4 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



dl dn __ tlm_ . dn 



so that they determine five of the quantities a, b, c, /, g, h, in terms of the 



remaining one, say 



dn 

 J == </ c 



dm dn 



dy *dy 



_ dn 



dl_ rf, 2 



rfwi rf- 



<M <"'/' 



r?// f/a; 



We also have 



F (v, x, y, z, I, m, n, a, b, c, f, g, h) = 0, 



so that, in general, there are six equations to determine the six derivatives of the 

 second order ; and if F is algebraical in a, b, c,f, y t h, there will be a limited number 

 of sets of values of these quantities, which can therefore be regarded as known along 

 the surface. 



In the same way, the derivatives of higher order can be deduced everywhere on 

 the surface, and so, taking any point as an initial point, we have the values of all the 

 derivatives of v known there ; we then have a series in powers ofx x , y y , z z , 

 which, in CAUCHY'S theorem, is proved a converging series when oc y^ is an ordinary 

 point in space for the equation : and consequently we infer the existence of the 

 solution as established by CAUCHY'S theorem.* 



This conclusion is justified only, however, if the equations do actually determine 

 sets of values of a, b, c,f, y, k. In the case where sets of values are not determined, 

 so that, e.g., the equation F = becomes evanescent on the substitution of the values 



* The most general form of the theorem may be stated as follows : 



If (*, y, z) = be an ordinary relation for the equation F = 0, that is, if it is not a solution of the 

 characteristic invariant equation, then a solution v of the equation F = exists satisfying the 

 conditions : 



(i) v is equal to a given arbitrary function of x, y, e, everywhere along the surface = 0; 



(n) one of the derivatives of v is equal to a given arbitrary function of .r, y, z, everywhere along 



the same surface ; 

 and a solution satisfying these conditions is uniquely determined by them. 



See also a paper, Proc. Lond. Math. Soc., vol. xxix, 1898, pp. 5-13, in particular, p. 12. 



