Page 91 - Foundations Of Differential Calculus

74 4. On the Nature of Differentials of Each Order
145. The symbol d does not affect only the letter immediately following
2
it, but also the exponent on that letter if it has one. Thus, dx does not
2
express the differential of x , but the square of the differential of x, so that
the exponent 2 refers not to x but to dx. We could write this as dx dx in
the same way as we would write the product of two differentials dx and
2
dy as dx dy. The previous method, dx , has the advantage of being both
briefer and more usual. Especially if it is a question of higher powers of
dx, the method of repeating so many times tends to be too long. Thus,
3
dx denotes the cube of dx; we observe the same reasoning with regard to
2 4
differentials of higher order. For example, d y denotes the fourth power
√
3 2
2
of the second-order differential d y, and d y x symbolizes the product of
√
the square of the differential of the third order of y and x.Ifitwerethe
3 2
product with the rational quantity x then we would write it as xd y .
146. If we want the symbol d to affect more than the next letter, we need
a special way of indicating that. In this case we will use parentheses to in-
2 2
clude the expression whose differential we need to express. Then d x + y
2 2
means the differential of the quantity x + y . It is true enough that if we
want to designate the differential of a power of such an expression, then
2 2 2
ambiguity can hardly be avoided. If we write d x + y , this could be
2 2
understood to mean the square of d x + y . On the other hand, we can
2 2 2
avoid this difficulty with the use of a dot, so that d. x + y means the
2 2 2 2 2 2
differential of x + y . If the dot is missing, then d x + y indicates
2 2
the square of d x + y . The dot conveniently indicates that the symbol
d applies to the whole expression after the dot. Thus, d.x dy expresses the
√
2
2
3
differential of xdy, and d .x dy a + x is the third-order differential of
√
2
2
the expression xdy a + x , which is the product of the finite quantities
√
2
2
x and a + x and the differential dy.
147. On the one hand, the symbol for differentiation d affects only the
quantity immediately following it, unless a dot intervenes and extends its
influence to the whole following expression; on the other hand, the inte-

gral sign always extends to the whole expression that follows. Thus,
2 2 n
ydx a − x denotes the integral of, or the quantity whose differential
2 2 n
is, ydx a − x . The expression xdx dx ln x denotes the quantity

whose differential is xdx dx ln x. Hence, if we wish to express the prod-

uct of two integrals, for instance ydx and zdx, it would be wrong to

write ydx zdx. This would be understood as the integral of ydx zdx.
For this reason we again use a dot to remove any ambiguity, so that

ydx · zdx signifies the product of the integrals ydx and zdx.
148. Now, analysis of the infinite is concerned with the discovery of both
differentials and integrals, and for this reason it is divided into two principal
parts, one of which is called differential calculus, and the other is integral
calculus. In the former, rules are given for finding differentials of any quan-