MINIMA SOLUTIONS IN THE CALCULUS OF VARIATIONS. 
109 
enough, we can always ensure that the above expansion holds. So far, nothing has 
been said as to the continuity of Sy, §y, &c., but, Sy, &c., being the successive fluxions 
of a single quantity Sy, they must all be continuous functions of x except Sy w , the 
highest fluxion, whose magnitude is not so restricted. For, if 8 y (r) changes suddenly 
from one finite value to another, for change of x from x to x + dx, its differential 
coefficient, 8 y {r+l \ would become infinite for that value of x. But, as Sy ( ^ +I> does not 
occur in f{x,y,y... y {n) ), the validity of the expansion will not be affected by its 
becoming infinite, and therefore By M may change from one finite value to another. 
14. Hence, if we discuss the problem of maxima and minima by the usual method, 
the variation we give is of necessity restricted as follows : Sy, Sy,. . . 8y ( " -1) must all be 
continuous functions of x. 8 y (n) need not be continuous, but the magnitude of each 
fluxion must be restricted with certain limits, which will vary with the nature of the 
problem under discussion, but it will in all cases be sufficient to make them infinitely 
small. It will be convenient to consider 8y as afix, where a is a small numerical 
coefficient, and cf>x is a function of x , such that it and any number of its fluxions may 
become zero, though in general they will be finite, while neither the function itself nor 
any of its fluxions up to and including the n th can become infinite for any value of x 
occurring in the integration. 
The coefficient a must be taken sufficiently small to ensure that, when considering 
only the sign and not the value of an expression involving it, we may neglect terms 
depending on a 3 or higher powers in comparison with those depending on a. Denoting, 
as usual, by SU, S 3 U, &c., the part of the expansion depending on the first, second, 
&c., powers of Sy and its fluxions, we may say that SU is of the order a, S 3 U 
of the order a 3 , and so on. Hence, in the absence of special determinations of the 
form of Sy, S 3 U, the part depending on a~ will exceed all terms depending on a 3 and 
higher powers of a, and then the sign of the whole variation will be the same as that 
of S 3 U (SU being zero when y has its synclastic value). 
It is convenient to have a geometric representation, and the function y will be 
taken as the ordinate of a curve of which x is the abscissa, and the curve corre¬ 
sponding to the synclastic form for y will be called the synclastic curve. 
15. We may now easily prove the following proposition:— 
Let fXi 
U = f{x,y,y... y U) ) dx, 
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y being any function of x. Let the second variation S 3 U be taken, subject to the 
condition that Sy, Sy, . . . Sy ( ' l_1) are zero at each limit. Then the sign of S 3 U is the 
same as that of the term involving Sy°' )3 in the integral, provided the range of the 
integration be sufficiently small. 
The second variation being written 
S 3 U = f (Y 00 Sy 2 -f 2Y 01 Sy Sy + Y u Sy 3 + &c. + 2Y„_ lj „ Sy ( " _1) S y 0l) + Y n>l S y m ) dx, 
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