FORMATION OF THE EQUATIONS OF CONDITION. 595 



when n m is even, 



n(b m b' m ) f r o ( a ~ a ') * n tn e equations for JT, 

 2~( a m + a'm) f r 2^ m + ^ /m ) * n tne equations for Y, 

 (b m + b' m ) for ~(# m + a' TO ) in the equations for ^; 

 and when ?i m is odd, by substituting 



n(b m + b f m ) for - (a m + a' m ) in the equations for X, 



1 f i 



~ o ( a m ~ a m) f r o (^m ~ ^'m) in the equations 



and 



-(b m b' m ) for ^(o^ a' m ) ni the equations for Z. 



Thus we have in the equations for X 



x ' m = 2 i ( a m- a ' m } or - (a m + a' m ) in the equation for # 



and x' m = - (b m - b' m ) or ^(b m + b' m ) in the equation for h% 



according as n m is even or odd. 



Also we have in the equations for Y 



y' m = ^(&m + &'m) or -(b m ~b' m ) in the equation for 



' Zj 



and 2/' m = - - (a, B + a' m ) or - - (a m - a' m ] in the equation for 



according as n m is even or odd. 



Similarly we have in the equations for Z 



z' B , = 2( a i + a ') or 2^""'"') in the e q uat io n for 9 r r> 



and z ' m = - (6 m + b' m ) or -(6 W -&',) in the equation for h', 



according as n m is even or odd. 



752 



