'u

=

dy3

dy (du dy - dy du) - 3 dy (du du-d'y du)

dxdx dx

dx

da dx

The general formula for this transformation will be found in the Memoir of Mr Murphy before referred to, but the result is of such extreme complexity, that it happens fortunately that we have seldom to employ these transformations for high orders of differentials; and where this is necessary, that the nature of the case usually gives us the means of simplification.

(2) The expression for the radius of curvature when a is the independent variable is

When y is made the independent variable, it becomes

into an equation in which y is the independent variable.

from y to a, when x = log {y + (1 + y2)*}.

(6) There is a very convenient formula by which we

can change generally the independent variable in y"

d" u

dy"

from

y to a when y = e. Taking the symbol of operation alone,

Now by the theorem given in Ex. 18, of Chap. 11. Sec. 1,

from y to a, having given a log (a + y).

d'u

dy

=

Instead of availing ourselves of the formulæ for expressing ď3 u

and in terms of the differentials of u and y with

dy

regard to a, we may effect the required transformation more

simply by differentiating successively and simplifying at each step. Thus, observing that

differentiating a third time and again multiplying by a + y,

into a function where s is the independent variable, having given that

into a function of r and 0, having given x = r cos 0, y = r sin 0. In this case we consider r to be a function

of 0; differentiating therefore x and y on this hypothesis,