208 On Analytical Geometry. 
Taking the difference of the two equations « of (12) and — 
‘ tee — —1_ qi2ev —1 -1 tov — -1_ ai? 
bate y peer ? _ gery seh, 5 aa "at | 
sequently n=—29’V —1. In the same manner by taking the differ- | 
ence of the two equations of (13) n’= aes ~—1; we have there- | 
fore for (12) and (13) 
» and con- 
) 
ue . 
= yaad 1} ny ee 
eae 23 nigh fav ah j . 
pant laa Ly=z0 gee’ Y =1 a 
<29V —1 4200 — 1 (13) 
aat~? +y=za— ? 
Multiplying the first equation of (12) by — a* ay = ink and ¢ 
ye. 
siarinup seh ithesecondot (13)’, and at(o+e eee - at”? 
or better a w= fay =1..We see from this ones 
that the three angles of the triangle ABC, are equal to the two 
BAC+B’AC, and consequently that BAC =o’ 4-9 =ABC+ACB. 
In a series of triangles whose sides are w, y and z, %,Y and 2’, 
etc. and included angles 9, 29, 39, etc.; we have 
429V — 1 i ae J] 
; xL-yo~ 
a 2— ya 1 ey 
es 
vy gtOeY =1 a qa etc. 
Ser +9V¥ — r a 
ckgicay when a Fe a root of the e uation 
different from ys the first members are factors of ety” ; 
have, therefore, x+y" nepal alittle HO" +O Y ese “ty ete. the 
theorem of Cones, when the tenets have one side « conn 
Cincinnati, April 15, 1831. 
