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The general total differential of two variables is given in terms of the general partial differentials by the formula,
the law of the coeficients being that
the law of the coefficients being that of Newton's Binomial Theorem.
(18) u = sino
There is a very important theorem (due to Euler) regarding homogeneous functions of any number of variables, which from the frequent applications made of it ought to be noticed in this place.
If u be a homogeneous algebraic function of n dimensions of r variables x, y, ?, ...; then
From this may be derived a series of equations of the form
In applying this theorem to transcendental functions of algebraical functions, it is to be observed that it is not sufficient that these last should be homogeneous, it is also necessary that they should be of zero dimensions, as, otherwise, in the development of the transcendental function the degree of each term would be different, and the function when expanded not homogeneous.
yx – XY
(24) If u be a homogeneous and symmetrical function of x and y of n dimensions, so that
and if it be expanded in terms of x so as to be of the form
E. (Qi ve yui),
and as it is symmetrical in x and y, we have
when x = y, so that - dx dy
• This extension of a property of Laplace's Functions was communicated to me by Mr Archibald Smith.
CHANGE OF THE INDEPENDENT VARIABLE.
Functions of One Variable.
If y = f(c) and therefore x = f-'(y), the successive differential coefficients of y with respect to a are transformed into those of x with respect to y by means of the formulæ,
and similarly for higher orders. The reader will find the demonstration of a general formula for the change of the nith differential coefficient in a Memoir by Mr Murphy, in the Philosophical Transactions, 1837, p. 210. The expression is of necessity extremely complicated, and the demonstration would not be intelligible without so much preliminary matter that I cannot insert it here, and I must therefore content myself with referring the reader to the original Memoir.
If u = f(y) and y = () so that u may also be considered as a function of a, the successive differential coefficients of u with respect to y may be transformed into those of u with respect to æ by the formulæ