A' = 



374 Mrs. Bryant on an Example in 



Then A', its reciprocal, is given* thus: — 



A/>ii Ap 12 Ap l3 Ap u 



Ap 21 Ap 22 Ap n Ap 2i 



Ap 3l Ap 32 Ap 33 Ap u 



Ap±i Ap 42 Ap 43 A Pu 



Hence, since each first minor of A f is equal to A 2 multiplied 

 by the corresponding constituent of A f, we have the series of 

 equations required for the proportionate values of the coefficients 



Pi P2 

 A' A 



£12 



A ? 



£13 

 A 



Consider, first, the Naples case. "We have 



Pl = Ax 

 P2 _ 



P22 P'2S p2± 

 PZ2 P33 PU 



P42 Pl3 PU 



■2013. 



= A< 



1 

 •71 



•76 



•71 

 1 

 •63 



■76 



63 

 1 



P33 Pte PSI 



= + 



1 



•63 



-•23 



P4S PU P*l 





•63 



1 



•27 



Pu Pu P11 





•71 



•27 



1 



= A 3 x-2013, 



= •3991. 



Similarly, for ^ and -^ we have the values of the other 



two principal minors of the original determinant, ?'. e. 

 (pB3y Pu, Pu) an( i (Piu P22, Pm) 5 eacn taken with its proper — 

 that is with the positive — sign J. For the signs of the remain- 

 ing minors which have to be equated to ~, &c, the following 



is a convenient rule. The minors of the constituents in the 

 first row are alternately positive and negative, beginning 

 with p n , whose minor is positive. The minors of the con- 

 stituents in the second row observe the same rule, beginning 

 with p 22 and proceeding in cyclic order p 23 , p 2 ±, p 21 . Similarly 

 for the third and fourth rows beginning with p 33 and p u . 

 Thus:— 



?12_ 



- (?23 ^34 P&) = — 



p23 P24 P21 1 = — 



•71 



•76 



•29 









p33 ^34 PBl J 



1 



•63 



-•23 



= -•1436. 







P 43 Pu Pa i 



•63 



1 



•27 





* See Prof. Edgeworth, on Correlated Averages, in Phil. Mag. August 

 1892, pp. 200, 201 especially, 

 t Ibid. 

 % See Salmon's ' Higher Algebra.' 



