OF SECONDARY MINORS OF A PERSYMMETRIC DETERMINANT. 



531 



Thus, in the case of n = 3 we have 



B„ 



B x 



B 2 



B 3 





Qi 



c 2 



c 3 





a 2 



& 



7 2 





a 3 



A 



y 3 





B„ 



A, 



A 2 



A 3 





c, 



c 2 



c 3 



+ 



a 2 



A 



72 





a 3 



A 



73 





A 1 A 2 A 3 



Bx 



a 



2 & 



3 /^3 



B 3 



72 

 73 



AxB 2 C 3 | (/3 3 -y 2 ) 



(31) A still more general theorem than that of § 27 is as follows : — 

 If two arrays B , B' be taken, B containing p + 1 rows and p + q — r columns and 

 B' containing p + q — 1 rows and q columns, and a new determinant, A p , of the 

 (p + q) tA order be formed having for its first p rows all the rows of B except the p th each 

 preceded by r zeros, and for the first column of the remaining space the last q elements 

 of the p th row ofB, and for the other columns the p + q— 1 rows ofB', — then 



2 



A„(-1K 



vanishes if the last coaxial minor of the q tft order in A be axisymmetric. 

 Thus, when 



B- 



(A, 

 B x 



A, 

 B 2 

 C 2 

 D, 



B 3 

 C 3 

 D, 



A5^ 



B, 



A 4 

 B 4 

 C 4 

 D, D 



5 / 



Pi & A A 



JJ' = J ^1 ^2 73 74 



8j S 2 8 3 S 4 



€., 



fl £l ^3 £4 I 



— that is to say, when p = 3 , q = 4 , r = 2 , — the four-termed aggregate 



2 



A, 

 A g 



A 4 

 A, 



Bj B 2 B 3 



0^ Co 3 



Dj D 2 D 3 D 



ft 7i S 



& 72 



& 73 



& 74 



B„ B, 



8., 

 8, 



'4 -^5 



C 4 c 5 



£3 



£4 



4 



vanishes when j 7i^ 2 6 s^4 I ^ s axisymmetric. 



Developing each of the seven-line determinants of the aggregate in terms of minors 

 formed from the first three rows and minors formed from the last four rows, we obtain 



Z |BxC 2 D 3 



£1 



A 4 

 A K 



a 2 € 2 £2 

 «3 C 3 £s 



U 



Z I B iC 2 D 4 



A 2 



A3 a 2 



A 4 a 3 



A 5 « 4 



1 8 i k 



8 2 4 



5 3 ^3 



54 £ 4 



