FOCAL LINES OF A REFRACTED PENCIL. 337 



and if V l = 



and V 3 



and we put D = (A, + a,) (B, + b i )-(C 1 + c,)*, 



(A t - o) D + p* (B 1 + b l ) + r i (A l + a,) - 2pr ( Ci + c x ) = 0, 



(O, - c.) D +pq (B, + b,) + rs (A, + a,) - ( ps + qr) (C, + c.) = 0. 



These equations enable us to determine A ly B t , C 2 when A lf B v C, are 



given. 



Here we must observe that the quantities A, B, C, &c., do not represent 

 lines, as in the first part of this paper, but the reciprocals of lines. 



VOL. II. 43 



