CuLVERWELL — Solutions in the Calculus of Variations. 381 



Then, according to the conditions laid down, we must have, at A 

 and JB, 



%2('-2) < k, (ro = 0, 1, 2, . . . cr,. - 1) i ' 



and the values for these fluxions are the same, whether we derive 

 them from the element AB, or from the contiguous elements ; and 

 this is to be true whether the points A and £ are or are not points 

 of discontinuity. 



Join A and 5 to P by two curves AF and BP which at A and B, 

 respectively, have the same values of 8yi, 8t/2, and the (r) fluxions of 

 those quantities (i.e. for the functions in (6) ) as those derived from 

 the element AB, and which shall also, from ^ to P and B to F 

 inclusive, satisfy 



Syi(<^i) <k{ai = ai, ai-¥ 1, . . . ?h) \ 



Sy2^°-2^ < ^ (02 = «2, «2 + Ij • • • %) ' 



as well as the further conditions 



j S^.W^O, {p,^0, 1,2, ... K-1)) 

 l\,ih) = 0, {p, = 0, 1, 2, .. .(%-!)) 



r> 



(8) 



which are the ordinary conditions of "fixed limits." 



It is evident that it is always possible to draw curves AF and BF 

 satisfying the conditions (6), (7), and (8) ; in particular, it is to be 

 observed that the existence of a " conjugate " point to F between A 

 and F, supposing the two to be joined by a stationary curve having 

 contact of the usual order with AF at F and A, does not affect the 

 possibility of the variation here supposed. 



§ 7. Let us now take the value of the integral all round the 

 triangle FABF. Evidently, 



I{ABFA) = I{AB) + I{BF)-I{AF). (9) 



We suppose that AB is an element in which the higher fluxions 

 yW^ y(o+i)^ ^(3., have variations of arbitrary magnitude, 



-(a) (a) -(o + l) (0 + 1) 



y -y , y -y , &c., 



so that we cannot expand them by Taylor's theorem. But as Fx < k, 

 we may write 



'B _ _ lA 



F{yi,y2) = j F{y„y,)Dx + {h),, 



