122 MR. E. P. CULVERWELL ON DISCRIMINATION OP MAXIMA AND 
equation, be able to make any one of these terms larger than the sum of the others. 
It is necessary, however, to show that such an equation has at least one real solution, 
and this can be done by the method of expansion in series just as it is usually done 
for the case of two variables onlv. 
*/ 
Thus, the criterion for a maximum or minimum value of the function has been 
found when the region of integration is small. It is the same as that obtained by the 
methods of transformation ; but the proof now given is free from the uncertainty 
which is connected with analytical proofs. 
21, The criteria for the case where the region of integration has any finite 
magnitude can be derived from the preceding by considerations depending on the 
continuity of the integrals. Remembering that we are still treating of the case where 
the limits are fixed, we may prove the following proposition :— 
If it is possible to take, around every point P in a region R of finite magnitude, a 
minor region (p), no matter how small, such that the integral U for that region is 
synclastic, then the integral for the entire region R will be synclastic, provided the 
further condition be fulfilled that it is impossible to take within the region R a second 
synclastic surface V', having at all points of its limiting intersection with the first 
the same values for the dependent variables and for all their fluxions, with the 
exception of the highest. If it be possible to find such a surface, the integral U will 
be anticlastic for the region R. 
Let us consider how it could happen that S 3 U became capable of either sign at 
pleasure when the region of integration is extended. Let S be the region for which 
this change of sign first becomes possible. Lienee (restricting ourselves to the case 
where the function U has a minimum value), when the integration extends over any 
region wholly contained in S, S 3 U is positive, while, if it be extended over a region 
including S, S 3 U can change sign. It is clear that this can only happen if the least 
value of S 3 LT is zero when the integration is extended over S. This mav be shown 
thus. The second variation being written 
S 3 U = a 3 j . . . | S yryy Az A z dx l . . . dx m , 
where Az, Az', &c., are finite quantities, it is evidently capable of being changed by 
an amount infinitely small compared to a 3 , by an infinitely small change in Az, Az', &c., 
and the new values of Az can be made to satisfy limiting conditions obtained from the 
previous ones by infinitely small changes. It follows hence that, if it were possible to 
take such values for Az, Az', &c., that S 3 U should be a negative quantity of the order 
a 3 when the integration extends over a region S + d S, greater than, but differing 
infinitely little from, S, we could, by an infinitely small change in the variations, 
obtain values making S 3 U negative and of the order a 3 within the region S — d S, 
which by the supposition is impossible, as within S the integration of S 3 U gives a 
result which is always positive. It follows that the greatest negative value of S 3 U 
