On Analytical Geometry. 289 



+. 



, ■ , a*+bV_i ^ 



From (8) we have — 7 v — z:z:=cf~^". Or developing, 



,Or putting cp for the real factor of the second member, and 71= 

 <pv — 1, (9 designating the angle BAG,) it appears, from the 



equation «-"'=+a+"V — l, (if X designate the angle B'AC) that 



f^±\<p-rX) — —V—i- the sum of these angles is thus seen to 

 be constant and to have opposite signs. Put 9', and 9^^, to repre- 

 sent the angles ABC, BAG, and equations (I), (2) and (3), become 



^.+2/._^y(«±2?^-"l4.«-29V=a^^^,^ ^^y 

 ^.4.^3_^^(^±29V--l_^^:p29V=l^^^,^ (2y 



2,.+^3_2,^(«±W-l_^^:,29-^/:=l)^^3. (3y 

 and (4), (5) and (6) become 



, , +29'/-! +29V3I 



and since A^B^= -=^ , (7) is changed to 



4v — 1 



^ +29^^ +2(r>V'^\ +29 v^^ +29'/iri 



z a —a^^ z a~ ^ —a^^ 



- — -— -__ ^==^ C7y 



j/ +23jV-l T29V-l'a: +29'V-1 t29'V-1 ^ ^ 

 a ^ —a^^ a- ^ -a^ ^ 



Resolving the first members of (1)' into two factors of the first 



degree and we have the following equations. 



+29-/ -1 +n ^ 

 x — ya- ^ =za- « 



^ — y (12) 



+29V — 1 xn \ 

 x—ya^ =za^ j 



— xa- ^ A-y=za- \ 



J — > (13) 



