238 



we have 



Report S.A.x\. Advancement of Science. 



1=4 

 r=l 



■/ 



— 111 <-2 



— 11 d.2 





d. 



+ 



+ 



= I{A,-B^) + iii{A^-C^) + u{A^-D^) 

 + t{Bz-C,) + q{B,-D,) 



where Ai, A2, ... stand for the cofactors of cii, a^, ... in P. The 

 vanishing of this expression when | aib^CzcU \ is axisymmetric is 

 assured by the fact that then \AiB2CzDi\ is axisymmetric also. 



8. As an example of the application of such theorems as the 

 preceding let us take the problem of evaluating the determinant of 

 a matrix which is the sum of two vanishing matrices, say the 

 determinant 



62 + (-2 

 ■ba-{-y 



■ca—a 



— ab + y 



c^ + a' 



— cb + a 



where 



— ca 



-cb ^2 + 62 



— ca + P 



— be— a 

 ir- + b^ 



/,2_j_^2 —ab — ca \ 

 — ba c^ + a^ —be = o 



—7 i^ 



Calling: these vanishing determinants P and O and viewing the 

 given determinant as having binomial elements in accordance with 

 the composition of its matrix, we may partition it into eight 

 determinants. Of these the first and last being P and O may be 

 neglected. Taking next the three, each of which contains two 

 columns of P and one of Q, we know that their sum vanishes by 

 reason of the theorem of §7. There thus only remains to be 

 considered the sum of the three, each of which contains two 

 columns of Q and one of P, and this sum by §2 is equal to 



An equivalent of the given determinant is thus found. 



