MINIMA SOLUTIONS IN THE CALCULUS OF VARIATIONS. 
121 
p 'o c/Az 
p 0 dx i 
dx = 0, 
where dAzjdx x is one of the highest fluxions, and P 0 and P' 0 are two points on the 
boundary for which the value of x x alone is different, those of x, 2 , . . . x m being the 
same. It follows from this that dAz/dx x must change sign once at least in passing 
from P 0 to P 0 ', and therefore, if it varied continuously, must everywhere he infinitely 
small. Since, however, there is nothing to prevent djd,x, dAz/dx x , being infinite (§ 18), 
dAz/dx x may have a finite value within the region. 
Remembering that the region is small, and that the order of magnitude of A<7, A b, 
&c., a, b, &c., being “highest fluxions,” does not depend in any way on that of the 
region of integration, we see that the sign of S 3 U will be the same as that of 
| ... | (Act) 3 -)- 2 (A a Ah) + &c.^ dx x , dx 2 , . . . dx m . . . (11) 
A a and A b representing highest variations. As the region is small, d 2 f/da 2 , drfjda db, &c., 
may be considered as constants throughout the integration, a supposition which again 
involves neglect of small terms. (Act, A b, &c., cannot be regarded as constants, however 
small the region may be, for their differential coefficients may be infinite.) Now, the 
part inside the bracket can be resolved into a sum of squares, and, if the coefficient 
of any one (or more) of these squares is negative, the expression can be given either 
sign. For, although the quantities Act, A b, &c., are not independent of each other to 
the extent that all the other square terms could be made to vanish, yet they are so to 
the extent that any one term may be made to exceed all the others; for instance, if 
the region of integration be that for which 
x x + x 2 + &c. ... — r' 2 = < 0 or <£' = < 0, 
and 
x x -f- x 2 2 + &c. . . . — r" 2 — < 0 or <j)'' — < 0, 
r and r" being small quantities of the order /3, we may assume 
and so on for the others. For these assumptions will satisfy the limiting conditions, 
whatever be the forms of f x , f 2 , &c., which may be regarded as quite arbitrary, and 
they will give finite values to the “ highest fluxions ” of the dependent variables. If 
we resolve the quadratic expression in (11) into a sum of squares, and substitute for 
Act, A b, &c., their values in terms of f x , f 2 , &c., we shall, by solving a differential 
MDCCCLXXXVII.—A. R 
