438 PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



1 



ber n in the two preceding formulae, let there be formed two series of equations, 

 as follows : 



^X^-2xzX^ + zX=0 zY^-2zzY^+zY=0 



^%-2xz'X, + z^X^=0 =rY-2xz'Y^ + ;^Y=0 



%'X^-2z^X, + z'X=0 ^Y^-2ts'Y3 4-«'Y,=0 



&c. &c. 



Put P=^X, + «% + sX, ..... +z-'X^_,+z"X^ + ,kc. 

 Q = jY, + ^Y.,, + ^% +«"-'Y„_, + «"Y„ + ,&c. 



Then, by adding into one sum each of the two series of equations, we have 



.(X„ + P)-2:.P + ^-X, = 0, 



s(Yo + Q)-22-Q + --Y, = 0. 



From these two equations we find 



p X z — Apg' „ ^ 1 1 ~ — Y(i« 



1-2XZ-Z'' " l-22:« + ^^ ■ 



Now, X„ = l, X =z, Y„=0, Y,=y: 



Therefore, the values of P and Q are 



p_ (,x-z)z Q_ ^z 



'l-2xz + z'-' ^~l-2xz + z'' 



Let the expansion of the fraction — ^ , be 



C„-i-C,« + C/ . . . -(-C_, *"-' + C„_,^"-'+C„«" + &c. 

 we have now 



P = «Cos + (.rCi-C„)s^ . . . +(^C„_,-C,_,)2" + &c. 



Q=yCoz+yCz . . . +t/C„_, z" + kc. 



But F = X,z + X,z' . . . +X.,^" + &c. 



and Q = Y,^+Y2«^ . . . +Y„z- + kc. 



Hence it follows that 



X„ = xC,._,-C„_,, Y„=3^C„_,. . . . 



To simplify, let us put u instead of 2 a; in the expression l — 2xz + z\ so that 

 it becomes l-(u-z) z, we have now by division = — 5 ,=^ — ; r- equal to 



^ ' *' l — 2xz + z- l—(u — z)z ^ 



l + z(u-z)+z'{u-zy . . . +s"-'(m-^)"-' + s"-'(„-s)-'+&;c. 



Let A, and B„ be the coefficients of c;° in the expansions of (m-s)""' and (u-z)"'', 

 and let A, and B, be the coefficients of z in the expansions of (m-?)""" and («-«)""", 



