PROCEEDINGS 



OF THE 



Cawlmtrge Iljitasopljkd Snrietjj, 



" Ignoration of coordinates" as a problem in linear substitutions. 

 By T. J. I'A. Bromwich, M.A., St John's College. 



[deceived 24 April 1901.] 



The method known in dynamics by the name of " ignoration 

 of coordinates" has received a good deal of attention at the 

 hands of writers on dynamics ; but it does not seem to have been 

 examined at all from the point of view of linear substitutions. 

 The object of the following is to explain how we are led to the 

 usual results by simple examination of the linear equations from 

 which we start. 



Suppose that we have (m + n) quantities £ 1; £>>•••> £ m > Vi,V-2,---> Vn 

 given as linear functions of (m + n) others x 1 , x 2 , ..., x m , y 1 ,y 2 , •■-,}/„, 

 by the equations 



_1 dV fr=l, 2, .... 

 '•"2 dy/ 1.9 = 1,2,..., 



where V is a quadratic function of the oc's and y's. We are now 

 going to express &,..., £„ , y } , . . . , y n in terms of w 1 , . . . , x m ,Vi,---, Vn ', 

 and to do so let the equations for the £'s and rj's be written in 

 the symbolical form 



% = Px + Qy, 



v = Rx + Sy , 



where P, S are square matrices of in and n rows respectively, 

 while Q is a rectangular matrix of m rows and n columns, and R 

 is one of n rows and m columns. 



1 dV 

 * r ~2 dx„ 



VOL. XI. PT. III. 



13 



