VOL. LXXXVI.] PHILOSOPHICAL TRANSACTIONS. 33 



waggon, to show him the strangest flash of lightning that ever was seen. From 

 his description, the range of it was from east to west ; and so low in the horizon, 

 that, had he not been on high ground, he could not have seen it." 



XVI. Newton s Binomial Theorem Legally Demonstrated by Algebra. By the 



Rev. Wm. Sew ell, A.M. p. 382. 



*» 

 Let m and n be any whole positive numbers; and (l -{- X Y a binomial to be ex- 

 panded into a series, as 1 + ax -f b* 2 •+- ex 3 + &c. where a, b, c, d, &c. are the 

 co-efficients to be determined. 



m 



Assume v m = (l + x)~" = 1 -f- ax + bx* -f ex 3 -f- d,t 4 -f- &c. 



m 



And z m = (1 + y)~ n = 1 + Ay + By 2 + cy 3 + dj/ 4 -f & c . 



Then will v n = 1 -f x, and z" = 1 -j- y ; .-. v n — • ' z n = » — y. 



And i/» — z m = a X (# — y) + b X (a? a — # 2 ) + c X (# 3 — y 5 ) + &c. 

 Consequently *„ ~ * „ = a + b X (x -f y) -f c X (* 2 + afy + y 9 ) + &c. 



Now */» — z m = (v — z) X : 7/*" 1 + t/"~" z -j- i;™- 3 z. 2 + &c z m ~\ 



Also u n — z n = (i; — z) X : v n ~' + i;"-* z -{- z/ 1 " 3 z 2 + &c z""'. 



Therefore — — — reduces to, and becomes = v — , „ z — i-i — c i — 



= A + BX(^ + y)+cX(^+^+/)+DX(^ + ^ + *y 2 + y 3 ) + &c. 



The law is manifest; and it is also evident that the numerator and denominator 

 of the fraction respectively, terminate in m and n terms. Suppose then x = y; 

 then will v = z; and our equation will become — — • or, — — = a + 2b# -4- 3ca? 2 

 -j- 4d* 3 + &c. 



But v" = 1 + x i therefore by multiplying we have — = a + (a + '2b)x -f- 

 (2B -f 3c) ar 2 -J- (3c + 4d) x 3 -f &c. Or v m = 



m 

 .7 ma . «a + 2«b , 2/?b + 3nc 2 , 3nc + 4nD 3 . ~ ■'*•.'• 



1+a: as x -\ x 1 -\ x 3 + &c. Compare this 



with the assumed series, to which it is similar and equal, and it will be, 



n\ n\ + 2»b 2«b + 3nc „ 



— = ] ; = A ; = B ; &C. = &c. 



m m m 



m m — n m — 2n Q 



.*. A = -; B = — j; — A; C = ■ „ a ■ B; &C 

 n ' 1.2.n ' 1.2. 3.« 



Therefore f| + »)" =: i + = » + ^^^ + ^~^^ *, + &c 

 the law is manifest, and agrees with the common form derived from other principles. 



Schol. In the above investigation, it is obvious that unless m be a positive whole 

 number, the numerator above-mentioned does not terminate: it still remains there 

 fore to show, how to derive the series when m is a negative whole number. In this 

 case, the expression (v m — z m ) assumes this form, vm — z*, or its equal 

 which divided by v n — z n , as before, gives 



z m — v m _ J_ — (v — z) X : f m "' + 



? * t,» — 2 » tJ ™ s m X (v — z) X : t> n " 1 + f" 

 VOL. XVIII. F 



Z — D" 



^V X * n - z" ~" i>"\s m (v - z) X : f tt " x + f n '" 2 z + *'""' 3 z 2 + ' ' ~ " »"z" 



