Page 181 - Foundations Of Differential Calculus

164 8. On the Higher Differentiation of Differential Formulas
2
for d y we write
2
2 dy d x dy 2
d y − − ,
dx y
3
for d x we write
2
2
dy d x dx d y 3dx dy 2
− + ,
y y y 2
3
for d y we write
2
2 2
3
2
2
2 2
3 3d xd y 3dy d x dy d x 4dy d y 4dy d x 3dy 3
d y − + 2 − − + + 2 .
dx dx dx y ydx y
2
2
2
IV. If the differential dx + dy is assumed to be constant, then for d x
we write
2 2
2
dy d x − dx dy d y ,
2
dx + dy 2
2
for d y we write
2 2
2
dx d y − dx dy d x ,
2
dx + dy 2
3
for d x we write
2 3 3
dy d x − dx dy d y
2
dx + dy 2
2 2 2 2 2 2 2
dx d y − dy d x 3dy d y − dx d y +4dx dy d x
+ 2 ,
2
2
(dx + dy )
3
for d y we write
3
2 3
dx d y − dx dy d x
2
dx + dy 2
2 2 2 2 2 2 2
dy d x − dx d y 3dx d x − dy d x +4dx dy d y
+ 2 .
2
2
(dx + dy )
277. These expressions, which include no constant differential, are given
in such a way that one has freedom to choose any differential to be constant.
Hence, differential expressions of higher order in which no differential is as-
sumed to be constant, can be tested to decide whether they have unsettled
or fixed significance. We choose arbitrarily some differential, for example
dx, to be constant. Then, with the rule provided in the previous paragraph,
the expression is reduced once more to a form in which no differential is
assumed to be constant. If this agrees with the given expression, then it has