406 Mr green, ON THE DETERMINATION OF THE 



the integral (14) will in consequence become 



and thus the equation (13) will take the form 



dx.dx, dx,u"^-^^ ~^'^-^ ^~V^ ) 



J {{x,-x; 



J + {x, - x;j +.... + (x, - x:j + m'^ } -^ r (■ 



w-l 



In this equation V '\?, supposed to be such a function of x^, x.^ x, 



and u, that the equation (2) and condition (9) are both satisfied. More- 

 over V'^O, and Vo is the particular value of F' for which 



Let us now make, for abridgment, 



dV 

 P = u"-' -r-, {when u = 0) (A), 



and afterwards change x into x\ and x" into x in the expression im- 

 mediately preceding, there will then result 



_- s f fi—s + V 



r dx^ dx2 . ...dx,'P,' ""^^'^ I 2 / „, ,,^^ 



/ ^^ rr — r; f^ •••U5), 



{{x,'-x,f+{x,'-x,y + ...+{'>':-^sY+u"]— r(^) 



--^(^).„ 



P' being what P becomes by changing generally Xr into x,', the unit 

 attached to the foot of P' indicating, as before, that the multiple 

 integral comprises only the values admitted by the condition {a), and 

 V being what V becomes when we make u = 0. 



The equation just given supposes u' evanescent; but if we were to 

 replace u with the general value u in the first member, and make a 

 corresponding change in the second by replacing F'' with the general 

 value F, this equation would still be correct, and we should thus have 



