498 DR G. PLARR ON THE DETERMINATION OF 



Thus we have for the first equation (I') : 



a: (X 2 + X' 2 ) + 2z'X X' - u*a\ W 2 + W 2 - 2B W W ] 

 + B'(X a + X'2) + 2s X X' - « W 2 [ B ( W 2 + W' 2 ) - 2 W W] = . 



The second equation (II') will contain the same terms as the first, with this 

 difference, that the terms in the second line will have changed signs because 

 of the change of sign of x', X' and W, and of B . 



Thus the terms of the first line will be =zero separately, and the terms of 

 the second line will be =zero separately also. 



The terms of the first line will be obtained by the sum 



(^Y /2 -tt 6 W 2 ' 2 i+(w 2 Z' 2 -w 6 W 8 ' 2 ) = (III.) 



the terms in the second line in question will be obtained by subtraction 



( V 2Y'2_^W 2 ,2 )-(«; 2 Z -tt 6 W 8 ' 2 ) = U (IV.) 



By the application of 



B 2 + B' 2 = l 



we may transform the terms of order zero in (III.) or the terms of the first 

 order in (IV.) in order to render the equations homogeneous as to B , B'. 

 The equations will then be respectively of the forms 



G B 2 + G 1 B B' + G 2 B'^0 ) 

 H B 3 + H 1 B 2 B' + H 2 B B' 2 + H 3 B'3 = . 



More explicitly it can easily be shown that these equations are of the forms 



G '(B « C') 2 + G^B^C'JB' + G 2 B'2 = 

 H ' B 3(« U C') 2 + H 1 'B 2 B'(a C) + H 2 B B' 2 + H 3 'B> C') 2 = . 



As to their degree the terms are complete rational functions of the tenth 

 degree in (III.), and twelfth in (IV.), in respect to both a ', a', and c ,c'. 



If we look on the whole question from a theoretical point of view, we may 



say that the question is now solved, because the elimination of g; from the two 



last equations will give us a relation between the two angles A and C, so that 

 one of them, A as for example, being looked upon as an independent variable, 

 will determine the other angle C in this hypothesis, and consequently B, and 

 mediately y and z will depend upon A. Of course this theoretical result, when 

 put to the practice, will lead to inextricably complicated multiple solutions, 



T> 



owing to the high degree of the resultant of the elimination of ^,. 



