440 PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



The expression 1 - 2 « ^ + 3- may have this form (a-- ^y -(:»:'- 1); now in the 

 hjTperbola x'-\=^cy''\ therefore, denoting this last quantity by «% we have 



1 — 2 X z + z'={x — z)- — u- —{x — u — z) {:x + ti — z). 



1 



And \-2xz + '. 



{x—u — z){x + u — z) 



= L\-1 l_l. 



2u\x — u — z x + u — z^ 



1 1 z zr z—" z°-' „_ 



Now ^:3^^3-^=:^^.+(^=^+(^3^' • ■ "*'(;.-«o-^(^-«)" 



1 1 z z' z"-' Z-' 



And ¥TlI^~~IT^ + (^^T^^^Fi^' ' ' ' + (a^ + «)'-' "^ (:^ + «)" 



2 It 

 Hence, by subtracting, and observing thatr^-w—l, and putting ^_2^^_^^» for 



- + , we have 



x—u — z x+u — :. 



2u i2u + {{x + u)^~{x-uY}z + {{x + uy'-{x-u)\z'' . . . 



1-2 xz+z'~^ + {{x + ii)"-^ -{x-uy'-'}z''--' + {{x + uf -{x-uflz'-' &c. 



Now the expansion of the fraction \_2xz + z ' ^^i^^g 



1 + C,z + C,z' + C,z' . . . +C„_^z-'- + C„_,z"-' + kc. 

 it appears, from what has been just now found, that 



^"-'~2u{^'^'^"^" -(=^-")" }• 

 And we found (Article 10, L) that 



/(w a) = 2: C„_, - C„_, ; F(na) =y C„_, : 



Therefore, fin a) = ^ | (x^ - 1 + « a;) {x + «)"-'_ (^^ - 1 _ « x) (x- uf-' I ; 



Butz' — l—u'; therefore ar-l+M a; = u^x + u), 

 and x^ — l — ux=—u(x—u); 



And hence .... 2/{na) = (x + uy + (a: — u)"; 

 and 2uF(na) = {{x + uy — ({r — uy}i/; 



Or, since .... u'=fc, and u=i/^c; 



2/(/Jo) = (a;+y Vc)"-|-(a;-y v/c)";) „ 



2F{na)sfc = (x + i/Jcy-{x-yJcy.} 



