Page 172 - Foundations Of Differential Calculus

8. On the Higher Differentiation of Differential Formulas 155
Then in a similar way for the higher differentials of y we let

dy = P dt, dP = Q dt, dQ = R dt, dR = S dt, . . . ,
so that
2 2 3 3 4 4
d y = Qdt , d y = Rdt , d y = Sdt , ....
When these substitutions are made we obtain an expression that contains
only the differential dt besides the finite quantities x, p, q, r, s, etc. y, P,
Q, R, S, etc. It follows that there is no unsettled signification.

268. If the first differential that is made constant depends on x or on y,or
if it depends on both at the same time, then it is not necessary to introduce
a pair of series of finite quantities p, q, r, etc. Indeed, if dt depends only on
x, then the letters p, q, r, etc. will be functions of x, and only the letters
P, Q, R, etc. will be present. The same thing will occur if the constant
differential dt depends only on y. However, if dt depends on both, then the
operation must be changed a bit. For example, we let the differential ydx
2
be constant, so that yd x + dx dy = 0 and
2 dx dy
d x = − .
y
Now let

dy = p dx, dp = q dx, dq = r dx, . . . ,
so that
2
2 pdx
d x = − .
y
When we differentiate further, we obtain
3 2 2 2
3 qdx p dx 2pdxd x
d x = − + 2 − ,
y y y
2
2
and when we substitute −pdx /y for d x we have
2
3 qdx 3 3p dx 2
d x = − + 2 .
y y
Furthermore,
3
4 rdx 4 pq dx 4 6pq dx 4 6p dx 4 3p 2 q 2 2
d x = − + 2 + 2 − 3 + 2 − 3dx d x
y y y y y y
2
2
and when for d x we substitute −pdx /y,wehave
3
4 −r 10pq 15p 4
d x = + − dx
y y 2 y 3