GENERAL INTEGRALS OF PLANETARY MOTION. 17 



we shall have 6n variables, represented by ^f, vj^, ^^ ^\ y}\, ^\, expressed as functions 

 of the 6n quantities 



Ci, Cj, C3 C3„, Ai, A2, A3 A3„. 



Let us now suppose these equations solved with respect to these last quantities. 

 We shall then have 6« equations of the form 



Ci=z^^; Ai = n^i, whence I^ — ^'i — hit, (26) 



^ and ^| being functions of ^, y;, ^, etc. The first and third of these expressions 

 are the 6n first integrals of the given equations, or, what we may call the integral 

 functions, being those functions of the co-ordinates, and the time, which remain 

 equal to arbitrary constants during the entire movement. 



Let us now, for generality, once more represent the 6n arbitrary constants by 



ftj, Cto, <^0?J» 



and let us consider the (6ny quantities of Poisson formed from the general ex- 

 pression-^ 



"6'a^ 6a, 6a^ 6a, 



, rda da, da da, -| „„ 



the symbol S't including, as in (14), the 3// values of ^, yj, and ^ in succession. Put- 

 ting the general expression (14) in the form 



^a,, aj) — ^, I J, 



Lo'a; 6a, 6a. 6a. 



''J * J 



forming by multiplication the product of this expression by (27), then putting 

 V =y, and forming the summation 



2 («f., a,) («i,«;), 



noticing also that the expression 



^=£6a^6_aj 

 j=i 6aj6y 



is equal to unity whenever x and y represent the same symbol, and to zero in the 

 opposite case, we find 



an expression which is itself equal to unity when fi = i, and which vanishes in all 

 other cases. 



Now Ui, aj, and a^ may here be any of the 6n arbitrary constants. Let us then 

 suppose a^, a^ to represent l^ and Z^ respectively, and aj to represent Cj. This equa- 

 tion will then become 



ih ci) [lu, cJ + (l,, c,) [?^, cJ + (?„ C3) [l^, C3] + etc. = 1 or ■ 



' It will be observed that the notations introduced by Lagrange and Poisson respectively, are here 

 reversed, a proceeding which was not intentional on the part of the writer 

 3 November, 1874. 



