Also, for the motion, when the heat is cut off we shall have an exactly similar series of equations, except that will represent the final or zero reading of the ordinate. In the case before us the values of ye are so small, in consequence of the rapid damping down of the oscillations, that we had better restrict ourselves to the first four of these equations. If we denote e o by æ, and eö by y, and if we eliminate A cos e and C, we obtain from each of the curves the two equations 142.0 0.1111 and - 4042 - 42 From the four curves in fig. 4 we find the following values of 40, 41, &c., with the resulting values of x – y and xy : фоф фарз ¢ – у ху m.m. m.m. m.m. m.m. 0-2032 0.1373 . 142.4 68.0 5.8 6.7 The resulting values of x and y from each of the four curves are, accordingly, Curve 1, 0.4449 0.2813 , 2, •4210 •2644 4858 .2826 " 4, . . .4151 •2677 Mean, · · · 0·4417 0.2740 These figures seem to show two things :-Firstly, that the damping effect is less when the instrument is heated than when cold, since the value of x just found is greater than that found from the curves of free oscillation. Secondly, it may be noticed that the values of both x and y are greater as derived from the 1st and 3rd curves than those deduced from the other two curves. The materials are, however, hardly sufficient to decide, whether this represents a real difference depending on the direction of the motion or is merely an accidental discrepancy. We have thus the following results, taking for é 8 the value found from the heat curves (the system being then in circumstances more closely analogous to those which take place during a transit of the sun than when cold): 1= 3-193, 6MT = 0-4417, T = 02740. From these we find 8 = 0.9838 or 56o.37, a = 0·2558, 1 = 0.4054 ; whence b2 = 1.0333. We thus find the coefficients of the differential equation on p. 303:2a + 1 = 0.9170, 2al + b2 = 1.2407, bol = 0·4189. Fig. 5. would be easy to compute the values of the differential coefficients from the differences of the readings of $ by means of the usual equation = {log (1 + $)}". Before applying this formula, however, it will be necessary to smooth the curve by removing the periodic part in the value of 0. The periodic terms are all contained in the expression Ae-at cos (8t - e), and since this value for 4 is a solution of the equation + b?lp = 0, it is clear that the resulting curve is still a solution of the differential equation (4). Curve a in fig. 6 represents the curve when the periodic terms have been removed. This has been done in a graphical manner, first with the eye, and then the curve was still further smoothed down by adjusting the differences in the way described by Sir John Herschel in the Memoirs of the Royal Astronomical Society, vol. v. We thus obtain the following readings corresponding to each second of time from the beginning of the transit. The values of the successive differential coefficients have also been computed by means of the formula given above, which, when expanded, become = A$ - $ 4'$+ 1438 44'$+... do |