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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.

(1) Change the formula

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into one where y is the independent variable.

The result is

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

d'y

dx When y is made the independent variable, it becomes

(dx) 21

{1+

ď « dy?

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into an equation in which y is the independent variable.

The result is

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(4) Change the variable in

du u

dy' (1 + y') "
from y to il', when x = log y + (1 + y)}.

The result is + u = (* +6=+).
(5) Change the variable in

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(6) There is a very convenient formula by which we can change generally the independent variable in y" from

dy" y to x when y = 6". Taking the symbol of operation alone,

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

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Change the independent variable in
du

du 2 a
dy

dy1 - y

- 200

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Instead of availing ourselves of the formulæ for expressing du du. dy and dys in

and in terms of the differentials of u and y with regard to x, we may effect the required transformation more

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

(n + mamma 1,

we have

du du

differentiating again and multiplying by a + y, we have

(a + y) alv. + (a + y) an der

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du du

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

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into a function of p and 0, having given x = r cose, y = y sin . In this case we consider po to be a function of 0; differentiating therefore x and y on this hypothesis,

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Substituting this expression for

s, we find

dx "

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