By Goursat E.

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35) Goursat proves that the same conclusion be reached without making any hypotheses whatever regarding the derivatives remain cFi/dXj of the functions { with regard to the x s. Otherwise the hypotheses exactly as stated above. It is to be noticed that the later theorems regarding the would not follow, however, without existence of the derivatives of the functions some assumptions regarding dFf /dXj. , un , . ,F,,) by proving the theorem in the special case of a two equations in three independent variables x, y, z and two unknowns u and v.

Application of these formulae, let us determine all those functions which satisfy the equation two functions y As an y=f(x) - " yy 3y"* = 0. Taking y as the independent variable and x as the function, becomes = this equation 0. Xj/> But the only functions whose third derivatives are zero are polynomials of most the second deree. Hence x must be of the form at C2 , C3 are three arbitrary constants. Solving this equation for y, see that the only functions y = /(x) which satisfy the given equation are of the form where Ci, we y = a _ V bx + c, FUNCTIONAL RELATIONS 42 where a, 6, whose axis c is two variables, Let us now consider an implicit function denned by the equation = 0.

This is one of the main advantages The equation (9) does not depend, in form, either upon the number or upon the choice of the independent variables and it is equivalent to as many separate equations as there are independent variables. of the differential notation. ; To calculate that the second u, v, d2 w, let us apply the rule just found for dta, noting of (9) involves the six auxiliary functions member w, du, dv, dw. w Cw ^y^M> dF ^ cw simplifying and using the same symbolism as above, or, d2 w = [7^- du+ 7 Vc/w , ^ dv ^^y + ^ + dw\ , CM; / +^ * TT- cu c?