MINIMA SOLUTIONS IN THE CALCULUS OP VARIATIONS. 
Ill 
It is, in general, a quantity of the order a 3 (Art. 14), and it is continuous if we can 
alter it by amounts infinitely small compared to a 3 . This is always possible. First 
let the limiting values of x be unchanged ; then, writing, for Sy, Sy + 8 3 y, where S 3 y is 
infinitely small compared to S y or a, it is evident that the change in the integral is 
infinitely small compared to a. If, still keeping the same variation Sy + 8 3 y, we change 
the limits by writing x } -f- dx } , x 0 + dx 0 , for x } and x 0 , the most important additional 
terms due to this change, 
(Y 00 8y 3 + 2Y 01 8 y Sy + . . . + Y mi 8 y m ) dx 
Jo 
are also infinitely small compared to the original integral. 
Now, as in the original expression for S 3 U we may suppose Sy, Sy, . . . Sy <n ~ l) zero 
at each limit ,x 1 and x 0 , so we may suppose S 3 y so determined that Sy -j- S 3 y, 
• • 
Sy + S 3 y, . . . Sy (n ~ r> -j- 8 3 y ( " _1) , shall be zero for the values x 0 + dx 0 , x 1 + dx v 
Hence, representing by S 3 U)? the value of the second variation when the value of 
Sy is such that it and all its fluxions, up to the n th , vanish for both limits of integra¬ 
tion x 0 and x L , we see that we can always alter the value of Sy so that 
s a u:::s - s 3 u 
8 3 U^ 
■ ! n 
*i 
x 0 
is infinitely small. (Of course we might also alter it so that it should be finite.) 
It will simplify the further explanation if we represent the values of y by ordinates 
of a curve of which x is the abscissa, # and we will suppose that the value of Y nn at 
the lower limit is negative, so that the curve obtained by making SU vanish is a 
maximum when the upper limit is very near the lower one. 
Considering the lower limit of integration x 0 as a fixed point A, and the higher one 
x l as an arbitrary point M, on the curve ABC, we know from § 15 that when M is 
sufficiently close to x x the integral is a maximum, i.e., S 3 U^/(aq—a? 0 ) is a negative 
quantity of the order a 3 . Suppose the curve first ceases to give a maximum when M 
coincides with C. Then we may easily see that C is the point to which it first 
* It is evident that what has been said about the admissibility of a variation may be expressed thus: 
By is the difference between the ordinates of the new curve and old curve, and any curve is admissible, 
provided By n is nowhere infinite. Thus the broken curve ABC is admissible provided that at D and E, 
the points of junction with AEC, it has contact of the (n — S)*' 1 order. Furthermore, the difference of the 
integrals taken along ABC and AFC is the same as the difference of the integrals along DBE and DFE, 
as is seen at once by regarding the sign of integration as one of summation. 
