[ 345 ] 



The law of continuation as far as regards the produds of z, 



and its coefficients a, i, c, &c. is evident from Prob. i. and agrees 



with that given by De Moivre. The law of the coefficients of 



" — I' ? '* * 

 thefe produds is alfo immediately derived. For let « bed 



y + 2 r _+ 3 J 



be any produd, then becaufe it occurs in the terms A, it followi 

 from Taylor's Theorem and Prob. i. that its coefficient is 



y+2r-f-3/X7 + 2r+ 3J — ix--_I^ 



, I xg.n — i.n— 2 --«—/>— IX — = =: 



2-3 ?T»»—r3' ?x?— ' X ■- I y.rr — i I xix*— iX - -i 



n. n — I, n— 2 - - • n — p~\ ('A or o + r 4- x terms) „,. ,- i 



= — = ^ ^ — " ■ 1l^ ^ . The fame as has 



y X y — I X - - I X r X r— I x - - i x j x s — i x - - - i 



been demonftrated by De Moivre for integral values of n. 



Example VI. From the equation (m) ax + i>z + cz ^ dz + 

 &c. zsgy + hj>' + />' + kj* -\- &c. to find the value of z when a 

 and^ begin together. 



Solution. Taking by Cor. Prob. i. the fucceffive fluxions 

 when z andj = o, andj is fubftituted forj'. 



[m) az =£_)> or z = ^— z= Ky putting A = — 



2 



(m) az + 1. 2 A' iy = I. 2 Ay' oi = -^ J>' = Bj" 



. I. 2 a 



3 



Vol. VII. X x (mj 



