantly used in Meteorology to show wThat points have equal temperature, pressure, magnetic variation, &c. Contour lines follow certain general laws which are best explained by regarding them as shore lines, as described above. Thus contour lines have no terminating points; they must either be ovals, or extend to infinity. Two contours never touch unless the surface becomes vertical, nor cross, unless it overhangs. A single contour line cannot lie between two others, both greater or both smaller, unless we have a ridge or gulley perfectly horizontal, and at precisely the height of the contour. In general, such lines should be drawn either as a series of long ovals, or as double throughout. There will be no angles in the contour lines unless there are sharp edges in the original surface. A contour line cannot cross itself, forming a loop, unless the highest point between two valleys, or the lowest point between two hills, is exactly at the height of the contour. The value of contour lines in showing the relation between any three connected variables, is well illustrated in a paper by Prof. J 16 PHYSICAL MEASUREMENTS. Thomson, Proc. of the Royal Society, Nov., 1871, also in Nature, V, 106. To acquire facility in using the Graphical Method, it is well to apply it to some numerical examples. Thus take the equation y = ax8 + b62 - cx + d, assume certain values of a, b, c and d, and compute the value of y for various values of x. We thus get a curve with two maxima or minima, and a point of inflexion. Find their position first by residual curves, and then by the calculus, and see if they agree. In the same way the curve yx2 - 2ayx +a2y = b, has the axis of X for an asymptote. Assume, as before, positive values of a and b, and determine the area between the curve and asymptote, first by construction and then analytically. PHYSICAL MEASUREMENTS. The measurement of all physical constants may be divided into the determination of time, of weight and of distance, the apparatus used varying with the magnitude of the quantity to be measured and the degree of accuracy required. 2lieasurement of Time. A good clock with a second hand, and beating seconds, should be placed in tlWe laboratory, where it can be used in all experiments in which the time is to be recorded. Watches with second-hands do not answer as well, as they generally give five ticks in two seconds, or some other ratio which renders a determination of the exact time difficult. The true time may be measured by a sextant or transit, as described in Experiment 16. This should be done, if possible, every clear day by different students, and a curve constructed, in which abscissas represent days, and ordinates errors of the clock, or its deviations from true time. Short intervals of time may be roughly measured by a pendulum, made by tying a stone to a string, or better, by a tape-measure drawn out to a fixed mark. We can thus measure such intervals as the time of flight of a rocket or bomb-shell, the distance of a cannon or of lightning, by the time required by sound to traverse the intervening spaceor the velocity of waves, by the time they occupy in passing over a known distance. After the experiment we reduce the vibrations to seconds by swinging our pendulum, and counting the number of oscillations per minute. MEASUREMENT OF TIME. 17 By graduating the tape properly, we may readily construct a very serviceable metronome. Where the greatest accuracy is required, as in astronomical observations, a chronograph is used. A cylinder covered with paper is made to revolve with perfect uniformity once in a minute. A pen passes against this, and receives a motion in the direction of the axis of the cylinder, of about a tenth of an inch a minute, causing it to draw a long helical line. An electro-magnet also acts on the pen, so that when thebcircuit is made and broken, the latter is drawn sideways, making a jog in the line. To use this apparatus a battery is connected with the electro-magnet, and the pendulum of the observatory clock included in the circuit, so that every second, or more commonly every alternate second, the circuit is made for an instant and then broken. Wires are carried to the observer, who may be in any part of the building, or even at a distance of many miles, and whenever he wishes to mark the time of any event, as the transit of a star, he has merely, by a finger key (such as is used in a telegraph office), to close the circuit, when it is instantly recorded on the cylinder. When the observations are completed the paper is unrolled from the cylinder, and is found to be traversed by a series of parallel straight lines, Fig. 3, one corresponding to each minute, with indentations corresponding to every two seconds. The time may be taken directly from it, the 4 3 34 5 a..-, fractions of a second being meas- o6 _ ured by a graduated scale. One 8 = great difficulty in making this ap- 9 Fig. 3. paratus was to render the motion of the cylinder perfectly uniform, as if driven by clock-work it would go with a jerk each second. This is avoided by a device known as Bond's spring governor, in which a spring alternately retards and accelerates a revolving axle when it moves faster or slower than the desired rate. The seconds marks form a very delicate test for the regularity of this motion, since in consecutive minutes they should lie precisely in line, and the least variation is very marked in the finished sheet. It is a very simple matter by this apparatus to measure the difference in longitude of two points. It is merely necessary that an observer should be placed at each station, 2 18 MEASUREMENT OF TIME. with a transit and finger key, a telegraph connecting them with the chronograph. They watch the same star as it approaches their meridian, and each taps on his finger key the instant it crosses the vertical line of his transit. Two marks are thus made on the chronograph, and the interval between them gives the difference in longitude. The advantage of this nlethod of taking transits is not so much its accuracy, as the ease and rapidity with which it is used. Observers can work much longer with it without fatigue, and can use many more transit wires, thus greatly increasing the number of their observations. It is called the American or telegraphic method, in distinction from the old, or "eye and ear" method of observing transits, where the fractions of a second were estimated, as described in Experiment 15. The chronograph is exceedingly convenient in all physical investigations where time is to be measured, and nothing but its expense prevents its more general application. A simple means of measuring small intervals of time with accuracy, is to allow a fine stream of mercury to flow from a small orifice, and collect and weigh the amount passed during the time to be measured. Comparing this with the flow per minute we obtain the time. A less accurate, but much more convenient, liquid for this purpose is water, using, in fact, a kind of clepsydra. Where very minute intervals of time are to be measured they are commonly compared with the vibrations of a tuning-fork instead of a pendulum. A fine brass point is attached to the fork which is kept vibrating by an electro-magnet. If a plate of glass or piece of paper covered with lampblack, is drawn rapidly past the brass point, a sinuous line is drawn, the sinuosities denoting equal intervals of time, whose magnitude is readily determined when we know the pitch of the fork. A second brass point is placed by the side of the fork and depressed from the beginning to the end of the time to be measured. The length of the line thus drawn, compared with the sinuosities, gives the time with great accuracy. Recently a clock has been constructed, in which the pendulum is replaced by a reed vibrating one thousand times a second. The clock is started and stopped, so that it is going only during the time to be measured, and the hands record the number MEASUREMENT OF WEIGHT. 19 of vibrations made. The reed produces a musical note, and any irregularity is at once detected by a change in its pitch. Measurement of Weight. This is done almost exclusively by the ordinary balance, whose principle is so fully explained in any good text-book of Physics that a detailed description is unnecessary here. We test the equality in length of its arms by double weighing, that is, placing any heavy body first in one pan and then in the other, and seeing if the same weights are required to countelpoise it in each case. The center of gravity should be very slightly below the knife-edges. If too low the sensibility is diminished, if too high the balance will overturn, and if coincident with them the beam, if inclined, will not return to a horizontal position. The three knife-edges must be in line, otherwise the centre of gravity will vary with the weight in the scale pans, and of course the friction must be reduced to a minimum. A high degree of accuracy may be obtained with even an ordinary balance by first. counterpoising the body to be weighed, then removing it and noting what weights are necessary to bring the beam again to a horizontal position. A spring balance is sometimes convenient for rough work, from the rapidity with which it can be used. It may be rendered quite accurate, though wanting in delicacy, by noting the weight required to bring its index to a certain point, first when the body to be weighed is on the scale pan, and then when it is removed. Measurement of Length. Distances are most commonly measuled by a scale of equal parts, that is, one with divisions at regular intervals, as millimetres, tenths of an inch, &c. This scale is then placed opposite the distance to be measured, and the reading taken directly. To obtain greater accuracy than within a single division, we may divide them into teiiths by the eye, as in Experiment 1. The steel scales of Brown & Sharpe are good for common measlurements, and may be obtained with either English or French graduation. Instead of dividirig into tenths by the eye, a vernier is frequently used. Thus to read a mnillimnetre scale to tenths, nine spaces are divided into ten equal parts, each of which will be a tenth of a millimetre less than the divisions of the scale, as in Expeimiient 2. One of the best devices for measuring very minute quantities is 20.MEASUREMENT OF LENGTH. the micrometer screw. A divided circle is attached to the head of a carefully made screw, so that a large motion of the former corresponds to a very minute motion of the latter. Thus if the pitch of the screw is one millimetre, and the circle is divided into one hundred parts, turning it completely around will move the screw but one millimetre, or turning it through one division only one hundredth of a millimetre. One of the best examples of this instrument is the dividing engine, which consists of a long and very perfect micrometer screw with a movable nut. See Experiment 21, also Jamin's Physics, I, 25. It is much used in engraving scales, but it has certain defects which are unavoidable, and have caused some of our best mechanicians to give it up. For example, it is impossible to make a screw perfectly accurate, and every joint, of which there are several, is a source of constantly varying error. For these reasons, and owing to its expense, the instrument described in Experiment 22 is for many purposes preferable. Two blocks of wood are drawn forward alternately step by step, through distances regulated by the play of a peg between a plate of brass and the end of a screw. As all joints are thus avoided, and the interval is determined by the direct contact of two pieces of metal, great accuracy is attainable by it. Where several scales are to be made with the utmost accuracy, one should first be divided as correctly as possible, and its errors carefully studied by comparing the different parts with one another, or with a standard. It may then be copied by laying it on the same support with one of the other scales, and moving both so that one shall pass under a reading microscope, the other under a graver. We may thus copy any scale with great accuracy, but the process is very laborious. A good way to construct the first scale is by continual bisection with beam compasses, as is done in graduating circles. The finest scales are ruled with a diamond on glass. M. Nobert has succeeded in making them with divisions of less than a hundred thousandth of an inch. The intervals are so minute that until within a few years no microscope could separate the lines. The method of making them is kept a secret. Mr. Peters, by a combination of levers, has succeeded in reducing writings or drawings to less than one six thousandth their original size. He exhibited some writing done by this machine, which MIN UTE MEASUREMENTS. 21 was so minute that the whole Bible might be written twenty-seven times in a square inch. Finally, it is claimed that Mr. Whitworth was able to detect differences of one millionth of' an inch with a micrometer screw he has constructed. To measure very minute distances a microscope is often used with a scale inserted in its eyepiece, which is used like a common rule. The absolute size of the divisions must be determined beforehand by measuring with it a standard millimetre, or hundredth of an inch. A more accurate method, however, is the spider-line micrometer, in which a fine thread is moved across the field of view by a micrometer screw, and small distances thus measured with the greatest precision. By using two of these instruments, which are then called reading microscopes, larger distances may be measured, or standards of length compared, as in Experiment 20. Small distances are also sometimes measured by a lever, with one arm much longer than the other, so that a slight motion of the latter is shown on a greatly magnified scale. Instead of a long arm it is better to use a mirror, and view in it the image of a scale by a telescope. An exceedingly small deviation is thus readily perceptible, and this arrangement, sometimes known as Saxton's pyrometer, has been applied to a great variety of uses. Where we wish to bring the lever always into the same position a level may be substituted for the mirror, forming the instrument called the contact level. Small distances are also sometimes measured by a wedge with very slight taper, but this plan is objectionable on many accounts. In geodesy all the measurements are dependent on the accurate determination in the first place of a distance of five or ten miles, called a base line. Most of the above devices have been tried on such lines; thus the reading microscope was used by Colby in the Irish survey, the wedge in Hanover, and by Bessel in Prussia, the lever by Struve' in Russia, and the contact level is now in use on our Coast Survey. The principle in all is to use two long bars alternately, which' are either brought in contact, or the distance between their ends measured each time they are laid down. Many other physical constants are really determined by a measure of length. Thus temperatures are determined by a scale of equal parts in the thermometer, and here sufficient accuracy is ob 22 &REAS AND' VOLUMES. tained by reading with the unaided eye. Pressures of air and water are also measured by the height of a column of mercury or water. Where great accuracy is required, as in the barometer, a vernier is commonly used. The instrument known as the cathetometer is so much used for measuring heights that it needs a notice here. It consists of a small telescope, capable of sliding up and down, a vertical rod to which a scale is attached. The difference in height of any two objects is readily obtained by bringing the telescope first on a level with one, and then with the other, and taking the difference in the readings. A level should be attached to the telescope to keep it always horizontal, but the great objection to the instrument is that a very slight deviation in its position, which may be caused by focussing or turning it, is greatly magnified in a distant object. A good substitute for this instrument may be made by attaching a common telescope to a vertical brass tube, the scale being placed near the object to be measured instead of on the tube, as in Experiment 12. Although the measurement of the following quantities is directly dependent on the above, yet their importance justifies a separate notice. Measurement of Areas. It is difficult in general to determine an area with accuracy, especially where it forms the boundary of a curved surface. If plane, any of the methods of mensuration used in surveying may be adopted. Of these the best are division into triangles, Simpson's rule, and drawing the figure on rectangular paper and counting the number of enclosed squares, allowing for the fractions. Another method sometimes useful is to cut the figure out of sheet lead, tin foil, or even card board, and compare its weight with that of a square decimetre of the same material. Measurement of Volumes. These are generally determined by the weight of an equal bulk of water or mercury, using the latter if the space is small. The interior capacity of a vessel is measured by weighing it first when empty, and then when filled with the liquid, as in Experiment 19. The difference in grammles gives the volume in cubic centimetres when water is used, but with mercury we must divide by 13.6, its specific gravity. In the same way we may determine the exterior volume of any body by ANGLES. 