MINIMA SOLUTIONS IN THE CALCULUS OP VARIATIONS. 
107 
are definite functions of x and the arbitrary constants introduced in the solution, it is 
evident that, x 0 being the initial value of ( x ), there will in general be some value x x 
at which it becomes impossible to determine the arbitrary constants, so that some one 
at least of the solutions z x . . . z n shall not have changed sign. Up to this point the 
transformation must hold, and the conditions for synclasticism derived from it must 
be sufficient and necessary. 
12. Confining our attention to integrals for which the transformation does hold 
(that is, integrals whose limits are not too widely separated), it is easy to see by the 
usual method that, unless Y nn dx retains the same sign throughout the integration, the 
integral cannot be synclastic. * For the integral 
S 3 U = |Y imi (SyM+a §y fe - 1) +&c.) 2 dx 
may then be divided into two parts, one negative and the other positive, and, as the 
form of Sy is arbitrary, we could make the numerical value of either of these parts 
exceed that of the other, and therefore S 3 U would be capable of either sign. 
But the condition that Y nn dx remains of the same sign throughout the integration 
is not sufficient to ensure that this integral shall be synclastic. This would be the 
proper place to examine the further condition if it could be derived from the preceding 
transformation, but it does not appear to me that we can avail ourselves of the 
analysis, for the following reasons :—Some of the quantities z r used in the tranforma- 
tion may vanish for some value of x included in the integration, and the investigation 
would not apply. Hence we should have to give an independent discussion to 
discover the limits of the integration within which the transformation does apply. 
But even then we should only have proved that the function was synclastic up to 
those limits at least, and we should have still to discover whether it might not be 
* It is usually stated that, unless Y m preserves its sign, the integral could not be synclastic; but 
this is a mistake arising from the supposition that, because dx increases from the lower to the higher 
limit, it must have the same sign throughout the integration. But it is evident from the figure 
that dx may change sign for a value of x between the limits, in which case there must be an even number 
of changes, or it may change sign an uneven number of times for values not numerically between the 
lower and higher limit, but yet passed through in going from the one to the other via the curve. It 
would be easy, by transforming the axes, to multiply examples of the latter, and in these Y, m and dx will 
change sign together. 
r 2 
