2 GENERAL INTEGRALS OF PLANETARY MOTION. 



h a function of the eccentricities, inclinations, and mean distances of the two 



planets, developable in powers of the two former quantities. 

 I, T the mean longitudes of the planets. 

 7t, 7t' the longitudes of their perihelia. 

 6, 6' the longitudes of their nodes. 

 i, j, Jc, numerical integer coefficients, 

 and in which i' -\- i -\- f -{- j -\- ^^' -|- A' = 0. 



The coefficient h is of the form 



Ae'e'J'<pY'' (1 + Ae'+ A^e"" + etc.), 

 while the circular function of which it is a coefficient may be put in the form 



si°i ^^'^ +'^"^' + ^'^^ +^>^''^') cos (i7' -f ^7) 



i co^ ^^'"^ '^^'^ + ^^^ + ^""'^'^ ^"^ *^*^' + ''^^• 

 As, these equations have hitherto been integrated the different elements are 

 developed in powers of the time, and we are thus led to expressions of the form 



But it is clear, that we shall get more general expressions if, instead of using 

 developments in powers of the time, we substitute the general values of the ele- 

 ments given by equations (1). The substitution will be most readily made by 

 reducing the circular to exponential functions. Putting in (1) for brevity 



^,^ + /3, = \- 



hit -f- Yi = /I'i 

 and 



n = e^v^^ 

 A = e^y/-^ 

 = f ev^-i, 



the equations (1) may be put in the form 



en =2,i^A 



en-^=2iiV;A-^ 

 ^0 = XiM^Mi 

 ^0-1 z=^^MiM-\ 



In the preceding differential to be integrated the coefficient of («T -|- H) is of 

 the form 



(1 + A,e^ + Ae'= + etc.) A^ e'-'" ^* ^^ "^^^ {jn -^j'yt! + W + W). 



If in the last factor Ave substitute the preceding exponentials for the circular 

 functions, its product bv e'e'-''^*^* in the case of a cosine reduces to half of the sum 



(eny (.'n')" mr w&r + Q' {g(lf (|.)". 



Substituting the values of these expressions in terms of the exponentials just 

 given, developing by the polynomial theorem, and then substituting for the expo- 



