On Aiiatytical Gtomeiiy. 287 



When the angles of the Uiangle ABC depend on the relative po- 

 sition of the points, A, B, and C, it is manifest that AB, AC, and 

 BC, are determined in position by the position of any two points 

 through which they pass ; let therefore AB' or x' be the continu- 

 ation of AB, and then we have in the triangle BB'C 



writing for z-, its value, x^ -\-y''- ■\-2{\ —2Pi)xy, and for (1 -2k') 



a? + (l-2AW , s ^ 



its value, — j drawn from (4) and 



s'3 z=x"^-\- If — 2(1 - 2A)x'y. 



We thus see that in passing from the triangle ABC to AB'C it is 



sufficient to change the sign of either a:, or 1 — 2A. 



In the same manner, it is shown, that to pass from the triangle 



ABC to ABC along AC, the sign of y, or 1 - 2A, must be changed. 



Let us make in x'- -{-y" — 2[l—2A)x'y=z'^ 



1 z' b'c 

 x'=y=^Ac=^l and then A^=^=-^. 



The magnitude and position of x, y, and z, being determined 

 by the position of their extremities, the surface S, comprised by 

 those lines must also depend on die same conditions. We ought there- 

 fore to have S=F{xy(p)=F'[xz(p'), cp and 9' designating the angles 

 BAC, ABC, but from the equations (7) x^y'^A'B=x'^z-A'B''f 

 from which it follows that (x^y-AB)" is the general term of F; put 

 therefore S=^C{x"y^ABY +p and then we shall have for the trian- 

 gle AB'C, S'=C{x''y^ABY+p' and for the triangle BCB', S''= 

 C({x'+xyz^-A'By-i-p"; butS"=S+S'and;3"A'B'=::?/2AB; we 

 have then 



C X ( {x'-{-x)^y^^ABy +p"=C X {y^^ABf {x'^-'' ^-x^-'' )+?+P' 

 and consequently, (since this equation is identical,) when x=x'f 

 {2xY"' =-2x^"' or 2'^"' =2 we have then a=^. In the same manner 

 may it be shown, that in the second, third, etc. terms of the devel- 

 opment of F, /3=J, 7=1, etc. and consequently we have, S= 

 pxy\/AB, p being a quantity depending on the nature of the surface. 

 Writing for AB, its value, and 



p - 



^=^'/{x-\-y+z) (x+y-z) {x-y-\-z) {-x+y+z)' 



The resolution of the first member of A+B=l into two fac- 

 tors of the same degree gives A^+B^v _ l=a-" / ,g. 



A^ "B^v'-l =«+■■■ ) 



