Mr. E. J. Nanson on the Relations between 



364 



where 



At = yt-BC-CA-AB. 



Multiply (8) by rf, £?, fy, and we find 



\tf +mC@+nBi;r) fcmnA=0, 



Z=l-A 



m«A = 0, -\ 

 bZB =0, > 

 -ZmC=0, ' 



(10) 



where 



m=l-B 2 



= 1~C 2 



Eliminating i& ££, £// from (8), (9), (10), we obtain the 

 required relations in the form 



\ 



IC 



IB 



A 



1 



mC 



X 



»tA 



B 



1 



nB 



wA 

 X 

 

 1 



wmA 

 nZB 

 ZmC 



=0. 



4. This result may be reduced to a more symmetrical form. 

 Substituting for I, m, n, \ their values, multiplying the fourth 

 row by BO, CA, AB, and adding to rows 1, 2, 3, we find 



D 

 C 

 B 

 A 

 1 



C 

 D 

 A 

 B 

 1 



B 

 A 

 D 



C 

 1 



A + BCD-A(B 2 + C 2 ) 

 B + CAD-B(C 2 + A 2 ) 

 C + ABD-C(A 2 + B 2 ) 

 D-ABC 



P 



= 0. 



Now substitute for p its value, multiply columns 1, 2, 3 by 

 BC CA, AB, and add to the last column ; thus 



D 

 C 

 B 

 A 

 1 



C 

 D 

 A 

 B 

 1 



B 



A 

 D 



C 

 1 



A + 2BCD 

 B + 2CAD 

 C+2ABD 

 D + 2ABC 

 k 



= 0, 



where 



Jfe=A + B + C-hD-£A-l. 



(11) 



(7) 



5. On expanding the determinant formed with the first 