23 immersing it and measuring its loss of weight, as when determin ing its specific gravity. An easier, but less accurate, method is by a graduated vessel. These are made by adding equal weights or volumes of liquid, successively, and marking the height to which it rises after each addition. The volume of any space may then be found by filling it with water, emptying it into the graduated vessel and reading the scale attached to the side of the latter. Measurement of Angles. Angles are measured by a circle divided into equal parts, the small divisions being determined by verniers or reading microscopes, as in measuring lengths. A great difficulty arises from the centre of the graduation not coinciding with that of the circle, and on this account it is best to have two or more at equal intervals around the circumference. By taking their mean we eliminate the eccentricity. The precision of modern astronomy is almost entirely due to the methods of determining angles with accuracy. This is dependent on two things; first, a good graduated circle, and secondly, a means of pointing a telescope in a given direction, as towards a star, with great exactness. The latter is accomplished by lacing cross-hairs at the common focus of the object glass and eye-piece, so that they may be distinctly seen in the centre of the field at the same time as the object. Most commonly two cross-hairs are used at right angles, one being horizontal, the other vertical. VVhen, however, we are to bring them to coincide with a straight line, as in the spectroscope, or in a reading microscope, they are sometimes inclined at an angle of about 600, that is, each making an angle of 30~ with the line to be observed. The latter is-then brought to the point of the V formed by their intersection. Still another method is to use two parallel lines very near together, the line to be observed being brought midway between them. The lines may be made of the thread of a spider, of filaments of silk, of platinum wire, or better for most purposes, by ruling fine lines on a plate of thin glass with a diamond, and inserting it at the focus. There are two methods of graduating circles with accuracy. The first, which is used in Germany, consists in a direct comparison with an accurately divided circle, as when copying scales as 24 GRADUATING CIRCLES. described above. That is, both circles are mounted on the same axis, and the divisions of the first being successively brought under the cross-hairs of a microscope, the graver cuts lines on the second at precisely the same angular intervals. In the second method, which is much quicker but less accurate, the circle is laid on a toothed wheel which is turned through equal intervals by a tangent screw. Both methods are really only means of copying an originally divided circle, as it is called, and the coistruction of this with accuracy is a matter of extreme difficulty. It is dependent on the following principles. Any arc or distance may be accurately bisected by beam compasses; the chord of 600 equals the radius, and the angle 85~ 20', whose chord is 1.3554, by ten bisections is reduced to 5'. By constructing an accurate scale, laying off 1.3554 times the radius on the circumference, and repeatedly bisecting- the arc, we finally divide the circle into 5' divisions. Where great accuracy is not required we may divide circles approximately by hand, as described under the Graphical Method, or more accurately by a table of chords and a pair of beam compasses. When the divisions of the circle are very large we may subdivide them by 0 1 2 3 4 a scale instead of a vernier. Thus if A s-JL 1 ~lLJ-[B AB, Fig. 4is part of a circle divided c0 30 D into degrees, we may attach a scale C D Fig. 4. - CD, divided to ten minutes, and subdivide these into single minutes by the eye. Thus in Fig. 4 the reading is 2~ 35'. Much labor is thus saved where the circles have to be divided by hand. Saxton's pyrometer, described above, is of the utmost value in measuring small angular changes. As the reflected beam moves twice as fast as the mirror, the accuracy is doubled on this account If the scale is flat, allowance must be made for the greater distance of its ends than the centre. To reduce the reading to degrees and minutes, the formula, tan 2a = s' d is used, or a -.5 tan's' d, in which a is the angle through which the mirror turns, s the reading, and d the distance of the scale taken in the same units. Instead of a telescope a light shining through a narrow slit is sometimes used, and an image projected on the scale by a lens, or the mirror itself may be made concave. This plan is adopted RADIUS OF CURVATURE. 25 in the Thomson's Galvanometer, and other instruments for measuring the deviations of the magnetic needle. Very small angles may also be measured by a spider line micrometer attached to the eye-piece of a telescope. This is used to determine the distance apart of the double stars, and other minute astronomical magnitudes. There are other methods, such as divided lenses, double image prisms, &c., but they will be considered in connection with the particular experiments which serve to illustrate them. Mleasurement of Curvature.'-To measure the radius of a sphere, as the surface of a lens, an instrument called the spherometer is used. It consists of a micrometer screw at the centre' of a tripod, whose three legs and central point aie brought in contact with the surface. By noting the position of the screw, the radius is readily computed, as in Experiment 14. When the surface is of glass, and the curvature very slight, a much more delicate method is as follows: Focus a telescope on a distant object, and then view the image reflected in the surface to be tested. If the latter is concave, it will render the ray less divergent, and hence the eye-piece will have to be pushed in. The opposite effect is produced by a convex mirror. The amount of change affords a rough measure of thee curvature. This method is so delicate as to show a curvature whose radius is several miles. GENERAL EXPERIMENTS. 1. ESTIMATION OF TENTHS. Apparatus. Two scales, 2tand XN are placed side by side, one being divided into millimetres, the other into tenths of an inch. Also a steel rule A, Fig. 5, divided into millimetres, and so arranged that it may be pushed past a fixed index B, by a micrometer screw, C. A spring,, D, is used to bring it back, when the screw is turned the other way. Experiment. Read the position of each tenth of an inch mark of scale 1/, in tenths of a millimetre, estimating the fractions by the eye. Thus if the interval is one half, call it.5, if a little less,.4, if not quite a third,.3, and so on for the other fractions. The.3 and.7 are the hardest to estimate correctly, as we are liable to imagine the former too great, the latter too small. They should always be compared with the fractions one and two thirds. Record your observations in five columns, placing in the first the readings of the scale 21, in the second the corre- A e sponding readings of N and in the third the first differences of N. Next, \\\\\\\\ subtract the first from the last number in column two, B and divide the difference. by the number of spaces measured, that is, the number of readings minus one. Fig. 5. This gives the average difference, and should be equal to each number of column three. Subtract it from these numbers, and place the results or errors, with proper signs, in column four. Next, compute the probable 28 ESTIMATION OF TENTHS. error (see page 3) of a single observation, using the fifth column for the squares of column four. In this way you can read any scale much more accurately than by its single divisions, and your computed probable error shows how closely you may rely on the result. Next bring one of the millimetre marks' of A, Fig. 5, opposite the index B. Read its position, as described on page 20. The scale E gives units, or number of revolutions, and the divided circle hundredths. Move the screw, set again, and repeat several times. Take the mean and compute the probable error of a single observation. Do the same with the next millimetre mark. Now move the scale until the reading shall be in turn.1,.2,.3, &c., of a millimetre, taking care to move the screw after each, so that you will not be biassed by your previous reading. Next compute what should be the true readings in these various positions. Thus let m' be the mean for the first millimetre, m" for the second; the reading for one tenth would be mn' + (m" - m') 10, for two tenths in' + 2(mn"- m')' 10, and so on. See how these readings agree with those previously found. If any differ by a considerable amount repeat them until you can estimate any fraction with accuracy. This work must be carefully distinguished from guessing, since there should be no element of chance in it, but atnaccurate division of the spaces by the eye. By practice one can read these fractions almost as accurately as by a vernier. 2. VERNIERS. Apparatus. A number of verniers and scales along which they slide are made of large size. The best material is metal or wood, although cardboard will do. By making them on a large scale, as a foot or more in length, there is no trouble in attaining sufficient accuracy. Several different forms are given in Gillespie's Land Surveying, p. 228, from which the following may be selected. 1st, Fig. 225, Scale divided to.1, Vernier reads to.01; 2d, Fig. 227, Same Vernier retrograde; 3d, Fig. 228, Scale.05; Vernier.002; 4th, Fig. 229, Scale 1~, Vernier 5'; 5th, Fig. 230, Scale 30', Vernier 1'; 6th, Fig. 233, scale 20', Vernier 30"; 7th, Fig. 239, Scale 30', Vernier 1'; Double Compass Vernier. Experiment. A vernier may be regarded as a simple enlargement of one division of the scale. Thus if the scale is divided INSERTION OF CROSS-HAIRS. 29 into tenths of an inch, and the vernier into ten parts, it will read to hundredths of an inch. Always read approximately by the zero of the vernier, taking the division of the scale next below it. The fraction to be added is found by seeing what line of the vernier coincides most nearly with some line of the scale. Thus in the first example, we obtain inches and tenths by seeing what division of the scale falls next below the zero of the vernier. If this is 8.6, and the division marked 7 of the vernier coincides with a line of the scale, the true reading is 8.6 +.07 = 8.67. To prove this, set the zero of the vernier at 8.6 exactly. Nine divisions of the scale equal ten of the vernier. Hence each division of the latter equals.09, or is shorter by.01 than one division of the scale. Accordingly the line marked 1 of the vernier falls short by.01 of the scale-division, the 2 line.02, and so on. If we move the vernier forward by these amounts these lines will coincide in turn. Hence when the 7 line coincides, as in the above example, it denotes that the vernier has been pushed forward.07 beyond the 8.6 mark. This method may be applied to reading any vernier. To find the magnitude of the divisions of the latter, divide one division of the scale by the number of parts contained in the vernier. Read and record the verniers as now set. Then set them as follows: 1st, 8.03; 2d, 29.9; 3d, 30.866; 4th, 4~ 10'; 5th, 0~ 17'; 6th, 20 58' 30"; 7th, 2~ 51'. The last vernier is a double one, reading either way, the left hand upper figures being the continuation of those on the lower right hand. This is best understood by moving it 5' at a time and noting what lines coincide. After each exercise the instructor should set all the verniers, and compare the record of the student with his own. 3. INSERTION OF CROSS-HAIRS. Apparatus. Some common sewing silk, card-board and mucilage, also a pair of dividers, ruler and triangle. Experiment. A great portion of the accuracy attained in modern astronomical work is dependent on the exactness with which we can point a telescope, or other similar instrument, in a given direction. This is accomplished by inserting tw.o filaments of silk 80 INSERTION OF CROSS-HAIRS. or spider's web at right angles to each other, at the point within the telescope where the image of the object is formed. In the astronomical telescope, where a positive eye-piece is used, this point lies just beyond the eye-piece, that is between it and the object-glass. A ring is placed at this point on which the lines are stretched. In telescopes rendering objects upright, as in most surveyor's transits, the lines are commonly placed between the object-glass and erecting lenses, and close to the latter. In the microscope, and other instruiners where a negative eye-piece only is used, the lines have to be placed on the diaphragm between the field- and eye-lenses. This pljn is objectionable, since the lines should be very accurately focussed, which can then only be done by screwing the eye-lens in or out. In the other cases the whole eye-piece may be slid in or out until the lines are perfectly distinct, and do not appear to move over the object when the eye is moved from side to side. It is comparatively easy to insert the lines on their ring, where a positive eye-piece is used. The following experiment therefore includes the others. Take a negative eye-piece, Fig. 6, from a microscope or telescope, and unscrew the eye-lens A. C is the diaphragm which limits the field of view, and on which the lines should be placed. Cut from the cardboard a ring, Fig. 7, whose inner diameter is a little greater than the opening of the diaphragm, and the outer diameter Fig. 6. such that it will easily rest on C. Mark on it two lines at right angles to each other passing through its centre. Unravel a short piece of the silk thread until you have separated a single filament. This is best done by holding the thread with the forceps over a sheet of white paper. We now wish to stretch two of these filaments over the lines marked on the cardboard circle. Put a little mucilage on the latter, dip one end of the silk into it, and press it Fig. 7. down with one of the radial strips of paper shown in Fig. 7. When this is nearly dry fasten the other end in the same way, taking care to stretch it so that it shall be straight, or the twist in the thread will give it a sinuous form. Attach the SUSPENSION BY SILK FIBRES. 