446 On the Calculus of Variations. [JUNE, 
sary not to confound a variation with a differential. Anelementary 
example will exhibit the proper distinction between these two de- 
artments of analysis. 
(Pl. LXVII. Fig. 8.) Let y* = a x represent the equation of a 
parabola A M, referred to the rectangular co-ordinates A P (x), 
PM (y), and of which the parameter is a. By drawing p m inde- 
finitely near to P M, and Mr parallel to the axis A V, the element 
P p or Mr will represent the differential (d x) of the abscissa, and 
the element 7 m the differential (dy) of the ordinate. The relation 
of the differentials d x, dy, is found by differentiating the equation 
adx _ adu 
Qy 2Vae° 
In the next place, let it be conceived that the equation 7? = ax 
varies by the indefinitely small increase 9a of the parameter a, 
which is one of its elements; and let a second parabola A N be 
constructed, which has a + da for its parameter. Then by sup- 
posing, first, that the abscissa A P remains the same for both para- 
bolas, it is manifest that the ordinate P N, of the parabola A N, 
will be represented by the primitive ordinate P M increased by the 
element M N, and will therefore represent the variation which the 
ordinate P M receives, in consequence of the variation of the 
parameter a. 
By representing, therefore, the variation of y by dy, as that of 
ais by Oa, it is necessary, in order to form the equation of the 
parabola A N, to substitute y + Oy for y, and a + 0a for a in the 
equation y? = ax, which gives (y + ?y)? = x (a+ a). Sub- 
tracting the primitive equation 7° = a x from the preceding equa- 
tion, and neglecting the variations of the second order, as in the 
y° = ax, which gives 2ydy = ada; ordy = 
. ee eee f4 x r ru xsa 
theory of differentials, we shall have 2y dy = xa, ordy= oes 
aoa ° : eye : . 
= Faq? am equation which exhibits the relation of the varia- 
ar 
tions da, Oy. 
2. If the abscissa A P be increased by the element Pp (3 x), the 
corresponding ordinate of the parabola A N will be gm, and the 
element s 7 will represent the variation of the primitive ordinate 
PM. To find the equation which expresses the relation of the 
variations da, dx, dy, in the equation 7? = aa, let y+ dy be 
substituted for y, x + da forx, and a@ + da for a, and we shall 
have (y + 9 y)® = (a + 0a) (2 +02) 5 from which subtracting 
the primitive equation y* = ax, and neglecting the variations of 
the second order, we then have 2y¢y = xda + ad «x; and there- 
fore. 3 y= ee oes oe — == ene =, an expression of the value 
of the actual variation sn of the ordinate. 
3. Scnotium I.—In this example, and indeed in all others of a 
similar nature, the parameter a, and its variation 0 a, are constant 
1 
