12 



GENERAL INTEGRALS OF PLANETARY MOTION. 



multiplied together, which renders it necessary to be more explicit in representing 

 the double summation we thus encounter. Having n of each of the quantities ^, 

 71, and ^ distinguished by writing the various values of the index i, which takes all 

 integer values from 1 to n, the quantities &, h, and N should all be affected with 

 this same index. But it is not necessary to write it after N or 5, because each 

 N is common to all the ^'s and jy's, or to all the ^'s, respectively. Again, we have 

 as many values of iV^as there are combinations of the coefficients ^l, 4, ig, etc., which 

 enter into it, while each A^has its corresponding coefficients Ic, i in number. We 

 must, therefore, consider h to be written 



"'( (''1? '^25 '^i ^3nJ? 



while h and N are affected with the same indices, the first excepted. In other 

 words, we have 



^ (ii, h, h : ■ . . . hn) = hh -r hh + + i-iAn 



^\h, 4, h iin) = HK + i-P-l + + hn'^Zn- 



Then, in the sense in Avhich we have hitherto used the sign of summation S we 

 have symbolically 



h=cc 



«i =— oc 

 To avoid the complication of writing so many indices we shall represent any one 



combination, as (y'l, v'a, 4,0 by the symbol j', and any other combination by 



1.1. We shall also put 



2 

 =— oc 



'3» =OC 



2 

 ,. = — oc 



8' = 



2 S. 



This summation includes all the terms in all the values of any one co-ordi- 

 nate, as £, Yi, or ^, respectivel5^ A sign for a summation including all 3w co-ordi- 

 nates is not here necessary, as Ic and iV^ are common to t, and >?, while the corre- 

 sponding quantities for ^, being of a different form, must be written separately. We 

 have, in fact, distinguished them by an accent. 



The co-ordinates and their derivatives which enter into the expressions (a,., aj) 

 will then assume the following form, the index t being understood after k and k'. 





= s^ 



evi_ 



- ,S' 



dak 





da. 



= Sfu 



^i = SJc^ cos N^ 

 rji = SfJCfj, sin iV^ 

 Ki = S,Ji!,j, sin N'^, 

 ^', = — S,(hkl sin AT 

 57'. ^= ,S, (bk), cos Ny, 

 ^', = S, (b'h'l cos A^', 



h 



(16) 



— ' cos a; — /•,, 



da^. da 



t sin N,^ 



die, 

 da^ 



sin A"^ + 7t:^ —^ t cos A"^ 





.a/^-^'^+^^^g^^-^^'^ 



(17) 