31 other thread in the same way, and bending the four strips of paper down lay the cardboard on the diaphragm. To hold it in place cut a strip of cardboard or brass, and bending it into a circle push it into the tube. By its elasticity it will hold the paper strips firmly against the sides of the tube. If the experiment has been well performed, on replacing the eye-lens we see two straight lines at right angles, dividing the field of view into four equal parts. The cardboard should not project beyond the diaphragm, or it will give a rough edge to the field of view, and we must be carGftil that no mucilage adheres to the visible portions of the threads. 4. SUSPENSION BY SILK FIBRES. Apparatus. The best method of suspending.a light object so that it shall move very freely is by a single filament of silk. The only apparatus needed is a stand seven or eight inches high, some unspun silk (common silk thread will do, but is not so good) and some fine copper wire. We also need two pairs of forceps, such as come with cheap microscopes, some bees-wax and a sheet of white paper. Experiment. Lay the silk on the paper and pick out a single fibre a little over six inches long. Bend pieces of the wire into the shapes A and B, Fig. 8. Pass one end of the filament through the ring of B, and fasten it with B, i a little wax, twisting or tying it to prevent slip- B ping. Fasten the other end to A in the same way, Fig 8. making the distance fiom A to B just six inches. Hook A into the stand, and lay the object to be suspended, as a needle on B. 5. TEMPERATURE CURVE. Apparatus. A beaker, stand and burner, by which water can be heated, a Centigrade thermometer, and a clock or watch giving seconds. Experiment. Place the thermometer in the water and record the temperature, dividing the degrees to tenths, as described in Experiment 1. Place the burner under the beaker at the beginning of a minute, and at the end record the temperature; repeat at the end of each minute, as the water is warmed, until the ther 32 TESTING THERMOMETERS. mlometer stands at 95~; at the end of the next minute remove the thermometer and the temperature will at first continue to rise, and will then fall rapidly. Record the time (in minutes and seconds) of attaining 95~, 90~, 85~, &c., taking shorter intervals as the temperature becomes lower, and the cooling less rapid. Record your results ill two columns, one giving times, the second temperatures. Finally construct a curve in which abscissas represent times, and ordinates temperatures, making in the former case, one space equal one minute, in the latter, one degree. When two students, A and B, are engaged in this experiment, the following system should be used. A observes the watch and records, while B attefids to the thermometer. Five seconds before the minute begins A says, Ready! and at the exact beginning, Now! B then gives the reading which A records. This plan saves much trouble, and greatly increases the accuracy of any observations which must be made at regular intervals of time. 6. TESTING THERMOMETERS. Apparatus. An accurate Centigrade thermometer is hung upon a stand, and close to it a Fahrenheit thermometer, which is to be tested, their bulbs being at the same height, and close together. A telescope with which they can be read more accurately is placed on a stand at a short distance, and their temperature may be altered at will by immersing their bulbs ill a beaker of water, which may be either cooled by ice, or heated by a Bunsen burner. Some arrangement is desirable for stirring the water to keep it at a uniform temperature. One way is to use a circular disk of tin with holes cut in it, which may be raised or lowered in the beaker by a cord passing over a pulley, so that the observer, while looking through the telescope, can stir the water by alternately tightening and loosening the cord. A simple glass stirring rod may be used instead, if preferred. -Experiment. The problem is to determine the error of the Fahrenheit thermometer at different temperatures, by comparing it with the Centigrade thermometer, which is regarded as a stand ard. By means of the telescope read them as they hang in the air, estimating the fractions of a degree in tenths. Do the same when their bulbs are immersed in water, then cool them with ice and read again. This observation is important, as it shows the absolute error of each instrument. Next heat the water a few ECCENTRICITY OF GRADUATED CIRCLES. 33 degrees with the burner, and then remove the latter. The temperature will still rise for a short time, then become stationary and fall. Read each thermometer at its highest point, stirring the water meanwhile. Repeat at intervals of about 10~ until thewater boils, and finally immnerse again in the ice water, and see if the reading is the same as before. We have now two columns of figures, the first giving the temperature of the Centigrade, the second that of the Fahrenheit, thermometer. Reduce the first to the second, recollecting that ~ C. = 32~ F., and 100~ C. - 2120 F.; hence F. -- C. + 32~, calling C and F the corresponding temperatures on the Centigrade and Fahrenheit scales respectively. Write the numbers thus found in a third column, and the errors will equal the differences between them and the readings given in column two. If the Centigrade thermometer does not stand at zero when immersed in ice water, all its readings should be corrected by the amount of the deviation, taking care to retain the proper sign. Now construct a curve whose ordinates shall represent the errors on an enlarged scale, and abscissas the temperatures. 7. ECCENTRICITY OF GRADUATED CIRCLES. Apparatus. A circle divided into degrees carries a pointer with an index at each end, which turns eccentrically, that is, the centres of the pointer and circle do not coincide. It may be made in a variety of ways. One of the simplest is to place a pivot on one side of' the centre of the circle, and on it a rod with a needle projecting fiolm each enlld. Another way is to let the circle turn and cover it with a plate of glass, on which are marked two fine lines, with a diamond or India ink. The indices may also be made of fine wire, or horsehair. Lines of considerable length must be used, since the edge of the circle advances and recedes as it is turned. If greater accuracy is desired the plan shown in Fig. 9 may be adopted. The c two indices (which mnay have verniers) are connected with the centre by the arms AC and CRB. The circle turns around the pin A -D, and a rod passing through the guides Fig. 9. EF, keeps the verniers in the proper position. Another good instrument for this experiment is the form of compass described under Magnetism in the latter part of the present work. 8 34 CONTOUR LINES. Experiment. Set the index A at 0~ by turning the circle, and read B. Repeat moving A 10~ at a time, until a complete revolution has been made. We have now two columns, giving the corresponding readings of A and B. Subtract 1800 from the latter, and'(A + B - 1800), or I(A ~+ B) - 90~ will be the true reading; write this in column three; in the same way the error of each index is (A — B) - 90~, which should be written in the fourth column. Construct a curve with abscissas equal to the numbers in column three, and ordinates equal to those in column four, enlarged. At the highest and lowest parts of the curve the indices differ most from their true position, or the absolute error, if we read one only, is here greatest. Find these points by Curves of Error, p. 14. On the other hand, where the curve cuts the axis the two indices are opposite each other, and the abscissa gives the azimuth of the line CD). As the ordinates alter most rapidly at these points, the error, when reading a small angle by one index, is here a maximum. Draw tangents, as before, by Curves of Error, and from their direction we can compute the amount of variation. It is a very good exercise to deduce by trigonometry the theoretical curve, and constructing it on the same sheet of paper to compare the results with those obtained by your measurement. We have heretofore supposed that the line connecting the indices passed through the axis around which they turned, or that D lies on EF. If, as often happens in practice, this is not the case, a second correction is necessary. 8. CONTOUR LINES. Apparatus. No apparatus is needed for this experiment, except ordinary writing materials. It is, in fact, an exercise rather than an experiment. Experiment. Mark in your note book nine rows of six points each, so as to form forty squares of about one inch on a side. Mark them with numbers taken from the adjoining table A. Now suppose these numbers represent the heights of the points to which they are attached, and we wish to draw contour lines to show the form of the surface passing through them. As the points are pretty near together we may assume that a line connecting any CLEANING MERCURY. 35 two that are adjacent will lie nearly in the surface. Now regard your drawing as a map, as on p. 15, and suppose the ground A B C 83 79 73 79 79 74 46 56 67 84 86 86 84 65 76 1 68 57 40 128 82 78 70 811 84 76 29 52 73 94 86 73 73 72 50 29 28 152 78 76 66 83 88 73 39 31 65 82 70 56 66 48 31 11 27 41 74 73 58 78 82 63 60 62 68 72 57 49 59 29 129 42 38 29 70 61 50 73 82 74 69 73 81 81 6548 38 46 38 7261 39 71 58 61 82 96 75 80 94 80 81 73 50 27 35 70 99 70 28 70 59 70 83184 72 80 58 58 65 70 49 21 4687 96 60 29 67 65 72 79 73 69 67 58 58 67 62 46 33 63 95 81 49 31 66 67 72 76 69 75 74 68 72 80 149 37 44 71 86 64 47 27 flooded with water to a height of 80. Evidlentlyr all the points in the upper line will be submerged except that on the left, and the shore line will come between 79 and 8-3, about a fourth way from the fornler. Also midway between 82 and 78 in the second line, two fifths of the way from 78 to 83, and a thilrd way from 79 to 82. Several points are thus obtained in each square through which the contour line passes. After obtaining as many as possible, draw a smooth curve nearly coinciding with them all, paying special attention to the rules given under the Graphical Method. Construct in the same way other contours at intervals of ten units. Do the same with the numbers in table B or C. This work is very well supplemented by procuring from the U. S. Signal Blureau at Washington, some of their blank maps (issued at $2.75 per 100), and filling them out fiom the weather reports for the (lay, according to their published directions. These maps may also be used for drawing isothermals, isogonals, &c., if a list is prepared in the first place of the temperalture, magnetic variation, &c., of a large number of stations in the United States. The method adopted for drawing these lines is essentially the same as that given above, only the points are irregularly spaced. 9. CLEANING MERCURY. Apparatus. But little appar';tus is neetled( fol this experiment, except such as is foiund in every chemical laboraltory. Sonime bottles, fiannels, &c., should be placed on the table, andll the student should try as many of the following methods of purification as he can, and record in his note-book his opinion of their comparative value. 36 CLEANING MERCURY. Experiment. Mercury is so much used in physical experiments that every student should know how to clean it. The impurities may be divided into three classes: first, mixture with metals, especially lead, zinc and tin; secondly, common dust and dirt; and thirdly, water or other liquids. Redistillation is almost the only way to remove the metals, and even this is not perfectly effectual, especially in the case of zinc. Moreover, by long boiling a small amount of oxide is formed, which is dissolved by the metal. The mercury used for amalgamating battery plates should therefore be kept separate from the rest and used for this purpose only. If but little of the metal is present it may be removed by agitating with dilute nitric acid. The best way to do this is to fill a long vertical tube with the acid and allow the mercury to flow into it fromn a funnel, in which is a paper filter with a fine hole in the bottom. The mercury falls through the long column of liquid in minute globules, and is thus readily and thoroughly cleaned. It may be drawn out below by a glass stopcock, or by a bent tube in which a short column of mercury shall balance a long column of acid. As the mercury collects it flows out of the end of the tube into a vessel placed to receive it. Instead of nitric acid a solution of nitrate of mercury may be used, if preferred. Another method is to fill a bottle about a quarter full of mercury, add a quantity of finely powdered loaf sugar, and shake violently. The metallic impurities are oxidized at the expense of the air, which must be renewed by a pair of bellows. A great variety of devices are used to remove the mechanical impurities of mercury. For example, pouring it into a bag of chamois leather and squeezing the latter until the mercury comes through in fine globules. Or, making a needle hole in the point of a paper filter, placing it in a funnel and letting the nlercury run through. The mercury may be washed directly with water, by shaking themn together in a bottle, or better, filling a jar half full of mercury and letting the water from the hydrant bubble up through it. This is an excellent way to remove most liquids. Next, to remove the water, pour the mixture into a small bottle, when the mercury will settle to the bottom, and the water overflow from the top. When the mercury fills the bottle transfer it CALIBRATION BsY MERCURY. 