42 SIXTH UEPOKT — 1836. 



Xi = ^, (s, a, a,, 



a„, a\, . . «'„), 1 

 «„, a'l, . . a'„) ; J 



(3) 



and, by the help of the initial equation analogous to (1), might then 

 eliminate a\, . . a'„, and deduce a relation of the form 



= \p (s, X,, . . x„, a, a,, ..«„); (4) 



that is, a relation between the initial and final values of the n + 1 con- 

 nected vaiiables s, x,, . . x„. Reciprocally, the author has found that if 

 this one relation (4) were known, it would be possible thence to deduce 

 expressions for the n sought integrals (3) of the system of the n diflFeren- 

 tiaJ equations (1) and (2), or for the n sought relations between 

 s, a^j, . . x„, and a, a,, . . c„, a',, . . «'„, however large the number n may 

 be ; in such a manner that all these many relations (3) are implicitly 

 contained in the one relation (4), which latter relation the author pro- 

 poses to call on this account the principal integral relation, or simply, 

 the PRINCIPAL RELATION, of the problem. 



For he has found that the n following equations hold good, 



fjjds) _ / {dx,) ^ ^ /> (dx„) , . 



^' (s) ^' (xj ■ ■ ^' ix„) ' ^ ^ 



which may be put under the forms 



a, = ^1 (a, s, Xi, . . x„, Xi, . . x'„), "1 



• • . , . \ <^) 



a„ = (p„ (a, s, Xi, . . x„, X i, . . x„), J 



and are evidently transformations of the n sought integrals (3)- 

 And with respect to the mode in which, without previously effecting 

 the integrations (3), it is possible to determine the principal relation 

 (4), or the principal function which it introduces, when it is conceived 

 to be resolved, as follows, for the originally independent variable s, 



s = tp (j7,, . . x„, a, «,, . . a„), (7) 



the author remarks that a partial differential equation of the first order 

 may be assigned, which this principal function ^ must satisfy, and also 

 an initial condition adapted to remove the arbitrariness which otherwise 

 would remain. In fact the equations (5) may be thus written, 



Sds Ss Sds Ss ._v 



S dxi S x^' " Sdx„ S x„' 

 in which 



dXi f (ds) oXi 



and since, by (1), there subsists a known relation of the form 



T- / S ds S d s /,„» 



o=F(s,x,...x„,—.—, (10) 



the following relation also must hold good, 



o = F(s,x„..xJ'..i±, (11) 



Xi x„ 



that is, the principal function ip must satisfy the following partial differ- 

 ential equation of the first order, 



