egy 
*% 
= WP y= Ap=1 and then Ars 
: bi = 3 
On Analytical Geometry. — 287 
When the angles of the triangle 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 
BC? =2/2 = (a! +2)? +2? 422(a' +2) (1—2A’) 
writing for z?, its value, v? +y?+-2(1—2A)ay, and for (1 - 2A’) 
— a-(1-2A)jy 
s 
its value, — 
» drawn from (4)-and 
2/2? 4/2 + y? —2(1—2A)z'y. 
We thus sce that in passing from the triangle ABC to AB‘C it is 
sufficient to change the sign of either x, 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 2/? --y? ~2(1—2A)a’y=2’? 
ce. oe 
* 
yy 
_ The magnitude and position of x, y, and z, being determined 
_ by the position of their extremities, the surface §, comprised by 
- those lines must also depend on the same conditions. We ought there- 
- fore to have S=F(xyp)=F'(«z'), o and 9’ designating the angles 
BAC, ABC, but from the equations (7) w*y? AB= 072? A'B’ ; 
from which it follows that («?y?AB)" is the general term of F; put 
therefore S=C(x2y2AB)* +p and then we shall have for the trian- - 
gle AB’C, S’=C(x’?y? AB)* +p’ and for the triangle BCB’, S”= 
C((0/4-2)?22 A’B’)* +p"; but S’=S+S and z?A’/B/=y? AB; we 
have then pa 
Cc X ( (2’-++a)*y2 AB)* +-p!’=C x (y2? AB)* (x/24 +-0?*)+p+p! 
and consequently, (since this equation is identical,) when 2=2", 
(2x)? —2a24 oy 22 =2- we have then a=4. In the same manner 
may it be shown, that in the second, third, utc. terms of the devel- 
opment of F, 8=$, y=4, etc. and consequently we have, S= 
Pry\/ XB, p being a quantity depending on the nature of the surface. 
Writing for AB, its value, and 
; PCE Fee ee 
S=3V (e+y+z) (@+y—2) (e—y+2)(—#+y+2) 
The resolution of the first member of A+-B=1 into two fac- 
, L if isa 
lors of the samesdegree gives A?+B?v —l=e (8) 
Sos i : iy ee 
AT=—BIY,-i=a" 
*. 
