MINIMA SOLUTIONS IN THE CALCULUS OF VARIATIONS. 
99 
dy 
a, = y> 
d 2 y , 0 d n y . , 
&c., &c., -- =y () , 
dx 3 ’ ’ dx n J ’ 
( 1 ) 
and the partial fluxions of a function (_/) of these quantities with respect to any of 
them, say y (n) , will be denoted by df/dy M . It will often be convenient to use a 
bracket [ ]‘ to denote the result of subtracting the value of the quantity in the 
bracket, when taken at the lower limit of integration, from its value for the upper 
limit; thus by [y Sy] 1 is meant y x 8 y x — y 0 8y 0 . In other cases, where it is unnecessary 
to write out the limiting terms, the letter L will be used as an abbreviation for the 
expression “ terms depending on the limits only,” as in the following equations :— 
J y dx = L — J xy dx = L + \ j x^y dx, .(2) 
where, though the letter L is the same in both equations, it does not necessarily 
denote the same quantity. 
2. To obtain a clear insight into the nature of the problem before us, let us examine 
it in the most simple and familiar case, that in which there is but one dependent and 
one independent variable. Writing 
11 = \f ( x > y, y, y • • • y M ) dx > .( 3 ) 
let us call Y.U the total variation of U due to a change of form in y, by which it 
becomes y + Sy (y being always a function of x ): 
v - u = Kf + f ^ + &c - + Js W”) dx 
_J_ 1 f Sy2 + 2 /-f 8 y 8 y + &c. + -ff- 2 8 y (n A dx 
w 1.2 J yqd j ^ d ydy J J 1 ^ dy in)i J ) 
+ i.b { (fyi V + &c -) dx + &c - 
. (4) 
which we may write 
Y.U = SU + i 8 3 U + 17^73 S 3 U + &c., 
where 8 U is the part of Y. U depending on the first powers of Sy and its fluxions, 
S e U that depending on their second powers, S 3 U on their third, and so on. As usual, 
these will be termed the first, second, and third variations of U, and so on. 
3. The form of U having been determined so that the value of 8 U vanishes inde¬ 
pendently of the value of Sy, the terms which we have to examine are those of S 2 U. 
This quantity, as obtained in the first instance, is explicitly a function of x, y, and Sy. 
But, as the value of y obtained by making SU vanish is known in terms of x, S 3 U 
can be expressed in terms of x and Sy. When so expressed, S 2 U may either remain 
of the same sign, whatever function 8 y may be of x, or else it may have one sign 
o 2 
