MINIMA SOLUTIONS IN THE CALCULUS OF VARIATIONS. 
119 
and at the same time fulfil the conditions supplied by the given limiting values of the 
dependent variables and of all but their highest fluxions, the difference between the 
integrals for the surface so determined and that for any other surface will vanish so 
far as regards terms involving only the first power of a, provided only that the second 
surface can be obtained from the first by a change such that none of the A variations 
for A become infinite when aq, . . . x m have the values corresponding to any point 
within the region of integration. This, indeed, is true whatever be the order of a, 
but, in order that the sign of the difference between the integrals shall be the same as 
that of the second variation (the part depending on a 3 ), it is, in general, necessary 
that the quantities a A z, a A z , &c., be small. 
19. To determine whether the integral is a true maximum or minimum, we have 
now only to find whether the terms of this order a 3 in (8) will be always of 
the same sign when the variations are given in any values consistent with the 
conditions given in Art. 18. For it is evident that, if we restrict ourselves to a less 
general variation in examining the sign of the second variation, we could neither 
be sure that the conditions obtained were sufficient to ensure that the integral 
was synclastic, though they would be necessary ; nor that the conditions that it 
should be anticlastic were necessary, though they would be sufficient. If, on the 
other hand, we were to admit a more general variation, the conditions for synclasticism 
would be sufficient, but not necessary, and those for anticlasticism would be neither 
sufficient nor necessary. In fact, it will be found that the conditions under which we 
are discussing the problem are really those necessary in order that it shall have a 
meaning. For instance, in the case of least action, when we say that the action in 
the free path is less than in any other, we imply that there is to be no sudden change 
