V=4(x+§t, y+nt, z+§t, ...) x, y, z ..., §, n, § .. form a system of independent variables, to show that 155. Hence we have the following identity of operators, viz.: and as the variables are all independent and the opera TAYLOR'S THEOREM. EXTENSION. 156. To expand 4(x+h, y+k) in powers of h and k. By Taylor's Theorem we obtain p(x+h, y+k)=4(x+h, y)+k©p(x+h, y) + k2 Ə2p(x+h,y) ду + ... 2! Əy2 228 = p(x, y) + (həd + + -1 (h+2hk Əy2 +. 2 +... φτ... Əy 2! Əx 157. Since it is immaterial whether we first expand with regard to k and then with regard to h, or in the opposite order, we obtain by comparison of the coefficient of hk in the two results the important theorem 158. Further Extension. Several Variables. The form of the general term in the preceding case and the further extension of Taylor's Theorem to the expansion of a function of several variables is more readily investigated as follows: be called F(t). Then Maclaurin's Theorem gives t2 tn n! F'(t) = F(0)+tF'(0)+5;F”(0) +...+ — ‚Fn((t), and by Art. 155 ә Fr(t) = ( § 2x+ny+ ...), p(x+§t, ...), and since the variables x, y, ..., are independent of t, we may put t=0 either before or after the operation has tn + n! Sax дх N + ...)" p (x + £t, y + not,...). Now, putting h= &t, k=nt, l= §t, ..., we obtain p(x+h, y+k; z+l, ...)= p(x, y, z, ...) +(k+k+1+...)(x, ...) +h; Эх Oy 159. Extension of Maclaurin's Theorem. Moreover, if we put x=0 and y=0, and then write x for h and y for k, we have an extension of Maclaurin's Theorem which, for two independent variables, may be written 160. If we now recur to Art. 136 we see that the true Эхду Əy 2 Sy2+etc., ду 2! x2 дх showing what error was made in that article in taking MR' as an approximation to the correct value. The student will find no difficulty in writing down the true values of the lengths of LQ or NS. EULER'S THEOREMS ON HOMOGENEOUS FUNCTIONS. 161. If u = Axayẞ+Вxa'yẞ'+...=ΣAxay3, say, where a+ß=a' + ß' =n, |