112 



hitherto employed the theory of Lagrange. The differential ^-j, 



in Lagrange's method, implicitly includes the differential of R with 

 regard to the mean motion. But it has been shown by Lagrange 

 himself, and since him by others, that we may finally omit the term 

 depending on the variation of », if we use fndt throughout instead 

 of n t. This reduction Professor Airy does not adopt, retaining ex- 

 plicitly the term , — . 

 r J an 



This difference in the formulas is equivalent to a change in the 

 meaning of the term epoch. In determining the longitude of the 

 epoch of the instantaneous ellipse, Laplace, and others who have 

 followed Lagrange, fix it by supposing this longitude to be the angle 

 which we must add to the angle described by the variable elliptical 

 mean motion since the origin of the time, in order to have the mean 

 anomaly in the ellipse. Professor Airy assumes the longitude of the 

 epoch to be that angle, which we must add to the angle described 

 by the instantaneous elliptical mean motion, considered constant 

 since the origin of the time, in order to have the mean anomaly : the 

 mean motion from the perihelion is, on the first supposition,/ n d t-\-s } 

 on the second, n t-\-s. In the results of the calculations in these 

 two ways there is no discrepancy, the difference of the formulae and 

 the difference of the suppositions necessarily balancing each other. 



It may be observed, that according to the method of the variation 

 of parameters, a large portion of the inequality, in the present in- 

 stance, falls upon the longitude of the epoch. The coefficient of this 

 inequality is something more than 2 ft ; which produces nearly the 

 same maximum amount in the longitude of the planet, the effects of 

 the variations of the other elements being insensible. 



In investigations of such extent and complexity as the one now 

 before us, the selection of notation is a matter of considerable im- 

 portance, in order to obtain the greatest possible degree of clearness 

 and brevity. In all cases when nothing is gained by the change, 

 it is convenient to the reader that the notation should conform to the 

 best established works already published. Professor Airy has in 

 general used the notation of the Mecanique Celeste. He has, how- 

 ever, introduced a new notation, in order to express in an abbre- 

 viated manner the differentials of the quantities C, taken m times 

 with regard to one major semiaxis, and n times with regard to the 

 other, and multiplied respectively by the ???th and nth. powers of 

 these semiaxes ; these products occurring so frequently, that the 

 adoption of a short symbol for them, (m, ») C, saves a great quantity 

 of very repulsive labour. 



Another abbreviation employed by Professor Airy respects the 

 angles. In the development of the terms arising from the suc- 

 cessive steps of the expansion of R, we obtain terms such as e- cos 

 2 F ana e' 3 cos 3 T, multiplied by others, such as cos (11 V— 11 T); 

 and hy the resolution of such products we obtain the cosines of the 

 sums and differences of these angles. But it appears that the sums 