37 to another vessel and repeat. If there is only mercury enough to half fill the bottle the second time, pour back some of the mercury already dried to displace the remaining water. Another way is to close the end of a funnel with the finger and pour in the mixture, drawing off the mercury below and leaving the water above. Care must be taken that the mercury does not spurt out on one side and escape. An inverted bottle, or better, a vessel with a tube and stopcock below, is more convenient for this purpose. When only a few drops of water are present they may be removed by blotting paper, or a camel's hair brush. Also by applying heat; but in this case a stain will be left when the water evaporates, unless it has been previously distilled. To see if the mercury is pure pour it into a porcelain evaporating dish. If lead is present it will tarnish the sides. A thin film will also, after a short time, form on its surface, due to oxidation; zinc and tin produce a similar effect. The surface when at rest should be very bright and almost invisible, and small globules, if detached, should be perfectly spherical, and not adhere to the glass but roll over it when the surface is inclined. 10. CALIBRATION BY MERCURY. Apparatus. The best way to perform this experiment is that given by Bunsen in his Gasometry, p. 27. This method is substantially as follows: Select a glass tube, about 2 cm. in diameter, and 40 cm. long, closed at one end. Fasten to it a paper millimetre scale. This is placed upright in a stand, at a short distance from a small telescope, by which the scale may be read with accuracy. On another stand is placed a vessel containing about two kilogrammes of pure mercury, covered with a layer of concentrated sulphuric acid, with a stopcock below, by which it may be drawn off. A small glass tube, also closed at one end, is used to receive it, which should contain, when filled, about 10 cm.3 Its open end is ground fiat, and it may be closed with a plate of ground glass, which is fastened to the thumb by a piece of rubber. Exrperimnert. Both mercury and tube should be perfectly clean, but if not, a few drops of water may be placed in the longer tube, provided great accuracy is not required. Fill the small tube with mercury, holding it with the fingers of the left hand, and remove the surplus by pressing the glass plate, which should be attached to the left thumb, down on to it. Take care that no air bubbles 88 CALIBRATION BY MERCURY. are imprisoned. Empty the mercury into the large tube, and read its height on the scale by the telescope, measuring from the top of the curved surface of the liquid. A clean wooden rod may be used to remove any bubbles of air or globules of mercury which adhere to the sides of the tube. Repeat this operation until the large tube is full of mercury. We now wish to know the volume of the small tube, as this is the unit in terms of which the larger one has been calibrated. The most accurate way to do this is to weigh the whole amount of mercury transferred, and divide by the number of times the smaller tube has been filled. But as it is generally difficult to weigh so heavy a body accurately, the contents of the smaller tube had better be weighed alone, repeating two or three times to see how much the quantity used will vary in consecutive fillings. The volume is then obtained by dividing the weight by 13.6, the specific gravity of mercury. Multiplying the quotient by 1, 2, 3, 4, &c., we obtain the volumes corresponding to our observed readings of the mercury column in the long tube. Represent the, results by a residual curve, as follows: Let s be the scale reading when the small tube has been emptied once into the long tube, and s' when the latter is full, or has received n times this volume of mercury, which we will call v. Then (n - 1)v of mercury will fill the space s' -'s, and the average volume per unit of length will equal (n - 1)v' (s' - s) = a. If the tube was perfectly cylindrical we could find the volume V for any scale reading S by the formula, V- = a (S - s) + v. In reality the tube is probably a little larger in some places than in others, it is therefore better to retain only two significant figures in a, and then compute by the formula the volumes corresponding to the various scale readings that have been observed. Subtract each of these from the corresponding volumes 1, 2, 3, &c., times v, and construct a residual curve in which ordinates equal these differences on an enlarged scale, and abscissas the scale readings. We can now obtain the volume with the greatest accuracy for any scale reading by adding to the value of V given by the formula, the ordinate of the corresponding point of the curve. A table may thus be constructed, giving the volume corresponding to each millimetre mark of the scale. But it is generally sufficiently accurate to make a simple interpolation from the original measurements, CALIBRATION BY WATER. 89 using only the first differences, as when employing logarithmic tables. 11. CALIBRATION BY WATER. Apparatus. A Mohr's burette B, Fig. 10, on a stand, and the vessel to be graduated A, which should be about six inches high, and an inch and a half in diameter. A paper scale divided into tenths of an inch should be attached to A with gum tragacanth, although shellac, or even mucilage, answers tolerably. A long string wound spirally around the vessel will keep the scale in place until the gum is dry..Experiment. Fill the burette B to the zero mark. This is done by adding a little too much water, and drawing it off by the stopcock C into apother vessel, until it stands at precisely the right level. Next, let the water flow into A until it reaches the one tenth of an inch mark, and read B. Do the same for each tenth of an inch, until the one inch mark is reached, and then for every half A inch to the top. Do not let the water level in B DJ fall below the 100 cm." mark, but when it reaches this point refill as before, and add 100 to Fig. 10. the volume measured. Care should be taken not to get too much water into A; should this happen, a little may be drawn out with a pipette and replaced in B, but a slight error is thus introduced. We have now a series of volumes corresponding to various scale readings. Construct a curve with these two quantities as coordinates. Find the point of the curve for which the volume is in turn 10, 20, 30, &c., cm.8, and record the corresponding scalereading. If the vessel is to be used for the measurement of volumes cover it with wax and draw horizontal lines on the latter, having the scale readings just found. Subject it to the fumes of fluorhydric acid, formed by mixing powdered fluor spar and concentrated sulphuric acid. The lines will thus be permanently etched on the glass. 12. CATHETOMETER. Apparatus. A Cathetometer may be made by using as a base the tripod of a music stand or photographer's head-rest, and screw 40 CATHETOMETER. ing into it a tube or solid rod of brass. To this is attached a small telescope with a clamp and set screw, and some form of slow motion. The latter may be obtained by placing the telescope on a hinge and raising and lowering one end by a screw. The slight deviation from a horizontal position will not affect the results, as the instrument is here used. At a distance of five or ten feet is placed a U tube, open at both ends, with one arm about ten inches long, the other forty. The bend in the tube is filled with mercury, and water is poiured into the long arm. We then have a long column of water sustaining a short column of mercury, the heights being inversely as the densities. By the side of this tube is a barometer, made by closing a common glass tube at one end, filling with mercury, and inverting over a cistern containing the same liquid. The precautions and details will be found under Experiment No. 55. By the side of this tube is placed a rod about ten inches long, sharply pointed at both ends, and capable of moving up and down so as to touch the surface of the mercury in the barometer cistern. A steel scale divided into millimetres is adjacent to both tubes, so that it can be read at the same time as the mercury columns. Experiment. Focus the telescope so that both scale and mercury are distinctly visible. Then raise it until it is nearly on a level with A, A. D the top of the column of water, and bring its horizontal cross-hair exactly to coincide by the slow motion. Read the scale, dividing the millimetres into tenths by the eye. Do the same at iB and C; then the difference in height of A and B, divided by that of Fig.ll. C and B, will equal the specific gravity of the mercury, which should be compared with its true value. As the surface of mercury is curved upwards, that of water downwards, the cross-hairs should be brought to the top of the former, and to the bottom of the latter. If great accuracy is required in this experiment, allow for the meniscus, or curved portion at the top of the HOOK GAUGE. 41 water, by adding one half its thickness to the height of the water column. Next raise the rod EF, and read the height, first of the top and then of the bottom. The difference will be its length. It is safer to test the result by moving it and repeating. Then bring the rod so that it shall just touch the surface of the mercury, that is, so that the point and its reflection shall coincide, and read the height of D, and of the top of the rod. Their difference added to the length of the rod gives the height of the column. Read the height of the standard barometer placed among the meteorological instruments. Reduce this to millimetres, and subtract from it the other measurement. The difference will be the depression caused by air and the other errors in the barometer D. 13. HooK GAUGE. Apparatus. A stand, Fig. 12, on which may be placed a vessel of water A, and a micrometer screw B, by which we can raise or lower a rod carrying two points, one turned upwards, the other downwards. Experiment. Fill up the vessel until the water just covers the point of the hook. Then turn the screw so that upon looking at the reflection on the surface of some object as a window sash, a slight distortion is produced by the elevation of the water above the hook. Make ten measurements, moving the screw after each, take their mean and compute the probable error of a single observation. When the point is raised it draws the liquid with it. Screw it down until it touches the liquid, and L read the micrometer, then raise it until the liquid Fig. 12. separates, and take ten readings in each position. Compute, as before, the probable error, and reduce to fractions of a millimetre, which is easily done if the pitch of the screw is known. This gives a measure of the comparative accuracy of the hook and simple point. Both are used for determining the exact height of any liquid surface, the hook being employed most frequently in this country, the point abroad. When the surface of a liquid is 42 SPHEROMETER. rising or falling, and we wish to know the exact time when it reaches a given level, we should use the hook when it descends, otherwise the point; because the former should always be brought up to the surface, the latter down to it. This instrument is so extremely delicate that it will show the lowering of a surface of water in a few minutes by evaporation. A variety of interesting researches may be conducted with it, by the different students of a class. Thus its comparative accuracy with water, mercury and other liquids, may be measured, their rate of evaporation, and the effect of impurities, such as a drop of oil.. The height to which a liquid may be raised by the point, is also a test of its viscosity. 14. SPHEROMETER. Apparatus. Two lenses, one convex, the other concave, a piece of thick plate glass and a spherometer. The latter consists of a tripod, with a micrometer screw in the centre, whose point may be moved to any desired distance above or below the plane of the three legs on which it rests. The most important qualities are lightness and stiffness, and on this account a very cheap, and quite efficient spherometer may be made with the nut and tripod of wood, using for legs, pieces of knitting needles. Exrperiment. Stand the spherometer on the sheet of plate glass and turn the screw until its point is in contact with it. There are three ways of determining the exact position of contact. The first method is dependent on the fact that if the point of the screw is too low the spherometer will stand unsteadily, like a table with one leg too short. The screw is therefore depressed until the instrument rattles, when its top is moved gently from side to side. An exceedingly small motion of this kind is perceptible to the hand. The screw is then turned up and down until the exact point of contact is found. The second, and probably the best method, is to turn the screw slowly, taking care that no greater pressure is exerted on one leg than on the other; as soon as the point touches the glass the pressure is removed from the legs, and the friction of the nut at once makes the whole instrument revolve. Care must be taken not to press on the top of the screw, or the tripod will be bent, and an incorrect reading obtained. The third method of determining contact depends on the sound pro SPHEROMETER. 43 duced when the instrument slides over the glass, which changes when the screw touches the surface. It should be moved but a short distance and without pressure, for fear of scratching the glass. Having determined this point with accuracy, read the position of the screw, taking the number of revolutions from the index on one side, and the fraction from the divided circle. Place the spherometer on each face of the two lenses and measure the position of the point of contact as before. Of course the screw must be raised when the surface is convex, and depressed when it is concave. Subtract each of these readings from that taken on the plate glass, and the difference gives the height of a segment of the sphere to be measured, whose base is a circle passing through the three feet of the spherometer. Call this height h the radius of the circle r, and the radius of the sphere R; then we have, Fig. 13, AB A - h, BID r, and A -R. But by sim- D< ilar triangles AB: ~DB = DB: BE, or r2 h h: r r2R: -A, or=R h+ 2 Compute in this way the radius of each surface of the lenses, remembering that a negative radius denotes a concave surface. To determine r, measure the distance of each leg of the spherometer from the axis of the screw, and take their mean. Measure also the distances of the three legs firom each other and take their mean. They form the three sides of an equilateral triangle; compute by geometry the radius of the circumscribed circle, and see if this value of r agrees with that previously found. Both r and h must be taken in the same unit, as millimetres or inches, and great care should be taken to make no mistake in the position of the decimal point. The reduction of h is effected by multiplying it by the pitch of the screw. Finally, compute the principal focal distance, F, by the formula F1_ =(n- 1) [R- + - —, in which R and R' are the radii of the two surfaces, as computed above, and n the index of refraction of the glass. The latter varies in different specimens, but in common lenses is about 1.53. 44 RATING CHRONOMETERS. 