110 
MR. E. P. CULVERWELL ON DISCRIMINATION OF MAXIMA AND 
the proposition will be proved when it is shown that throughout the integration 
Sy {n ~ 1} /Sy M , 8y <Jl ~ z) /8y in ~ l) , &c., are all negligible, for then the term Y nn 8y {n) ~ is 
obviously the most important. 
Now r x 
%r‘’= Si f>dx, 
j x 0 
no constant being added, as Sy (ll ~ 1) vanishes when x = x Q . Let the numerically greatest 
value of x — x 0 in the integration be (3, and that of 8y (n) be y ; then, numerically, 
8y {n ~ l) < fiy. 
A fortiori, 8y (n ~ 1) < fi' 2 y, for, if /3~ be the greatest value of Sy ( " _1> , 8y (n ~ : < ft ft. 
But ft < (3y ; similarly 8y (,l ~ S) < fty, and so on. Hence 8y { "~ l) /8y M is of the order /3, 
and similarly for each of the fractions.* 
It follows from this that the only term of the order a 2 in the expression for 8 2 U is 
It follows that, if Y nn dx does not change sign in passing from x 0 to x x , neither can 
S 3 U change sign, whatever be the form of 8y, and it is clear that S 2 U/U is of the 
order a 2 , exactly as if the integral were taken over a finite portion of the curve ; 
and it should be remarked that the value of S 2 U is of exactly the same order as 
if the limiting variations 8y . . . 8y {n ~ l) had not been zero—that is, the order of S 2 U 
is the same as if the most general variation possible had been given to y, although 
the actual variation is zero, as also are all its fluxions up to, but not including, the n ih . 
It might possibly be objected that, as 8y u l) is zero at each limit, therefore 
is also zero. Hence, as dx may be taken to increase uniformly, 8y {n) must change sign 
between x = x () and * 1C ~ ; and, as these values are very close, 8y (n) is everywhere very 
near its vanishing point, and is therefore everywhere very small. But this proceeds 
on the idea that 8y (n> must be continuous. There is no difficulty when it is remem¬ 
bered that 8y M may change suddenly from a positive value to a negative value. 
16. Thus, without any analytical transformations, it has been shown that if Y„„ be 
positive the integral is a true minimum, and if Y nn be negative it is a true maximum 
when the integration is extended over a very small range. We have still to consider 
how far the integration may be extended without annulling this property. With¬ 
drawing the restriction that x x — x Q is small, let us consider the continuity of the 
second variation 
8 2 U = 
P (Y 00 % 2 + 2Y 01 By 8y + &c. + Y m 8y «*) dx. 
J.r 0 
* It follows similarly that, provided (3, the range of the integration, be not too great, the sign of c :; U 
is the same as that of I 1 (d 3 //Sy (Ml3 <Bw (,i)3 ) dx. that of oRJ the same as that of | ^dYldy ln)i (Sw ( ' {) ) 4 cfoj, and 
J*o ' J*o 
so on; and this does not in any way depend on y having its synclastic value. 
