16 GENERAL INTEGRALS OF PLANETARY MOTION, 



we should have in (23) and (24) by putting E = Q. — Ho, 



' (9c, 



§ 5. Fiindd mental Relation between the Coefficients of the time, b^, 6,, etc., considered 



as Functions of c^, Cj, etc. 



In the preceding section we have found ourselves able to express the first approxi- 

 mate values of the variables in terms of 3?? pairs of arbitrary constants 



in which the two members of each pair are conjugate to each other ; or possess the 

 property that the expressions (14) all vanish except when a^ and a^ represent the 

 two members of a conjugate pair, in which case we have 



(Z„ c,) = + 1. (25) 



The distinguishing characteristic of the integrals we have been investigating is that 

 Ihey do not contain the time, except as multiplied by the 3?i factors 6, which are 

 functions of the 3ri constants c. This characteristic will enable us to deduce a 

 fundamental relation between the diiferential coefficients of h with respect to c. In 

 the first place, we remark that each c has a 6 to which it stands in a peculiar rela- 

 tion, in that the latter, multiplied by the time, is added to the I, which is conjugate 

 to c to form the corresponding X. The theorem in question is this : each b being 

 supposed to be marked with the index of its corresponding c, we shall have for all 

 values of i and j from 1 to S;?, 



dCj dCi ' 

 in other words, the expression 



2 bidCi 

 will be an exact differential. 



It is quite possible that this theorem may admit of being deduced immediately 

 from the preceding theory, but I have not succeeded in doing so, and have there- 

 fore been obliged to consider the problem in the reverse form. We have, in start- 

 ing, supposed ourselves to have completely expressed the 3?i co-ordinates ^, vj, ^, as 

 functions of the 6n quantities 



and we have just shown how to replace the first Zn quantities by the quantities 

 c^, c, Cjjj. If we add to these the first derivatives of the co-ordinates (16) 



