HAMILTON'S CHARACTERISTIC FUNCTION 



The conditions of conjugate pencils may therefore be written, either 



0,8,= 



(20), 



or in a form derived from this by exchanging the suffixes t and ,. 

 If the axes of co-ordinates are turned so that q and r vanish, 



A-8&-o 1 a g 8-2y 1 yjw-&&/*+j>V (21), 



and oA-fcp' 1 



yA--r\f[ (22). 



A = a/ J 



If we write a l = X l i>, &=F,s, y^ 



a, = X t p, A=l>. r, = 

 then X lt Y lt Z^ will be inverse to X t , Y lf Z t , and will satisfy the equations 



x t x 3 +z t z t =i, zA+r.z.-oi 

 z& + r.r,- 1, j^z.+^r,- oJ 



Fig. 2. 



Fig. 3. 



B i 



The relations between the quantities X lt T lt Z, and X it Y 3 , Z 3 , are she\\n 

 in the annexed figure (Fig. 2). 



Let AR = X! and RB=Y l in the same straight line, and RO = Z l perpen- 

 dicular to AB. With as centre and unity as radius describe a circle. 



Draw CB 1 the polar of A, and C A' the polar of B, with respect to this 

 circle. These lines meet BO and AO in B' and A' respectively. Join J'//' 

 cutting J20 in R. Then RK = X,, RA'=Y,, and OR' = Z 1 . 



