54 Scientific Proceedings, Royal Dublin Society. 



We may also put 



sin a = a, sin p = p, and sin to = zs. 



If in equation A we write yR for y and zR for 2, so that the 

 new y and z may be expressed in parts of R, and if we substitute 



R 8 



the above values of a, b, c, &c, we shall have - - multiplying 



every term in equation A. 



It is therefore only necessary to consider the factors of a, b, c f 

 &c, within the brackets. We thus obtain 



a = (0 + to) 2 - o- 2 , b = 1 - <r 2 , c = 1 - a 2 , 



h = <p + to, I = a - (<j> + to), m = - 1 + o- 2 - (<j> + to) cr, 



d = - [fo + to) - a] 2 , 



, 1 , eosV 1 



to sin to to^ 



We thus obtain 



a -6= (<p + to) 2 - 1, «-c=(0+to) 2 - 1, 



(0 + to) 2 ff 2 

 1-aV = — + a - (<jt + zj), 



to to 



1 <T 2 



1-bV = + <r - (<j> + zj), 



ZJ ZJ 



I - cl' ' = h cr - (<£ + to), 



to to 



ZJ 2 zj 1 L J 



^^_(*±f>_2 + ^_<*±^. 



TO" TO TO 



Whence 



(« - &) 2 + 4A 2 = 1 + terms of the second degree in 0, to, &c, 

 (a - c) 2 = 1 + terms of the second degree, 



