118 
MR. E. P. CULVERWELL ON DISCRIMINATION OF MAXIMA AND 
For our purposes it will be necessary to take a of such a magnitude that the part 
depending on a 2 is greater than all the subsequent terms. (It seems well to observe 
that this does not, in general, imply that a is infinitely small.) But there are restric¬ 
tions to the values of the quantities A 2 , A z, &c., and in order to find them it will be 
necessary to give an outline of the method which is usually employed in the calculus. 
The part of (8) depending on the first power of a is reduced by successive integra¬ 
tion by parts, so that the part of the integral not solely depending on the limiting 
values contains only the variations A y x , A y 2 , &c., and does not contain their fluxions. 
Thus, if D represent the operation by which 2 is got from yfi we shall get from the 
term 
such terms as 
df 7 
v A z dx i 
dz 
. . . dx 
m 
a |.... | (B 1 dx 2 dx 3 . . . dx m + B., dx x dx ?j . . . dx m + &c.) 
where the first part of the right-hand side depends only on limiting values. Applying 
similar reductions to all terms containing fiuxions of variations, we get an expression 
of the form 
SU = a | . . . | {L : dx x dx z . . . dx m + L 3 dx x dx. z . . . dx m + &c. } 
+ a J . . . | (A x A y x + A 3 Ay 2 + &c.} dx x clx 2 . . . dx m 
• (9) 
This integration by parts depends for its validity on the supposition that no one of 
the quantities As becomes infinite for any values of x x . . . x m within the limits of 
integration. But it is very important to remark that the integration is legitimate, 
whether the variations of the highest fluxions are or are not discontinuous in the 
sense of suddenly changing from one finite (properly, not-infinite) value to another. 
It is to be observed that, in discussing the sign of S 2 U, we introduce no limitations 
except those already implied in the usual treatment of SU. 
Again, it is to be observed that the terms in the limiting integrals in (9) will 
contain the limiting values of all but the highest fluxions of the variations. Hence, 
when we say that the limits are given, we mean that the values of the dependent 
variables and of all but their highest fluxions are given for all points on the boundary 
of the region of integration. Hence, if we determine the forms of the functions 
y x . . . y K so as to satisfy the equations— 
Ihus, when 