15. ESTIMATION OF TENTHS OF A SECOND. Apparatus. A heavy body carrying a small vertical mirror is suspended by a wire, so that it will swing by torsion, about once in half a minute. A small telescope with cross hairs in its eye-. piece, is pointed towards the mirror, and a plate with. a pin hole in it is placed in such a position that when the mirror swings, the image of the hole will pass slowly across the field of view of the telescope, like a star. It may be made bright by placing a mirror behind it and reflecting the light of the window. The whole apparatus should be enclosed so as to cut off stray light. A good clock beating seconds is also needed.,Experiment. Twist the mirror slightly, so that it shall turn slowly. ()n looking through the telescope a point of light or star will be seen to cross. the field of view, at equal intervals of about half a minute. Note the hour and minute, and as the star approaches the vertical line take the seconds from the clock and count the ticks of the pendulum. Fix the eye on the star and note its position the second before, and that after, it passes the wire. Subdividing the interval by the eye we may estimate the true time of transit within a tenth of a second. Take twenty or thirty such observations and write them in a column, and in a second column give their first differences. Take their mean and compute the probable error. It will show how accurately you can estimate these fractions of seconds. This is called the eye and ear method of taking transits, which form the basis of our knowledge of almost all the motions of the heavenly bodies. It is still much used abroad, although in this country superseded in a great measure by the electric chronograph described on p. 16. 16. RATING CHRONOMETERS. Apparatus. Two timekeepers giving seconds, one, which may be the laboratory clock, to be taken as a standard, and a second to be compared with it. For the latter a cheap watch may be kept expressly for the purpose, or the student may use his own. If the true time is also to be obtained, a transit or sextant is needed in addition. Experiment. First, to obtain the true time. As this problem belongs to astronomy rather than physics, a brief description only RATING CHRONOMETERS. 45 will be given. It may be done in two ways, with a transit or a sextant; the former being used in astronomical observations, the latter at sea. A transit is a telescope, mounted so that it will move only in the meridian. With it note by the clock the minute and second when the eastern and western edges of the sun cross its vertical wire, and take their mean. Correct this by the amount that the sun is slow or fast, as given in the Nautical Almanuac, and we have the instant of true noon. The interval between this and twelve, as given by the clock, is the error of the latter. The sextant may be used at any time when the sun is not too near either the meridian or the horizon. A vessel containing mercury is used, called an artificial horizon, and the distance between the sun and its image in this is measured. Since the surface of the mercury is perfectly horizontal, this distance evidently equals exactly twice the sun's altitude. If the observation is made in the morning, when the sun is ascending, the sextant is set at somewhat too great an angle, if after noon at too small an angle, and the precise instant when the two images touch is noted by the clock. The sun's altitude, after allowing for its diameter, is thus obtained. We then have a spherical triangle, formed by the zenith Z, the pole P, and the sun S. In this, PZ is given, being the complement of the latitude; PS, the sun's north polar distance, is obtained fiom the Nautical Almanac, and ZS is the complement of the altitude just measured. From these data we can compute the angle ZPS, which corrected as before and reduced to hours, minutes and seconds, gives the time before or after noon. The practical directions for doing this will be found given in full in Bowditch's Ncavigctor. By these methods we obtain the mean solar time, which is that used in every day life. For astronomical purposes sidereal time, or that given by the apparent motion of the stars, is preferable. It is found by similar methods, using a star instead of the sun. In an astronomical observatory it is found best not to attempt to make the clock keep perfect time, but only to make sure that its rate, or the amount it gains or loses per day, shall be as nearly as possible constant. We can then compute the error E at any given time very easily by the formula E= EE' + tr, in which E' was the 46 MAKING WEIGHTS. error t days ago, and r the rate. By transposing we may also obtain r, when we know the errors X and E', at two times separated by an interval t. Take the last two observations of the clockerror, *hich should be recorded in a book kept for the purpose, and compute the error at the time of your observation, and see how it agrees with your measurement. If the day is cloudy, or no instruments are provided for determining the true time, the experiment may be performed as follows: Compute, as above, the rate and error of the clock. Next take the difference in minutes and seconds between the clock and the watch to be compared. To obtain the exact interval, a few seconds before the beginning of the minute by the watch, note the time given by the clock, and begin counting seconds by the ticks of the pendulum. Then fixing your eyes on the watch, mark the number counted when the seconds' hand is at zero. Repeat two' or three times, until you get the interval within a single second. Now correcting this by the error of the clock, taking care to give the proper signs, we get the error of the watch. The next thing is to set the watch so that it shall be correct within a second. For this purpose it must be stopped, by opening it and touching the rim of the balance wheel very carefillly with a piece of paper, or other similar object. Set the minute hand a few minutes ahead'to allow for the following computation. Subtract the clock-error. from the time now given by the watch. It will give the tinle by the clock, at which if the watch is started it will be exactly right. A few seconds before this time hold the watch horizontally, with the fingers around the rim, and at the precise second turn to the right and then back. The impulse starts the balance-wheel, and the watch will now go, differing from the clock by an amount just equal to the error of the latter. 17. MAKING WEIGHTS. Apparatus. A very delicate balance and set of weights, some sheet metal, a pair -of scissors, a millilnetre scale, and a small piece of brass, A, weighing about 18.4 grammes. The weights are best made of platinum and aluminiuln foil; but where expense is a consideration, sheet brass may be used for the heavier, and tin foil for the lighter weights. To improve the appearance of the brass and prevent its rusting it may be tinned, or dipped in a silvering PROPER METHOD OF WEIGHING. 47 solution, or perhaps better still, coated with nickel. Some steel punches for marking the numbers 0, 1, 2 and 5, a mallet and sheet of lead should also be provided. PROPER METHOD OF WEIGHING. A good balance is so delicate an instrument that the utmost care is needed in using it. The student should thoroughly understand its principle, and know how to test both its accuracy and delicacy. See Measurement of Weights, p. 19. The beam should never be left resting on its knife-edges, or they will become dulled. It is therefore commonly made sb that it may be lifted off of them by turning a milled head in front of the balance. A second milled head is also added to raise supports under each scale-pan. To weigh any object the following plan must be pursued. To see if the balance is in good order, lower the supports under the scalepans, then those under the beam, by turning the two milled heads. The long pointer attached to the beam should now swing very slowly from side to side, and finally come to rest at the zero. Replace the supports, and open the glass case which protects the balance from currents of air. The object to be weighed, if metallic and perfectly dry, may be placed directly on the scale-pan, otherwise it should be weighed in a watch-glass whose weight is afterwards determined separately. Now place one of the weights in the opposite scale-pan, and remove the supports first from the pans and then from the beam. This must be done very slowly and carefully. Students are liable to let the beam fall with a jerk on the knife-edges, by which the latter are soon dulled and ruined. An accurate weighing is necessarily a slow process and should never be attempted when one is in a hurry. Moreover, by removing the supports. quickly the scale-pans are set swinging, and the beam itself vibrating through a large arc, so that it will not come to rest for a long time. It is better while using the larger weights to lower the supports a very small amount onfy, and notice which way the index moves. As it is below the beam it always moves towards the lighter side. The smaller weights mlust be touched only with a pair of forceps, as the moisture of the fingers would soon rust them. Those over 100 grms. may be taken up in-the hand by the knob, but no other part of them should be 48 PROPER METHOD OF WEIGHING. touched. Weights should never be laid down except on the scalepans, or in their places in the box. Now try weighing the piece of brass A. Lay it on one scale-pan, and a 10 gr. weight on the opposite side. The index moves towards the latter when the supports are removed, as described above. Replace the 10 grs. by 20 grs. This is too heavy, and the index moves the other way. Try the 10 grs. and 5 grs. - too light; add 2 grs. - still too light; another 2 grs. -too heavy; replace the latter by 1 gr. - too light. The weight evidently lies between 18 -and 19 grammes. Add the.5 gr., or 500 mgr. - too heavy; substitute the 200 mgr. - too light, and so go on, always following the rule of taking the weights in the order of their sizes, and never adding small weights by guess, or much time will be lost. Having determined the weight within.01 gr., the milligrammes are most easily found by a rider. This consists of a small wire whose weight is just 10 mgr. It is placed on different parts of the beam, which is divided like a steelyard into ten equal parts, which represent milligrammes. Thus if the rider is placed at the point marked 6, or at a distance of.6 the length of one arm of the balance, it produces the same effect as if 6 mgrs. were placed in the scale-pan. It is generally arranged so that it can be moved along the beam without opening the glass case, which protects the latter from dust and currents of air. By taking care to lower the supports of the beam slowly, as recommended above, the swing of the index is made very small; it is sufficient to see if it moves an equal distance on each side of the zero, instead of waiting for it to come absolutely to rest. To make sure that no errror is made in counting the weights, their sum should be taken as they lie in the scale-pan, and also from their vacant places in the box. Decimal weights are made in the ratio of 1, 2 and 5, and their multiples by 10, and its powers. To obtain the 4 and the 9 it is necessary to duplicate either the 1 or the 2. The English adopt the former method, the French the latter. Comparing the two mathematically, we fifi' that using the weights 5, 2, 2, 1, we shall, on an average in ten weighings, remove a weight fiom box to scale-pan 34 times, of which it will be put back 17 times during the weighing, and the remaining 17 times after the weighing is completed. In the English method, with the weights 5, 2, 1, 1, PROPER METHOD OF WEIGHING. 49 under the same circumstances the weights are again used 34 times, replaced 15 times during the weighing, and 19 after it. There is therefore no difference in rapidity of one plan over the other. The French system has, however, the great advantage that we may at any time test our weights against one another, since 1 + 2 + 2 should equal the 5 weight, and sometimes in weighing, if a mistake is suspected a test may be applied by using the additional weight instead of putting back all the small weights, and adding a larger one, as is necessary in the English system. To meet this difficulty a third 1 gramme weight is sometimes added by English makers. Erperiment. To make a set of weights for weighing fractions of a gramme. Four are needed of platinum or brass weighing 500, 200, 200 and 100 mgrs., and four of aluminum, or thick tin foil, weighing 50, 20, 20 and 10 mgrs. The latter should be made first, since being the lightest they are the easiest to adjust. Cut a rectangle of the foil about 3 or 4 centimetres on a side, and weigh it within a milligramme. Now determine its area by measuring its four sides and taking the product of its length by its breadth. If the opposite sides are rot equal, take their mean. Let A equal the area, and lWthe weight of the foil. Evidently W. A will equal w, the weight per square millimetre, and 50, 20 and 10 divided by w will give the areas of the required weights. Cut pieces somewhat too large and reduce them to the proper size by the Method of Successive Corrections, p. 10. This is accomplished by weighing each and dividing its excess by w. The quotient shows how much should be cut off. As they cannot easily be enlarged if made too small, and the thickness of the foil may not be the same throughout, pieces should be cut off smaller than the computed excess. Small amounts may be taken from the corners, and when completed one of the latter should be turned up to make it easier to pick them up with the forceps. Finally, lay them on the plate of lead, and stamp their weight in milligrammes on them, with the steel punches and mallet. Do the same with the heavier foil, thus making the 500, 200, 200 and 100 mgrs. weight. More care is needed with them, and the last part of the reduction should be effected with a file. Unless great care is taken, two or three will 4 50 DECANTING GASES. be spoiled by making them too light, before one of the right weight is obtained. 18. DECANTING GASES. Apparatus. A pneumatic trough, which is best made of wood lined with lead, and painted over with paraffine varnish. A graduated glass tube D, Fig. 14, closed at one end, and holding about 100 cm.8, a tabulated bell-glass B containing about a litre, with stop-cock C attached, and two or three dry Florence flasks. The mouths of the latter should be ground, so that they may be closed by a plate of ground glass; to remove the moisture they should be heated in a large sand bath, or over steam pipes. A thermometer is also needed. Experiment. Measure the temperature of the air in the flask A by the thermometer, also its moisture, or rather its dew-point. The latter may be Assumed to be the same as that of the D c room, and obtained from D the student, using the meteorological instruments. Now close the flask with i= — _ _ | the plate of glass, and immerse it neck downwards in the pneumatic trough. Fig. 14. It may be kept in this position for any length of time, as the water prevents the air from escaping. Next fill the large graduated vessel B with water, by opening its stop-cock C, and immersing, then close C and raise it. Now decant the gas into it by pouring, just as you would pour water, only that it ascends instead of falfling. When all has been transferred read very carefully the volume, as given by the graduation, also the approximate height of the water inside above that outside the jar. Dividing this difference by 13.6 the specific gravity of mercury, and subtracting the quotient from the height of the barometer, gives the pressure to which the enclosed air is subjected. Its temperature may be assumed equal to that of the water, and it may be regarded as saturated with moisture. Next, to transfer it into the graduated tube D, attach a rubber tube to C, and after fill REDUCTION OF GASES. 