and con- 



290 On Analytical Geometry. 



Taking the difference of the two equations of (12) and 



+23-/ — 1 ■ t2(p'v/^ +2(p'/^1 x2(p\/ — 1 



y~ +29V— 1 +2/-/— i~ +n . TV, ' 



a ^ — a+ ' —a- +fr 



sequently w= — 2$'V' — 1. In the same manner by taking the differ- 

 ence of the two equations of (13) n'= —^^"V - 1 ; we have there- 

 fore for (12) and (13) 



. +2(pv'-l T2(pViri^ 



/— /— > (12)' 



T2(pV - 1 +2(p'v—l I 

 x — ya^ =za-- ^ j 



+2(p^/-l , z2(p"^/-l)' 



— xa- ^ -\-y=za^ ^ f 



T2(pV -1 , +2/V — li ^ 



— xa^ ^ ■\-y=za~ ^ ) 



Multiplying the first equation of (12)' by - a^ ^-v/— Ij and com- 

 paring with the second of ( 1 3)', and a+ ^"^ "^''^ ^ = — a- ^ 

 or better a'^^^ ~rf J __-^ _ j^ -yy^ ggg f^.^^^ ^^us equation, 



that the three angles of the triangle ABC, are equal to the two 

 BAC+B'AC,and consequently that B'AC=9'+9''=ABC+ACB. 



In a series of triangles whose sides are x, y and z, x, y and z% 

 etc. and included angles (p, 2(p, 3^, etc.; we have 



X — ya~ ^ =:za'^ ^ 



+49 -/"l , t29'V- 1 

 x — ya- ^ =z'a^ ^ 



+6^^/■^ „ t2(p"VII1 ^^ 

 x — ya- ^ —z"a^ ■ , etc. 



and consequently when a- is a root of the equation r" + l =0, 



different from unity, the first members are factors of a:"+?/"; we 



have, therefore, x"±ij''=zz'z"a+"^^ ^^ ^^ ' , etc.; the 



theorem of Cotes, when the triangles have one side x common, 

 Cincinnati; April 15; ISol. 



