378 Proceedings of the Royal Irish Academy. 



For instance, in the case of a plane curve, if OP be the stationary 

 curve, and Oabc . . . P the varied one, 

 the condition is that the integral treated 

 as taken along Oabc . . . P shall be treated 

 as the sum of the integrals taken along Ar 

 Oa^ ab, be, &c. ; or, as we may write 

 it— 



I{Oabc . . . P) = I{Oa) + I{ab)-v&c} (1) 



It willalso be supposed that the limiting ysluQ^ of all those quantities 

 which are only permitted to have small variations are given ; that 

 the increment of the independent variable is always positive, except 

 where specially stated to be capable of either sign ; and that all the 

 quantities involved in the determination of the stationary value are 

 continuous. 



§ 1. The resulting criteria maybe stated in general terms thus: — 

 Let the integral in question be written as 



/= I ...i^{^i,a?2,.--yi, y3.-.yi(''' "'•••', &c., yi(».^---), &c.} dx^dx^ . . . 



where Xi, x^, «S;c., represent the independent variables ; yi, y^, &c., the 

 dependent variables; and where y^i''^^'---), &c., is meant to include all 

 fluxions such as 



(a/X \ CtOC 2 • • • 



which are permitted to take small variations only, while ?/i("' ^' • • •), &c., 

 includes all those functions which are permitted to take variations of 

 any finite magnitude. 



Let also y'^'^' ^' • • •) typify the lowest of the (a, j3, . . .) fluxion, -i.e., let 

 it typify those fluxions which, though they are themselves permitted 

 to have arbitrary and finite variations, do not arise from the differen- 

 tiation of similar fluxions, but from the difl'erentiation of the y'"" • "^ 

 fluxions. 



Let also F<jj . . . and Ya^ . . . typify the functions 



dF ^ dF 



and 



dy(a,b,...? dy(a.,fi 



respectively. 



^ The cases iu which this equation is not admissible are evidently those in which 

 we have to take account of the value of the integral at the point of discontinuity — 

 in other words, where we have to regard the discontinuous variation Oab ... P as a 

 limiting case of a continuous variation. Such cases are excluded in this Paper. 



