THE CURVE ON ONE OF THE COORDINATE PLANES. 499 



For the present we have contented ourselves with the treatment of several 

 particular cases, namely, 



(a) whenB 2 =+l, B' = 0, 



that is, when the angle B has its extremity at the end of any of the four 

 quadrants ; 



(b) when B o = 0, B'a=+1, 



when the extremity of B lies in the bisecting lines of the four quadrants ; 



(c) when the angles A and C answer to the values 



3cos 2 A = l, 0'a=+l, C = 0, 



in which case the relation (IV.) decomposes itself into the three linear factors 



(6 2 + C 2 )B + (6 2 -c 2 )B'a C' ) 

 a 2 B + (a 2 + 2e 2 )B'a C', 

 a 2 B -(a 2 + 25 2 )B'a C, 



and the equation (III.) becomes an identity. 



