Page 173 - Foundations Of Differential Calculus

156 8. On the Higher Differentiation of Differential Formulas
and so forth. Then, since dy = pdx,wehave

2
2 2 2 p 2
d y = qdx + pd x = q − dx ,
y
2 2
and again substituting −pdx /y for d x we have
3
3 4pq 3p 3
d y = r − + dx
y y 2
and
2 2 4
4 7pr 4q 25p q 15p 4
d y = s − − + − dx ,
y y y 2 y 3
and so forth. When these values are substituted for the higher differentials
of x and y, a given expression is transformed into a form of the kind which
no longer contains higher differentials. This is accomplished by considering
some differential to be constant.

269. Frequently in the application of calculus to curves it may happen
2
2

that the first differential dx + dy is assumed to be constant. For this
reason we now show the way in which for this case the second and higher
differentials should be eliminated. At the same time, by using the same ar-
gument, the way will be opened to show the process if any other differential
is assumed to be constant. Now we let
dy = p dx, dp = q dx, dq = r dx, dr = sdx, ....
2
2
2

Then the differential dx + dy takes the form dx 1+ p . Since this is
constant, we have
2
2 2 pq dx
d x 1+ p + 2 =0,
1+ p
so that
2 pq dx 2
d x = − 2 ,
1+ p
2
and we already have the value of d x. Furthermore, we have
3 2 3 2 2 3 2
3 pr dx q dx 2p q dx 2pq dx d x
d x = − 2 − 2 + 2 − 2
2
1+ p 1+ p (1 + p ) 1+ p
2
2
2
2 2

pr dx 3 q dx 3 4p q dx 3 pr dx 3 3p − 1 q dx 3
= − − + 2 = − + 2 .
2
2
1+ p 2 1+ p 2 (1 + p ) 1+ p 2 (1 + p )