'In the second place it is to be observed, that the whole coefficient of any power of x in the products of multiplication A, may be reduced to the regular binomial form, established in the 13th article. Thus n . "—^ + mn + m /~—1 the whole coefficient of** by actual , ■ »• .1 1 n1+ma + 2m«—n—m n + m—1 multiplication becomes .—n+m . 1 1 ■ ir r * -2 2 «, n—I n — 1 n — 1 m—1 m — 1 m—2 Also n. . T + ms • n-.— + m • • .. • n + m . . .... • -n— , 23 2 T '23 the whole coefficient of by actual multiplication become* n*+m'-in1 — <<m1 + lnIm . wl n—6mn+2n+7m *» , n+m - 1 2 ii-—-i-— + - ; ~ n + m . • 6 b 2 n+m—And from the preceding observation it is evident, that we may, in the same manner, reduce the whole coefficient of any otherpower of x, in the products of multiplication A to the regular binomial form. • (16) But in proceeding, as above, to change the form of the coefficients prefixed to any power of x, in multiplication A, into the tegular binomial form, we ajre not under the necessity of supposing a and nt to be whole numbers. The actual multiplications will end in the same powers of n and m, the same combinations of them, and the same numerals, whether we consider n and 01 as whole numbers, or as fractions. 1 * We are therefore at liberty to suppose n and m to be any two fractions whatever, in the two series multiplied into one another in multiplication A, and the same two fractions will take the place of o and m respectively in the regular binomial scries i + u + ntx + n + m. n+m—-I , -T n+m — 1 n+m—2 . . —— n+m — I x + n+m . .— . x' + n+m . • 2 23 2 n + m 2 ^ n__m—1 xt ^c wj,j(,j1 eXpresse8 the product of the 1 two series into one another. '(17) If therefore r be any positive whole number, we can raise 1 _t 1 the binomial scries 14 x + • ~ — * + 7~' * . x* + &c. to any pro 3 ♦ posed power hy successive multiplications; or we can express any power of it by supposing the multiplications actually to have been gone through. Thus, calling the last mentioned series the root, if it be multiplied by itself, and if the coefficients in the product be ex pressed in the regular binomial form, its square vftH be I + -2. * + _2 l _2 r _2 z JL — l I —X T i 2 r r , 2 r r 2 r 2 3 r » 3 2 3 -xl + &c. Again, if this *erie* be multiplied by the root, 4' , and Sad the coefficients in the product be expressed in the regular bino* _J f rr.hl form, the cube of the root will be i + ~ x+ — . Mt + ■ X_i -3 -2 JL_. J_-2 -L-3 s+A..—— -1 x* + &c. for this series being raised to the rth power becomes equal to I + x. * As by the general principles of involution the nth power of i n ''s i + x| r » 'c tberefore follows, from the last obievration n and the preceding article, that TTxj' — '+-** + -~~ . A • n n _n * X1 + . . X5 + • . * ',> i r a ——a 3 j: . J: x* + &c.'— 3 j >——3 x*-^ &c.; and as this equation holds in every possible value of or, and as, by the general principles of involution, 14 xi" is etjual to 1, when m is equal to 0 then-^=- or =i—nx - n-' x*—11 :. - x'—n . . . 1 x'— &c.' ,» . 3 * 3 + We consider it as almost an affront to point out, to any person of moderate mathematical attainments, the similarity,' or rather the sameness, of these two proofs, -tiller, Art. 5., multiplies the series i-f-«x + &c. by the series i+w* + &t; and he makes the same remark in Art. 6. with regard to the coefficients of the product that is made by Dr. R. in his ;6'h Article. Again i in Art. 7. Fuler sutes that, when m and n are whole numbers, \m\ = (1+*)" anrf ["] - (' 4*)"; or, according to his own explanation, that the real tx^auded form we have 1+ — x + —-—( —1 \ x1 + &c. = (1+*) * • 'a 2 .a V" I If we look to Art. 17. of Dr. R. we shall find a like process conducted on exactly the same principle: »*+ "~7 — 1 j.x-f- &C. multiplied twice, thrice, n times into itself, produces a series which results from the above by putting fin J-, _! 4- —, or —, ( — + — ) or or generally r r r r V r r J r —: if «=rr, then 1 + — x + —'-— \ 1 Vl+&c- mu^ t rip lied r times — 1 + w, or 1 + — x -f- &c- = (1 + *) ' • The only difference, that can be pointed out between these two parts, is that Professor R. puts n=r, whereas Euler in fact puts n—mr. If in Dr. R.'s proof we put nzzmr, then 14. + ~{~ 1 )**+ *c. multiplied n times into itself, =1+«x-f &e.=:(r+x)»; or i +—x-f-&c. = (i+*)"» , which is the last part of Dr. R.'s 18th article; and which, it is clear from what we have just shewn, he might have had without going through that part in which m is put = r, or in which the series I -f-mx -f &c. is reduced to 1 +x: but the insertion of one unnecessary step is not the sole objection which we have to make against Dr. R.'s mathematics'*. The sameness of principle and process is preserved also in the fast parts, in which the theorem is to be proved when the index is negative: but, as we may possibly tire our readers by stripping off the mysterious symbols which concealed the? likeness of Euler's proof from Dr. R. we shall vary the proceeding, and clothe the latter in the habiliments of the-" former. ^ , -■ - ■■ ,. ,,, 1 J1 ■ • If in Euler's proof we put i= i, we have 1 + — x+ $cc.=(i+x) 1 » and this agrees exactly with Dr. R.: but then in order to obtain i the series for (1+*} * wc must make a*eth«r~etepthet»gb whether we do or do not introduce an additional step, we see sot the slightest difference of principle and method. .' |