1817.] On the Calculus of Variations. 447 
for the whole extent of the two parabolas, whilst the co-ordinates 
A P (a) and PM (y) continually change. The changes relative to 
the same parabola are related to DIFFERENTIALS; while those 
which result from the passage of one parabola to another are related 
to VARIATIONS. * 
Each of the variations da, 3x, }y, may be taken arbitrarily : 
thus, for example, we may suppose da = dx; but when this con- 
dition is once made, the values of the other variations are subor- 
dinate to it ; and it is not allowed afterwards to make dy = d y, or 
dy =a. 
"s. Scuotium JI.—There is no particular difficulty in determin- 
ing the variations of all orders of algebraic, exponential, and cir- 
cular functions. ‘To obtain the variation of a function, it is merely 
necessary to write 3 in place of d, the symbol of differentiation ; 
and in this respect the calculus of variations corresponds with the 
differential calculus. But the principles of the differential calculus 
are not sufficient, when it is required to determine the variations 
of functions, which contain the signs of integration, when those 
integrations are not capable of being effected. For example, let 
SV dx bean expression in which V is a given function of x, y, x, 
&c. and constant quantities; we differentiate by omitting the 
symbol /; which gives V d x for the differential ; but the expression 
of the variation Of V dx is very different. Now the principal 
olject of the calculus of variations is to determine the variations of 
integral formule of this species; and we proceed, therefore, to 
establish the principles which ought to serve as the basis of this de- 
partment of analysis. 
FIRST PRINCIPLE. 
The variation of a differential is equal to the differential of the 
variation, and reciprocally ; that is to say, 3(d 11) = d (311). 
For let it be supposed that the variable function IT represents the 
ordinate of a curve; this ordinate changes by differentials in the 
same curve; but by variations in passing from the proposed curve to 
another indefinitely near to it. Let IV’ be the consecutive value to 
II, for the first curve, and consequently I’ = II + dIl, ord = 
IV’ — Il. ‘Taking the variations of this last equation, it will become 
d(d 11) = oI’ — OTL. But since I and IV’ are consecutive quan- 
tities in the series of II, we may regard OT], 3 II’, as consecutive 
quantities in the series of 11; so that O11’ = 3IL + d(d11); or 
d(¢1%) = oI — dt. Thus by equating these two values of 
dil’ — oT, there will arise 3 (d 11) = d (31M). 
Coro.tiAry.—lIf, therefore, a function containing any number 
of d’s and @’s, which affect the same variable. we may make these 
characteristics change place at pleasure ; for we have found 3 (d IT) 
= d (311); consequently (d* 11) = d (3 (d T1)) = d* (3 11); and 
@(d3 11) = d (Od? Tl) = d* (3d Tl) = d* (311); &e. &e. 
* This remark is important to the student. 
