159 



De Moivres Multinomial Theorem will easily follow from 

 the Binomial Theorem of Newton, which was given above : 



for, in (Qc + R2 + S? + T2; + &c.)^ the co-efficient of x7 



>;.(^-l).&c.(^-^l) ,(Q,^R,Vs.'.T.%&c.)" 

 1. ii. &c. . « ^ '' 



IS 



234- '" n—ri 



and in (Qs+Rz + S;r + T2 + &c.) the co-efficient of (Q2) is 

 1. 2. &c. d ^ (R~^ + s^+ r. + &c.) 



And in (Rz+Szh-T^V &c.) the co-efficient of (r/) 

 1. 2. .^cc. e ^x(S^~+T.f&c.) 



rf— <• 

 is 



Therefore in {x + Qz + Rz + Sz -l-Tz + &c.) r 



the co-efficient of .rr~"x (Qz)"~ x (Rz ) ~'x (Sz )'~" x &c. 



will be ^.(^— l).&c.(^— ?I3I) n.(?i— l).&c.(7t— rf^) 

 ] . 2 . 3 . &c. 71 ^ 1 . 2 . 3 . &c. d ^ 



(/■((/— l).&c.(rf—e—l) e.(e— l).&c.(g— ./— 1) X &c. 

 1 , 2 . 3. &c. e ^ 1 . 2 . 3 . &c. / 



Now, for/-^;^''-'^.'^'-^-'x&c. pdt /"+'-^+-'+^^-^ 

 and simplify the co-efficient, and you will have this gener?tl 

 expression for any term : 



^ . (^'-1) .(^-2) . &c &c.(^-^=i:) 



1.2..&c.(7J— d) X (1.2..&c.(f/— e) X 1.2.&c.(e— /; ^ 



.t>- xQ xK xS x&c. x;: ^ The 



