(3 x y3 + 2y3) x2 + 2xy (3x2y + 3xy2 + y3) − x3y2 (2xy + y2)} = 2 (2 xy + y). (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 a so as to be of the form Σ. (Q¡ x3 yr −i), then will {(2i – n) Q¡} As u is homogeneous of n dimensions, we have and as it is symmetrical in ≈ and y, we have Substituting the expansion of u in this equation, we get * This extension of a property of Laplace's Functions was communicated to me by Mr Archibald Smith. CHAPTER III. CHANGE OF THE INDEPENDENT VARIABLE. SECT. 1. Functions of One Variable. IF y = f(x) and therefore x = f(y), the successive differential coefficients of y with respect to a are transformed into those of a 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 nth differential coefficient in a Memoir by Mr Murphy, in the Philosophical Transactions, 1837, p. 210. The expres sion 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 = (x) so that u may also be considered as a function of x, the successive differential coefficients of u with respect to y may be transformed into those of u with respect to a by the formulæ 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 |