49(5 DR G. PLARR ON THE DETERMINATION OF 



We remark that by the value of u 2 and by 



w 2 + x = a 2 a 2 + a' 2 (b 2 c 2 + c V 2 ) 



+ a 2 *' 2 + «o 2 (& V + c 2 c 2 ) + &V 2 + c V , 

 « 2 +a* = a 2 + & 2 + c 2 = t , 1 . 



we have 



Then 



W 2 = a 2 & 2 c 2 



If we put 



and by analogy, 



we have 



W 2 = a'b (a Q b c - b'c')( - a b Q c' - b'c ) 



= a%[ - a Q \\c' + b' 2 c c' + a b b'(c' 2 - c 2 )] 

 4W 2 = a\[ - « 2(l + B )C + (1 - B )C - 2« B'C<J 

 = ab [a' 2 C + B C'( - 1 - a 2 ) - 2B'a C ] . 



W' = B C'(l + « 2 )+2BVO , 

 W =a' 2 C, 



4W 3 =«7/(W +W'>. 

 As to W/=«, ^ «?, it is independent of B , B', namely, we get 



4W 1 '=-2a a' 2 C\ 



For the calcul of Y'Z', we may prepare the following values. First, we have 

 already 



4b 2 c, = a' 2 0'-W, 

 4& 3 c 3 = «' 2 C' + W' 



Secondly, 



Thirdly, 



Substituting into 

 we get 



c 2 a 2 =-((i b c' + b'c )a'b 



= - a '( b o 2a o c ' + h b ' c o) 

 4c 2 « 2 = - 24(1 + B>/ + B'c ] 



4c 2 a 2 = ( - 2a')[a c' + (B c'a + B'c )] 

 4c 3 a 3 = ( - 2a')[a c- (B c'« + BV )] . 



a 2 b 2 = a'b (a d b c -b'c) 



=MVo(l + B )-BVJ. 

 4a 2 J 2 = (2a0[a c + (B « c - BY)] 

 4« 3 Z> 3 = (2a')Kc -(B tf c -B'c')] . 



4y' = «% 3.[a'2C'-W , ] 



+ b\ - a c »)( - 2a')[a c' + B c'<z + B'c ] 

 + C W 3 )(2«')K Co + B « oCo -B'c']. 