51 ing D with water and inverting' it in the trough, let the air bubble into it from the tube by opening C and lowering B, When D is nearly full, close C so as to prevent the escape of the air, and read and record the volume as given by the graduation on D. Now decant the air from D into the flask A. Great care is necessary in this operation to prevent spilling, and it is best to practise a few times beforehand, until it can be transferred without allowing a single bubble to escape. Continue to empty B until all the air has been passed into A. The latter will then be nearly full of air, unless some has been lost. In the latter case do not give up the experiment, but keep on, retaining as much air as possible. Now holding the neck of the flask in the hand press the ground glass against it with the thumb, so as to retain what water is still in it, and taking it out of the trough stand it on the table right side up. Wipe the outside dry, and weigh it in its present condition; also when full of water, and when empty. Call the three weights m, n and o, respectively. The volume of air in cn.', at the beginning of the experiment, will equal n - o in grammnes; that at the end n - m. There are now four volumes of air to be compared. First the volume at the beginning of the experiment, when the air was moist and the dew-point was given; secondly, when transferred to B; thirdly, that found by adding the readings of D; and fourthly, that at the close of the experiment. Reduce all of these to the standard pressure and temperalture by the method given below, when they should be equal if no air has escaped, otherwise the difference shows the amount of the loss. Great accuracy must not be expected, owing to the absorption of the air by the water, and for various other reasons. REDUCTION OF GASES TO STANDARD TEMPERATURE AND PRESSURE. 1: Dry Gas. Given a volume VTr of dry gas at templerature t, and barometric pressure P, to find what would. be its volume V0a if cooled to 0~ C, and the pressure altered to the standard H - 760 m.m. Suppose that it is first cooled to 0~, without changing the pressure, and call its new volumnle Vop. We have by Gay Lllssalc's law for the expansion of gases, VT- = o-p (1 + at), In which a = -f, the coefficient of expansion of gas. Again, by 52 STANDARDS OF VOLUME. Mariotte's law we have, Vp: VH H: H P. Hence op = ~ H VOH'P, or substituting, Vtp = VOH(1 + at)-p or, P 273 P yOH ___ _ 2 (1). t(1 + at)H Vt(27 + t)76 For any other temperature t', and pressure P', we have, 273 P P 273 +tV0x - Vttp, (273 + t') 760' hence Vtp,- Vp P- 273 + t (2). The first formula is used to determine the true quantity of gas present, that is, the volume at the standard temperature and pressure. The second, to compute the new volume when we alter both temperature and pressure. II Gas saturated with Moisture. Call p the pressure of aqueous vapor at the temperature t. Then of the total pressure P we have p due to the vapor, and P - p to the gas; substitute, therefore, P — p for P in equation (1), and we have, 273 (P -p) (3) V P (273 + t)7 60 III: Gas moist, but not saturated. Let the gas be gradually cooled, until the temperature becomes so low that the moisture can no longer be retained as vapor, but begins to condense on the walls of the vessel. This temperature T is called the dew-point; letp' be the corresponding pressure of the vapor. Then p: p' - +at: 1 + aT, or p = p'(1 + at). (1 + aT), and substituting this value in equation (3), we have V (273+ t) 760 [ - P'(1+ aT).. (4). 19. STANDARDS OF VOLUME. Apparatus. A balance AB, Fig. 15, capable of sustaining 5 kgrs. on each side, and turning with a tenth of a gramme under this load. Remarkably good results may be obtained with common balances, such as are used for commercial purposes, by attaching a long index to the beam, as in the figure. Several pounds of distilled water should be provided, a thermometer, a set of weights, and a rubber tube and funnel. Instead of a scale-pan, a counterpoise C is attached to one arm of the balance as a method of double weighing is to be used. The standard to be graduated, which we will suppose to be a tenth of a cubic foot, consists of a glass vessel D, whose capacity somewhat exceeds this amount. A STANDARDS OF VOLUME. 53 i" steam valve is screwed into the cap closing the lower end, which also carries a sharp brass point to form the lower limit of the volume. A ring is attached to the cap closing the upper end of the vessel, by which the whole is supported. A brass hook with the point turned upwards passes through this cap, in which a hole has been drilled to allow the air to pass in or out. The hook may be raised or lowered, and clamped at any height by a conical nut surrounding it, or by a set screw. Finally a millilnetre scale should be attached to the upper end of D. Experiment. Note the height of the barometer, the temperature of the room, also that of the distilled water. Fill D, by attaching the rubber tube, as in the figure, opening E A and pouring in the water. When the vessel is full, close E and remove the rubber tube. Take care that no air bubbles adhere c _ to the side of the1 -- glass. Open E and Fig. 15. draw off the water until it stands just on a level with the top of the scale attached to the glass. Counterpoise by adding weights to.D the scale-pan F, until the index stands at zero, first reading the directions for weighing, given on page 47. Draw off enough water to lower E its level just one centimetre, counterpoise again, and repeat until the surface reaches the bottom of the scale. If too much water is removed at any timerefill the vessel above the mark, and draw off the water again. Now bring the water level just above the point of the hook, and close E, so that the flow shall take place drop by drop. IUse the hook as in Experiment 13, and as soon as the point becomes visible close E. Read the level of the water and counterpoise as before. Repeat two or three times, adding a little water after each measurement. Now open E, and let the water run out until the lower point just touches the surface. Measure the temperature of the water as it escapes. To counterpoise the beam nearly three 64 STANDARDS OF VOLUME. kilogrammes additional must be added to PF. Make this weighing with care, and repeat two or three times, as when observing the upper point. Subtract each of the weights when the vessel was full, from the mean of those last taken, and the difference gives the weight of the water contained between the lower point and each of the other observed levels. Now to determine the volume, we have given by Kater, the weight of 1 cubic inch of distilled water at 620 F., and 30 inches pressure, equals 252.456 grains, and 1 gramme equals 15.432 grains. From this compute the weight of one tenth of a cubic foot. Two corrections must now be applied, the first for temperature, the second for pressure. Water has an expansion of about.000,09 per 1~ F. when near 620, and glass.000p08 linear, or three times this amount of cubical expansion at the same temperature; of course the apparent change of volume is the difference of increase of the water, and of the glass. Evidently at a high temperature less water would be required, hence this correction is negative if the temperature is above 62~. Practically in making standards it is best to keep the temperature exactly at 62~, adding ice or warm water if necessary, as this correction is a little doubtful, owing to the unequal expansion of different specimens of glass. The vessel D) is buoyed up by the air, by an amount equal to the weight displaced, and this weight is evidently proportional to the barometric pressure H. Now 100 cubic inches of air at 30 inches weigh 2.1 grms., hence at 1 inch it would be 21, and if the pressure is changed from 30 to;, the change in weight would evidently be 2.1 X (30 -H)' 30. The weights, however, are also buoyed up in the same way, but as the specific gravity of brass is about 8, the effect is only one-eighth as great. The true correction is then seven-eighths of this amount. The higher the barometer the greater the buoyancy, and the lighter the water will appear, or this correction will be negative for pressures above 30 inches. Both the corrections will be small, and in most cases can be neglected; but it is well to make them, in order to be sure to understand the principle. Having thus computed how much the tenth of a cubic foot ought to weigh, see if the distance between the points is correct, and if not, determine by interpolation READING MICROSCOPES. 55 where the water level should be in order to render the capacity exact. 20. READING MICROSCOPES. Apparatus. Three cheap French microscopes mounted on moveable stands, as in AB, Fig. 16. Two should have cross-hairs in their eye-pieces, while the third should contain a thin plate of glass with a very fine scale ruled on it. An accurate scale divided into millimetres is required as a standard of comparison, and since the division marks of those in common use are too broad for exact measurements, it is better to have one made to order, with very fine lines cut on the centre of one face instead of on the edge. The best material is glass, but copper or steel will do, especially if coated with nickel or silver. Several objects to be measured should be selected, as a rod pointed at each end, the two needle points of a beam-compass, and a scale divided into tenths of an inch, whose correctness is to be tested. Under the microscopes is placed a board D, on which the object to be measured C, is laid, and which may be raised or lowered gradually by screws, or folding wedges. Another method of supporting the microscopes, superior in some respects, will be found described under the Experiment of Dilatation of Solids by Heat. Experiment. If a measurement within a tenth of a millimetre is sufficiently exact, use the two microscopes with cross-hairs. Place them at such a distance apart that each shall be over the end of the object t6 be ineasured, which should be laid on D. They should be raised or.lowered until in focus, and then set so that their cross-hairs shall exactly coincide with the two given points. Remove Fig. 16. the object very carefully, so as not to disturb their position, and replace it by the standard scale, bringing the zero to coincide with one of the cross-hairs. Now looking through the other microscope read the position of its cross-hairs on the scale, esti. mating the fractions of a millimetre in tenths. If the image of the scale is not distinct it may be focussed by slowly raising or lowering the board on which it is placed, taking great care not to disturb the microscopes. To get the whole number of millimetres, a needle may be laid down on the scale, and the right division distinguished by its point. If greater accuracy is desired, use the third microscope, find.. 56 DIVIDING ENGINE. ing the magnitude of the divisions of its scale in the following manner; focus it on the steel scale, placing it so that two divisions of the latter shall be in the field at the same timne. Read each of them by the scale in the eye-piece, and take the difference; the reciprocal is the magnitude of one division in millimetres. Repeat a number of times and take the mean. To make any measurement, place this microscope with one of the others over the points to be determined, and take the reading with its scale, estimating tenths of a division; then substitute the steel scale as before, and read the millimetre mark preceding, also that following. By a simple interpolation the distance is obtained frontom these three readings with great accuracy. Try both these methods with the objects to be measured, and then test the scale of tenths of an inch by measuring the distance of each inch mark from the zero, and reducing the millimetres to inches. Measure also in the same way the ten divisions of one of the inches. One of the best ways to measure off a large distance, as ten or twenty metres, with accuracy, is by means of a couple of reading microscopes. A steel rule is used, the ends being marked by the microscopes, as they are in rough measurements, by the finger. In all cases where the graduation extends to the end of the rule it is better to use the mark next to it, both as being more accurate, and as affording a better object to focus on. 21. DIVIDING ENGINE. Apparatus. This instrument rests on a substantial stand ABED, Fig. 17, like the bed-plate of a lathe. A carefully constructed micrometer screw moves in this, and pushes a nut C fiom end to end. The screw should have a pitch of about a millimetre, or a twentieth of an inch, if English measures are preferred. The head of the screw is divided into one hundred parts, and turns past an index which is again divided into ten parts, as in Fig. 4, p. 24. The screw may be turned by a milled head or a crank. The nut must have a bearing of considerable length, a decimetre is scarcely too much, as any irregularities are thus compensated. It should be split so that it may be tightened by screws, or better, by a spring, and slides along two guides, AB formed like an inverted V, and DE, which is flat. A scale is cut on the latter to give the whole number of revolutions of the screw. The nut DIVIDING ENGINE. 57 should move with perfect smoothness from end to end, but not too fireely. A certain amount of back-lash is unavoidable (that is, the screw may always be turned a short distance backwards or forwards without moving the nut), but this does no harm, as when in use it should always be moved in the same direction. A second screw similar to the other, but smaller, and at right angles to it, is attached to C so that its nut may be moved backwards or forwards about one decimetre. It carries a reading microscope R, made of a piece of light brass tubing, by inserting an eye-piece above, and screwing a microscope objective into the lower end. It may be focussed by sliding the tube up and down by a rack and pinion. Cross-hairs should be placed in the eye-piece, but in some cases a fine scale, or eye-piece micrometer, is preferable. To use this instrument as a dividing engine, the microscope must be made movable, so that it can be replaced by a graver for metals, or'a pen for paper. The micrometer head ]F has ten equidistant holes cut in it, in which steel pins can be inserted. These strike against a stop which they cannot pass unless it is pushed down by the finger. A sheet of thick plate glass D)STE serves as a stand on which to lay objects, and under it is a large mirror to illuminate themn, but it may be removed when desired. -Experiment. This instrument may be applied to a great variety of purposes. Several experiments with it will therefore be described. 