404 Mk green, on THE DETERMINATION OF THE 



dV 

 — fdxidXi dxi—r—u'"-' . U' 



.fdx,dx,...dx,duu''-'.u\^r'^,+^ + '^^^ (12). 



Supposing therefore that U like V also satisfies the equation (2'), 

 each of the expressions (11) and (12) will be reduced to its upper line, 

 and we shall get by equating these two forms of the same quantity : 



idx^ dx2...dxs-j~ u'"-' V = fdxi dXi...dxs -y- «'"* U' : 

 au au 



the quantities bearing an accent belonging, as was before explained, to 

 one of the extreme limits. 



Because V satisfies the condition (9), the equation immediately pre- 

 ceding may be written 



dU' dV 



fdxidx2...dxs-j — u'"~' V = fdxidxi...dx,—y^u'"-' U,'. 

 du du 



If now we give to the general function U the particular value 



u= {{x, - x,"y + {x, - x,y + + {x, - xjy + u']^-, 



which is admissible, since it satisfies for V to the equation (2), and gives 

 U" = 0, the last formula will become 



dVi 



/dxidx-i dxsu'"'' —j-^ 

 du 

 {{x, - x^y + {x, - x:j + + (a;, - xlj + m'^}^ 



_r dxydx^ c?;g,.(l-w) «'"-'+' V , 



\{x, - xlj + (ar, - xij + + {x, - x:j + u''\'^ 



in which expression «' must be regarded as an evanescent positive 

 quantity. 



In order now to effect the integrations indicated in the second 

 member of this equation, let us make 



