340 Mr. Lubbock on the Variation of 



d c = [c 9 a] -3 — d t + [c 9 ©I -5 — d * + &c. 

 L J da LJ dco 



The quantities between brackets may be considered as constant 

 in the first approximation, and such that 



[c 9 a] = -4 [0, e] [to, c ] == — [>, ©]. 



Similar theorems exist with respect to all the constants, and it is 

 upon this property that the theorem in question may be said to de- 

 pend. If a, e, y, c, ft), a, have the same signification as in Poisson's 

 memoir (Memoh'e de VInstitut 9 torn, xiii.) 



0, e]=0 [>, y] = [>, y] = 



[>, o)] = [c,y] =0 [ft), a] =0; 



but these latter theorems do not appear to influence the proposition 

 of which the proof is required. 



In differentiating the disturbing function with regard to c 9 as in- 

 dicated in the expression 



d a = =- u t 9 



an dc 



a being the semi-axis major, R must be differentiated only with re- 

 spect to c, inasmuch as it was contained in It primitively, and not 

 as it is introduced in further approximations by the variations of 

 the elliptic elements. As, however, the secular variations and 

 those multiplied by m and m° 9 which I have already included in 



<*>• (")■ ( d aD. *• 



do not contain i or c 9 and as therefore c only occurs in the quanti- 

 ties 



as it was introduced primitively *, 



f 



§5, $a 9 8 e, &c, denoting here only that portion of the variations 

 of f, a, e, &c, which consists of periodical terms not independent 

 of /, and therefore multiplied by the square of m at least. 



* This remark applies solely to the constant c, and to no other. 



