440 PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



The expression 1 — 2 a: 5; + 2:' may have this form {x—zy — (z^ — l); now in the 

 hjrperbola x'-l = cf; therefore, denoting this last quantity by «', we have 



1 — 2 X Z + Z^=:(x — zy — U^=(x — U — z) (X + U — s). 



1 



And 



{x — u — z) (x + u — z) 



l-2xz + z' 



=J-i_l ^l. 



2u\x — u — z x + u — z^ 



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



Now x-u-z^^^^(x^^^'^(x-uf • ■ '^ix-uy-''^(x-uy^^- 



1 _ 1 z z' ■ z"-' -g"-' 



And a; + M-^~« + M'^ (« + «)' "^(3: + t<y " ' ' ■*" (a; + «)"-' "^ (ar + «)" ^' 



2 M 



Hence, by subtracting, and observing that x'-ti'=l, and putting -^_2^^^-, 2 for 

 + , we have 



X—U — Z X + u — z 



2u (2u + {{x + uf-{x-uy]z + {{x + uf-{x-u) ]^^ 



! 



1 - 2 ;«: ^ + ^^' ) + {(a: + «^)"-' - (ar - m)"-' }2!"-^ + {(a; + «)» _ (a; _ m)'']^"-' &c. 



1 



Now the expansion of the fraction -i_2xz+ ^ ^^i^S 



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

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





And we found (Article 10, L) that 



f{n a)=x C„_, - C„_, ; F(na) =y C„_, : 



Therefore, /(^ a)=2^ { (a:"- 1 + ««)(« + ur-'-{p^-\ -ux) {x-uf-' \ ; 



B\itiif-l=u'; therefore a^'-l+Mx^ u(x + u), 



and x^ — l — ux=—u(x — u); 

 And hence .... 2/(na)=(x + uy + (a;-u)''; 

 and 2uF (na) = {(x + uy-(x-uy]y; 



Or, since .... u^=y% andM=yVc; 



2/{na) = ia; + t/s/cy 



y + (w-ys/cy;\ 

 y-{a;-t/^cy. J 



N 



2F(na)s/c = {a: + i/^cy ' 