1st. To test the screw. Lay a glass plate divided into tenths of a millimetre on DSTE, and bring the microscope over it. Use a maderately high power, as a i1" objective, and focus on the scale; the want of a fine adjustment A may be partly remedied by varying the distance of the eye-piece from the objective. Bring the first division of the scale to coincide with the cross-hairs of the microscope by turning the micrometer- Fig. 17. head F. Read the whole number of turns from the scale on DE, and the fraction from F. Move it one or two turns to the right, and set again; repeat several times, and compute the probable error 58 DIVIDING ENGINE. Of one observation. It equals the error of setting. Turn the screw the other way, and bring it back to the line. The difference between this reading or the mean of ten such readings, and that previously obtained, gives the back-lash. Set in turn on several successive points of the scale. The first differences should be equal. Mark two crosses on a plate of glass with a diamond, three or four centimetres apart. Measure the interval between them with different portions of the screw, and see if they agree. If not, the defect in the screw must be carefully examined, and corrections computed. The screw 2Mshould be similarly tested. 2d. Determination of the pitch of the screw. Procure a standard decimetre (or other measure of length) and measure the distance between its ends. The temperature should be nearly that taken as a standard, or if great accuracy is required, allowance made for the difference of expansion of the screw and decimetre. From this measurement, which should be repeated several times, compute the true pitch of the screw, and the correction which must be applied when distances are measured with it. 3d. To measure any distance. Lay the object on the glass plate and bring the cross-hairs of the microscope to coincide first with one end of it, and then with the other. The difference in the readings is the length. Apply to it the correction previously determined. 4th. To determine the form of any curved line. For example, use one of the curves drawn by a tuning fork, in the Experiment on Acoustic Curves. Bring the cross-hairs to coincide with several points in turn of one of the sinuosities, and read both micrometer heads. These give two coordinates, fiom which the points of the curve may be constructed on a large scale, and compared with the curve of sines, the form given by theory. The relative positions of a number of detached points may also be thus determined, as in the photographs of the Pleiades and other groups of stars by Mr. Rutherford. 5th. Graduation. For a first attempt, make a scale on paper with a pencil or pen. Replace the microscope by a hard pencil with a flat, but very sharp point. It must be arranged so that it can be moved backwards or forwards a limited distance, but not sideways. Every fifth line should be longer than the rest, which RULING SCALES. 59 should be exactly equal to each other in length. Fasten the paper securely on the glass plate so that it shall not slip. Suppose now lines are to be drawn at intervals of half a millimetre. Insert a pin in one of the holes in F, and turn the latter to the stop. Draw a line with the pencil for the beginning of the scale, depress the stop to let the pin pass, give FP one turn, bring the pin again to the stop and draw a second line, and so on. If the lines are to be a millimetre apart, draw one line for every two turns. In. the same way, by inserting more pins a finer graduation may be obtained. Instead of using the pins a table may be computed beforehand, giving the reading of the screw for each line to be drawn, allowing for the errors of the screw, if great accuracy is required. The scale is then ruled by bringing the nut successively into the various positions marked in the table, and drawing a line after each. A most important application of this instrument is to the measurement of photographs of the sun taken during eclipses. The position of the moon at any instant is thus obtained, with a degree of )recision otherwise unattainable. In this, and other cases where angles must also be measured, the plate of glass ES should be removed, and the object laid on a rotary stand, with a graduation showing the angle through which it is turned. 22. RULING SCALES. Apparatus. In Fig. 18, two strips of wood A and B, rest on a smooth board, and are held in place by the weights C and D. The ends of a string are attached to them, which is stretched by means of a weight F, so that if C and ) are raised A and B will slide. A peg is inserted in B, which moves between two steel plates fastened to A, one being fixed, the other movable by means of a screw G. If, then, either weight is raised, the strip of wood on which it rests will be drawn forward by F, but will be free to move through a space equal to the difference of the diameter of the peg and the interval between the two steel plates. If desired, G may be a micrometer screw, by which this interval may always be accurately determined. It may be fastened in any position by a clamp or set screw. A steel rod H is used to draw the division lines. It is fixed at one end, and carries at the other a pencil, pen, graver or diamond, according as the lines are to be drawn on paper, metal or glass. By this arrangement there is little or no 60 RULING SCALES. lateral -motion of the graver, but unfortunately it draws a curved line. To remedy this defect, the rod may be replaced by a stretched wire, to the centre of which the graver is attached, or the latter may slide past a guide against which it is pressed by a spring. Experiment. For many purposes in using a scale, it makes but little difference what the divisions are, provided that they are all equal, and this is especially the case in all accurate measurements, sinc.e as a correction must always be made for temperatlure, we can readily at the same time correct for the size of the divisions. The instrument here described will probably give divisions more nearly equal than those obtained by a micrometer screw, but it is more difficult to make thein of any exact magnitude, since any deviation is multiplied by the number of divisions. To draw a scale, lay a piece of paper on -B and fasten it with tacks or clips. To secure uniformity in the length of the long and short division marks, rule three parallel lines as limits, attach a sharp flat-pointed pencil to H; and slide A and B until the beginning of the scale is under ff. Draw a line with the latter, and make one stroke Fig. 18. with the machine. This is done Fig. 18. by raising C, when F will draw A forward a distance equal to the interval between the two plates near G, minus the thickness of the peg. Lay C down and raise D. A will now remain at rest, but B will move through the same distance. Draw a second line with the pencil, and repeat, making every fifth line about twice as long as the others. They will be found spaced at distances which may be regulated by the screw G. Try making short scales in the same way, with large and small divisions. It is always safer to keep the hand on one weight while the other is lifted. The magnitude of _F should be such that the strain on the cord will be greater than the friction of repose when the weights are up, but less than the friction of motion when they are down. If F is too light, when C is -raised A will not start; if too heavy, it will strike so hard that it will move B. To test the accuracy of the machine draw a single line, take a hundred RULING SCALES. 61 strokes and draw another. Then without moving G push the slides back and draw a third line close to the first; take a hundred strokes and draw a fourth line near the second. Measure the interval between the first and third, and the second and fourth. They should be equal, but if not, the difference divided by an hundred gives the average difference in length of a stroke the second time, compared with the first. Instead of a pencil, a pen may be used to draw the lines, or a graver, if a metallic scale is desired. The finest scales are ruled on glass by a diamond. Instead of using the natural edge of the gem, as when cutting glass, an engraver's diamond should be employed, which is ground with a conical point; the direction in which it should be held, and the proper pressure, being obtained by trial. Scales may also be etched by covering the surfice with a thin coating of wax or varnish, and the lines marked with a graver. If metallic, it is then subjected to the action of nitric acid; if of glass, to the fumes of fluorhydric acid. It is possible that the new method of cutting glass by a sand-blast may prove applicable to this purpose with a great saving of time and trouble. MECHANICS OF SOLIDS. 23. COMPOSITION OF FORCES. Apparatus. Two pulleys A and B, Fig. 19, are attached to a board which is hung vertically against a wall. Two threads pass over them, and a third C, is fastened to their ends at D. Three forces may now be applied by attaching weights to the ends of the cords. The weights of an Atwood's machine are of a convenient form, but links of a chain, picture hooks, cents, or any objects of nearly equal weight may be used. Small beiads are attached to the three threads at distances of just a decimetre from ). Erxperim.ent. Attach weights 2, 3 and 4 to the three cords, and let D assume its position of equilibrium. Owing to firiction it will remain at rest in various neighboring positions, XB X their centre being the true one. Now measure the distance of each bead from the other two with a millimetre scale, and obtain the 1: D | angle directly from a table of chords. If these are not at hand, dividing the distance by two, gives the natural sine of one half the required angle. By the law of the parallelograin of forces, the latter are proportional to the sides of a triangle having the directions of the forces. But these sides are proportional to the sines of the opposite angles, hence the sines of the:angles included between the threads should be proportional to the forces or weights applied. Divide the two larger forces by the smaller, and do the same with the sines of the angles, and see if the ratios are the s:ale. The angles themselves should first be tested by taking their slum, which should equal 360~. If either angle is nearly 180,' it cannot be accurately measured in this way, but must be found by subtracting the sum of the other two fiom 3600, or measuring one side fiom MOMENTS. 63 the prolongation of the other. It is well to draw the forces from the measurement, and see if a geometrical construction gives the same result as that obtained by calculation. Repeat with forces in several other ratios, as 3, 4, 5; 2, 2, 3; 3, 5, 7; taking care in all cases to include in the weights the supports on which they rest. 24. MOMENTS. Apparatus. A board AB, Fig. 20, is supported at its centre of gravity on the pin C. It should revolve freely, and come to rest in all positions equally. Two forces may be applied to it by the weights D and E, attached by threads to the pins F and G. Their magnitudes may be varied fiom 1 to 10 by different weights, and their points of application by using different pins, as H, I and J. To measure their perpendicular distances from the pin C, a wooden right-angled triangle or square is provided, one edge of which is divided into millimetres, or tenths of an inch. Experiment. Various laws of forces may be proved with this apparatus. 1st. When a single force acts on a body AB fixed at one point, as C, there will be equilibrium only when it passes through this point. Remove PFD and attach a weight F to G. It will be found that the body will remain at rest only when the point G is in line with Xland C. 2d. A force produces the same effect if applied at any point along the line in which it tends to move the body. Apply the two weights D and E, which tend to turn the board in opposite directions. Make their ratio such that MG shall be in line with G, 1- J. Now transfer the end of the thread firom G to H, I and A c~ B Jin turn, when it will be found that' the position of the board will be un- E changed. It should be noticed, howFig. 20. ever, that in the last case the board is in unstable equilibrium, since FJfalls beyond the point of support C. 3d. The moment of a force, or its tendency to make a body revolve, is proportional to the product of its magnitude by its perpendicular distance from the point of support. Make D equal 2, and attach it to R so that the thread rests over the edge of the board, which is the arc of a circle with centre at C, and radius.6. Its tendency to make the board revolve is therefore the same, what 64 PARALLEL FORCES. ever the position of the latter. Make E successively 1, 2, 3, 4, 5, 6, and measure the perpendicular distance of the thread to which it is attached in each case from C. This distance is measured by resting the triangle against the thread and nmeasuring the distance of C by its graduated edge. In each case the moment of E will be found to be the same, and equal to 2 X 6, the moment of D. 4th. When two forces hold the body in equilibrium their resultant must pass through the fixed point. Make D equal 2, and attach it to F, and F equal 3, applied at G. Lay a sheet of paper on the right hand portion of AB, making holes for F, C and J to pass. Draw on it with a ruler the direction of the two threads prolonged, and then removing it, construct their resultant geometrically by means of the parallelogram of forces. It will be found to pass through (C. Repeat two or three times with different weights and points of application. 25. PARALLEL FORCES. Apparatus. The apparatus used is shown in Fig. 21. AB is a straight rod about two feet long, with a paper scale divided into tenths of an inch attached to it. It is supported by a scale-beam CD with a counterpoise, so that it is freely balanced, and remains horizontal. Weights formed like those of a platform scale may be attached to it at any point, by riders, as at E, F and G. Taking each rider as unity, four sets of weights are required of magnitudes 10, 5, 2, 2, 1,.5,.2,.2,.1. Two other beams, like C)D, should also be provided, to which these weights may be attached, as at E, so as to produce an upward force of any desired magnitude. All these scale-beams may be very roughly made, even a piece of wood supported at the centre by a cord, being sufficiently accurate. English beams of iron may, however, be obtained at a very low price. Experiment. The resultant of any system of parallel forces XD lying in one plane may be found by 4nr T 01 pa this apparatus. Thus suppose we have a force of 15.7 acting upwards, and two of 8.3 and 1.4 acting downwards, and distant from the first 6.4 and 8.7 inches respectively. Produce the upward force by addFig. 21. ing the weights 14.7 to E, and the PARALLEL FORCES. 