122 Mr Brill, On the Application of Quaternions [Nov. 24, 



As we have here twelve equations, and only eight quantities 

 to be determined, it is obvious that the equations imply four 

 relations between the differential coefficients of a, /3, 7, B. It is 

 to be remarked that the group of twelve equations consists of 

 four sets containing three equations each, the three equations of 

 any one set containing only a single pair of the above-mentioned 

 eight quantities. The pairs are as follows : %i and v, f and I, g 

 and m, h and n. Thus each set of three equations will furnish 

 us with a single relation, and enable us to determine, subject to 

 that relation, the values of a pair of the eight quantities required. 



The four relations connecting the differential coefficients of a, 

 /3, 7, S are easily shewn to be identical with those contained in 

 Article 1, and consequently the existence of equation (2) is justi- 

 fied. Thus we have : 



da dy , d8 



= ^ L = — q — m — m + 2g = g-2m = ^-, 



dz dx y y J dy' 



dB da n7 7,0 7 , 98 



dx dy oz 



ox ay oz 



It is also easily proved that the values of R and 8 can be 

 expressed by the formulae 



« B -(*4-*l>- M -(4 : -*s)-- 



3. Let p and q be two independent quaternion solutions of 

 equation (1), then according to the preceding Article we have two 

 equations of the form 



dp = dp.T + d<r . U, 



dq = dp . V+ da . W. 



From these we obtain 



dp . U' 1 = dp . TU~ l + da, 



dq. W~ l = dp.VW- x +d<r; 



whence by subtraction 



dp . U' 1 - dq . W- 1 = dp (TU' 1 - VW' 1 ), 



and therefore 



dp = (dp . U~ x - dq . W- 1 ) (TU- 1 - VW'T- 



