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would be easy to compute the values of the differential coefficients from the differences of the readings of o by means of the usual equation

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Before applying this formula, however, it will be necessary to smooth the curve by removing the periodic part in the value of p. The periodic terms are all contained in the expression Ae-at cos (St - e), and since this value for is a solution of the equation

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it is clear that the resulting curve is still a solution of the differential equation (4).

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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

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It will be observed that from the 5th second to the 125th the values

dt3

become so small that we may reasonably reject them. Similarly

from the 16th to the 116th second the values of

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to be of importance. Between the 130th and 131st second the sun passes off the instrument, and there is near here a point of inflexion on the curve, in consequence of which the method of computing the values of the differential coefficients by means of the differences is inappropriate. This is, of course, of no importance from our present point of view, as this part of the curve merely shows the motion of the couple when in process of cooling.

If now we multiply the values of the differential coefficients given in the above Table by the corresponding coefficients of equation (4) already obtained (p. 310), we find the following quantities :

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