180 I>E MOIVRE. 



fraction of which the denominator is 1 - ma + pa^. We have 

 to find the scale of relation of the Series of which the general 



term is u^Vn {ro)'\ 



We know by the ordinary theory of decomposing Recurring 

 Series into Geometrical Progressions that 



where p^ and p^ are the reciprocals of the roots of the equation 



and a^ and a^ are the reciprocals of the roots of the equation 



1 —ina -\-pa^ = ; 



and R^, R^, A^, A^ are certain constants. 



Thus u^v^ = R^A^ (p/^X + ^1^2 {pi^:T 



this shews that the required scale of relation will involve four 

 terms besides unity. The four quantities p^a^, p^a^, p^a^, p^7^ will 

 be the reciprocals of the roots of the equation in z which is found 

 by eliminating r and a from 



1 —fr + gr"^ = 0, 1 — ma + pa^ = 0, ra — z\ 



this equation therefore is 



1 -fmz + {pP + gm^ — 2^p) z^ —fgmjpz^ -^-g^fz^ — 0. 



Thus we have determined the required scale of relation ; for 

 the denominator of the fraction which by expansion produces 

 w„t;„ (ra)" as its general term will be 



1 —fmra + {pf^+gm^ — ^gjp) ^V — fgmj^r^d ■\- g^j^r^a^. 



De Moivre adds, page 229, 



But it is very observable, that if one of the differential Scales be the 

 Binomial \ — a raised to any Power, it will be sufficient to raise the other 

 differential Scale to that Power, only substituting ar for ?•, or leaving 

 the Powers of r as they are, if a be restrained to Unity; and that 

 Power of the other differential Scale will constitute the differential 

 Scale required. 



