j dym represents the mth differential of u, formed by making y to increase m times succes (dru \ dxmdyr~TM' ^y*" m^i" cates the rth differential of u, formed by making x to increase m times successively by equal increments dx, and y to increase (r—m) times successively by equal increments dy. The principles on which this notation is constructed are so immediately applicable to functions of more than two variables, that it is unnecessary to do more than state the results as they occur in the following Articles. Here however I must remark on a defect in the notation. If u = F (x, y), for instance, the same symbol d2u in the numera tors of i<Pu\ i d*u \ id2u\ \dx2'' \dxdyl' \dy*'' means processes which, although of the same kind, may lead to results altogether different. In the first it means the second differential of u springing from two successive variations of x; in the second it represents d2u springing from a variation of x, taking place on the back of a previous variation of y; and in the third, d2u as originating in two successive variations of y. The brackets therefore are used in conjunction with the different denominators to indicate these variations of the same symbol, and are marks sufficiently distinctive to prevent confusion. But before we proceed further, we must prove the following proposition. 79.3 If a function is differentiated many times in respect of independent variables which it contains, the result is the same, whatever is the order of the variables with respect to which it is differentiated: provided that it is differentiated the same number of times and with respect to the same variables. Let u = r(x,y,z, ); ~ dy dx'-1 dz< dx dy-i ~(99) Of this property, thus proved in the general case, some particular examples arc subjoined. the brackets indicating that the derived-functions within them are partial. Similarly, *-(£)-+»(wfc)«'*+«(j&)**,+(3»*' Similarly may other total differentials be fouud; but the general term is too complicated to be of any practical use. Now let the results be simplified by making x and y equicrescent, neither of which assumptions is inconsistent with the given equation; then d2x = d3x = ... = 0; d2y = dhj = ... = 0; and we have the following series of equations, u = Y{x,y), »« = +(|J)rfy>. and so on; the law of the coefficients being the same as that of (1 + x)"; which may be proved to be true for positive integral values of the exponent, by a train of reasoning similar to that in Art. 55; whence the rath differential is |