6.5 two downward forces by weights 7.3 and.4 (allowing 1 for each of the scale-pans) at F and G, setting them at the points of the beam marked 3.6 and 18.7. They are then at the proper distance firom C, which is at 10 inches from the end. We now find that A goes down and B up; by placing the finger on the beain we see that it can be balanced only by applying a downward force to the right of C. Now place a rider in this position, and move it backwards'and forwards, varying the weight on it until the beam is exactly balanced. The magnitude of this weight will be found to be 6, and its position 16.8, or 6.8 inches from C. The resultalnt of the three forces will be just equal and opposite to this. Had the force required to balance them acted upwards, we should have used one of the auxiliary scale-beams. To test the correctness of this result we compute the resultant thus: R - 15.7 - (8.3 + 1.4) = 6, and taking moments around C we have 8.3 X 6.4 - 1.4 X 8.7 - 6 X x, or x - 6.8 the observed distances. Determine the position and magnitude of the resultant in several similar cases, as for example the following, in which Umeans an upward, and D an downward force, and each is followed first by its magnitude, and then by the point on the bar at which it is to be applied. 1. D, 5.0, 4.3; U; 10.0, 10.0. 2. D, 2.6, 3.2; uT 7.8, 10.0. 3. D), 7.4, 3.7; U; 17.1, 10.0. 4. -D, 11.1, 2.1; -D, 6.5, 5.6; U; 2.3, 18.4. 5. 2D, 5.2, 1.9; U, 15.2, 10.0; XD, 8.4, 12.6; U; 3.0, 18.1. Two equal parallel forces acting in opposite directions and not in the same line, form what is called a couple, and have no single resultant. Thus apply the two forces )D, 12.0, 5.0, and U, 12.0, 10.0. No single force will balance the beam. Equilibrium is obtained only by a second couple having the same moment, and turlning in the opposite direction; thus the moment being 12.0 X 5.0 = 60.0, the beam may be balanced by two forces of 10.0, each distant 6 inches from one another, placing the upward force to the left. Find in the same way some equivalent to D, 4.3, 7.6, and 0, 4.3, 10.0, and notice that it makes no difference to what part of the beam the two forces are applied, provided their distance apart remains unchanged. 5 66 CENTRE OF GRAVITY. This same apparatus may be applied to illustrate the case of a body with one point fixed, acted on by parallel forces, as, for example, the lever, by using a stand H with two pins, between which the beam may turn. This stand is also useful in finding the point of application of the resultant in the above cases. 26. CENTRE OF GRAVITY. Apparatus. Several four-sided pieces of cardboard (not rectangles) and a plumb line, made by suspending a small leaden weight by a thread, from a needle with sealing wax head. Experiment. Make four holes in the cardboard, two AB, Fig. 22, close to two adjacent corners, the others in any other part not too near the centre. Pass the needle through A and support the cardboard by it; the thread will hang vertically downwards, and the centre of gravity must lie somewhere in this line, or it would not be in equilibrium. Mark a point on this line as low down as possible, and connect it with the pin hole. Do the same with B; the intersection of the two is the centre of gravity. Turn the cardboard over and repeat with the other holes. This gives two determinations of the centre of gravity. To see if the two points are opposite one another, prick through one and see if the hole coincides with the other. By suspending at any other points, the same result should be obtained. Be careful that the holes are large enough to enable the card to swing freely. Next, lay the card down on your note book and mark the four points A, B, C, D. Connecting them with lines gives a duplicate of the cardboard. On this construct the centre of,B gravity geometrically. Divide into two triangles by connecting A C. Bisect AD in E, and C.D in F. The centre of gravity of A CD -x-G > must lie in AF, also in CE, hence at G. Ob\g tain G' by a similar construction with AB C. The centre of gravity of the whole figure must Fig. 22. lie in GG'. Make a second construction by connecting BD, making the triangles ABD and B C(D; the intersection of G G' and its corresponding line gives the centre of gravity. Lay the piece of cardboard on the figure and prick CATENARY. 67 through the two centres of gravity previously found. They should agree closely with that found geometrically. 27. CATENARY. Apparatus. A chain three or four yards long, each link of which is a sphere, known in the trade as a ball link chain. Every tenth link should be painted black, and the fiftieths red. A horizontal scale AB C, Fig. 23, attached to the wall, also a number of pins to which the chain may be fastened by short wire hooks, and its length altered at will. A graduated rod _B-D is used to measure the vertical height of any point of the chain. Experiment. First, to determine the average length of the links. Let the chain hang vertically from A1, measure the length of each hundred links, and take their mean. A simple proportion gives the number of links to which A C is equal. Suspend the chain at A and C, making the G F E flexure at the centre about half a foot. Measure it exactly, and increase the original length 10 links at a time to 100. Increase it also i D by 17 links, by 63 and by 48, and Fig. 23. measure as before. Write the Xesults in a column and take the first, second and third differences of the first measurements. Now obtain by interpolation the three values for 17, 63 and 48 links, and compare with their measured values. Next suspend the chain as in ADE, and measure the deflection at intervals of five inches horizontally. This is best done by passing a pin through the graduated rod at the zero point, letting it hang vertically, then measuring by it. Taking differences as before, those of the first order will be at first negative, then increase until they become positive. Where the first difference is zero, is evidently the lowest point of the curve. By the method of inverse interpolation find this point, treating the first differences as if they were the original variable, and recollecting that each difference belongs approximately to. the point midway between the 68 CRANK MOTION. two terms from which it was obtained. Thus the difference obtailled from the 5 and 10 inches corresponds to 71. Obtain this point also by measurement, by laying off BF equal to CUE, prolonging ET to G and measuring GF. A C minus one-half GE will equal the required distance. Repeat with several points below E, and compare with the computed position of the lowest point. 28. CRANK MOTION. Apparatus. A steel scale AB, Fig. 24, divided into millimetres, slides in a groove so that its position may be read by an index E. It is connected by the rod AD to the arm of the protractor, whose centre is C. On turning CD, which carries a vernier F, AB moves backwards and forwards. Several holes are cut in AD so that its length may be altered at will. Experiment. Make AD as long as possible. Measure CD by turning it until D is in line with C and A, and read E; then turn it 1800, and read again. One-half the F difference of these readings equals C-D. Next, to find the reading of C. [ __the vernier when CD and DA are -A B. I] in line. Make ACD about 900 and read E and F. Turn CD unMig. 24. til the reading of E is again the same and read F. The mean of these two readings gives the required point. Repeat two or three times, and take the mean. Let AB represent the piston rod of an engine, and CD the crank attached to the fly-wheel. The problem is to determine the relative positions of these two, during one revolution. Bring D in line with CA, and move it 10~ at a time through one revolution, reading E in each case. Do the same, using a shorter connecting rod, so that AD shall be about two or three times C-D. To compare these results with theory, first suppose the rod CD infinitely long. The distance of AB from the mean position will then always equal CD X cos A CD. This is readily computed from the accompanying table of natural cosines. If, as is most convenient, CD is made just equal to 1 decimetre, the distances are given directly in the second column of the table by moving the HOOK'S UNIVERSAL JOINT. 69 decimal point two places to the right. Compare these results with- your observations. Construct a curve in which abscissas represent the computed positions of AB, and ordinates the difference between the observed and Angle. Cosine. computed results, enlarging the scale ten times. 00 1.000 If a smooth curve is thus obtained it is probably 100.985 200.937 due to the short length of AD. The correction 30~.866 due to this is readily proved to be AD - 40~.766 500.643 IJAR2 - CD2 sin2 A(2D, or calling the ratio 60~.500 AD' CD -- n, it is AD (n - /n2 _ sin2A CD). 700.342 80o.174 Compute this correction for every 300, knowing 900.000 that sin2 30~ =.25, sin2 60~ -.75. The points thus obtained should lie on the residual curve found above. Do the same with the shorter arm AD. 29. HooK's UNIVERSAL JOINT. Apparatus. A model of this joint with graduated circles attached to its axles. The latter should be so connected that they may be set at any angle. Experiment. Set the axes at an angle of 45~, and bringing one index to 00, the reading of the other will be the same. Now move the first 5f at a time to 180~, and read the other in each position. Record the results in columns, giving in the first the reading of one index, in the second that of the other, and in the third their difference, which will be sometimes positive, and sometimes negative. Construct a curve with ordinates taken from the first colunmn, and abscissas from the third, enlarging the latter ten times. It shows how much one wheel gets behind, or in advance of, the other. To compare this result with theory, let Fig 25 represent a plan of the joint, A C and CB being the two axes. Describe a sphere with their intersection C as a centre. The great circle GD is the path described by the ends of one hook, A CE that described by the other. D and E must, by the construction of the apparatus, always be 90~ apart. Then in the spherical Fig. 25. triangle CDE we have given DE — 90~, ECD = 45~, the angle between the axes, and one side as iJD, and we wish to compute 70 COEFFICIENT OF FRICTION. (YE. But by spherical trigonometry, tang C-E = tang CD cos ECD. Substituting in turn C-D = 50~, 100, 150~, 200, &c., we compute the corresponding angle through which the second wheel has been turned. Construct a second curve on the same sheet as the other, using the same scale. Their agreement proves the correctness of both. Experiments like Nos. 28 and 29 may be multiplied almost indefinitely. Thus various forms of parallel motion, the conversion of rotary into rectilinear motion by cams, link motion, gearing, and, in fact, almost all mechanical devices for altering the path of a moving body may be tested and compared with theory. 30. COEFFICIENT OF FRICTION. Apparatus. AB, Fig. 26, is a board along which a block C is drawn by a cord passing over a pulley D, and stretched by weights placed in the scale pan E. The friction is produced between the surfaces of C and AB, which should be made so that they may be covered with thin layers of various substances as different kinds of wood, iron, brass, glass, leather, &c. C is made of such a shape that by turning it over the area of the surface in contact may be altered The pressure on C and the tension of the cord may also be varied at will, by weights. E.xperiment. Weights are added to E in regular order, as when weighing, and the tension in each case compared with the friction of C. Friction may be of two ~] C kinds; first, that required to start a body at rest, called the friction of IA ff B l repose, and secondly, the friction of motion, or that produced when the bodies are moving. To measure the friction of repose, see if the Fig. 26. weight is capable of starting the body when at rest, if so, stop it and repeat, varying the weight until a tension is obtained sufficient sometimes to start it and sometimes not. This friction is very irregular, varying with different parts of every surface, and with the time during which the two substances have been in contact. It is but little used practically, since the least jar converts it into the friction of motion. The latter is much less than the friction of repose, and more uni ANGLE OF FRICTION..1 form. It is found by tapping the body so that it will move, and seeing if the velocity increases or diminishes. In the first case the weight in E is too large, in the second too small. The first law of friction is that the friction is proportional to the pressure. The ratio of these two quantities is called the coefficient of friction. Make the load on C, including its own weight, equal to 1, 2, 3, 4, 5 kgs. in turn, and measure the friction. The latter equals the weight of E plus the load added to it minus the friction of the pulley. If great accuracy is required, a table should be prepared, giving the magnitude of the latter for different loads. Compute the coefficient of friction from. the observations, and if the law is correct they should all give the same result. Measure, in each case, the friction of repose and of motion, and notice that the latter is always much the smalley-. Secondly, the friction is independent of the extent of the surfaces in contact. This law follows from the preceding, but it is well to prove it independently by turning C on its different sides, so as to vary the areas in contact. The friction will be found to be the same in each case. Finally, measure the coefficienis in a number of cases, and compare the results with those given in the tables of fiiction. 31. ANGLE OF FRICTION. Apparatus. In Fig. 27, AB is a stand with an upright BCC. AD is a board hinged at A, which may be set at- any angle by a cord passing over the pulley C. The hinge is best made of soft leather held by a strip of brass, and its distance from the upright should be just one metre. A scale of millimetres is attached to the uprio'ht, and a wire parallel to A-D serves as an index. Evidently the reading of the scale gives the natural tangent of the angle of inclination DAB. A cord attached to D passes over the pulley C, around the wheel F, and is stretched by the counterpoise G. F may be clamped in any position, or turned by a crank attached to it. AD) may thus be set at any angle, and its position is to be determined when so inclined that any given body, as E, is just on the point of sliding. E may be~ made exactly like the' sliding m:ass in Experiment 30, but to measure its friction of motion a fine wire should be attached to it and wound around the axle of F. When the crank is turned raising AD, the body E is thus drawn slowly down the inclined plane. In order that it may not move too rapidly this portion of the axle should be much B BREAKING WEIGHT. smaller than that around vwhich the cord -CF is wound. EF should be a wire, as if a thread is used it will stretch, giving E an irregular motion, alternately starting and stopping. EExperiment. To measure the coefficient of friction of repose, turn the crank until the body begins to slide; the reading of the C scale gives the tangent of the angle of inclination. But decomposing the weight of E into two parts, paral/ - lel and perpendicular to the C/ plane, their ratio will equal the coefficient of friction, A. and also the tangent of the inclination. Hence the coefficient of friction is given Fig. 27. directly by the scale. Measure again the coefficients found in Experiment 30, and see if the results agree with those then obtained. [TC mainder of this work, which -"k be issued shortly, wel con E xperiments in Mechanics of _Liuaids and Gases, Sound, Light, Heat, and Electricity, also an Appendix giving, in detail, plans for establishing and conducting Physical -Laboratories.]