390 Proceedings of the Royal Irish Academy. 



Whence, from (23), 



= f (^(y) - F{y) dx + /^ Y, hj - r M^ydx, (24) 



where now i^^ stands for (y + A Jl + ?/-), and Zi and Mare quantities 

 derived from i^ according to known rules. But since M= for the 

 stationary curve, (24) may, neglecting small quantities, be written as 



[fQ) - F{y) - Y,(y - y)\Dx^ Edx. 



Hence, the general rule is, that whatever be the orders of the 

 highest fluxions in the two integrals, the function E is that formed in 

 the usual way from the function under the integral sign in Euler's 

 method. 



§ 14. When we require to make J udx a minimum subject to an 

 equation of condition «; = 0, a precisely similar mode of treating the 

 equation shows tbat we obtain the function E by writing [xi + \v) 

 for F\ but in that case, of course, the highest fluxions y^"\ z'"^, &c., 

 cannot be all arbitrary, because v = is to be everywhere satisfied. 

 But in this case it is very necessary to observe that the problem is 

 unmeaning, unless the conditions admit of our taking a variation. 

 OABP, in which, while the variations are finite in AB, they are 

 zero in OA, and indefinitely small in BP. If, for instance, the 

 problem be the old one of the shortest line in space of given curva- 

 ture, then if OP be the stationary solution, it will be found impossible 

 to take any line OABP of constant curvature, and such that while 

 d-yldx"- and dh/dx~ axe finite in AB, they are indefinitely small in BP, 

 for this would involve a discontinuity in the curvature at B. 



§ 15. Weierstrass shows that, if x, y, x and y only appear in the 

 function F^ the independent variable being t, and if the question be 

 really one relating to a plane curve, there can be no maximum for 

 discontinuous variations such as are here dealt with. For he finds 

 that ^ is a quadratic function multiplied by xdt, and as x can change 

 sign arbitrarily, the function E can change sign. But as xdt = dx, 

 this only means geometrically that if dx can arbitrarily change sign, 

 there can be no true minimum or maximum, as is at once evident, 

 because, if the independent variable, which we may take as x, may 



