Primary Minors of an Axisymmetric Determinant. 509 

 every row 



of the latter taken along with itself producing the row 



V, V, ■ • -, V, 2A n _^ w , 2A„_ 2 /< 



>ij • 



'llij\. 2 



- 2 

 ~1 



«*2 



~2 



3V 



- 2 



'#2*3 



of the former, and, taken along with any one 



/i 1? A' 2 , . . ., k' n 



of its fellows, the row 



/?!&!, h 2 k 2i . . ., /i n £„, hn-ihi + hnhi-i, • • •, ^A+^A; 

 and the result in question is that the determinant thus derived 

 is equal to the (?i + l)th power of the original. A case of 

 this, viz. : 



2#2# 3 2# 3 #! 2^!^ 2 



23/23/3 %s^i %iy 3 



2£ 2< sr 3 2^3^ 2^2 2 



,7l~l #2*2 3/3^3 #2^3 +#3*2 #3*1 + ^1 3/1-2+^13/2 



J z x x x z^x 2 z z x z z 2 x z + x 2 z z z d x 1 + x 5 z 1 z l x 2 + x 1 z 2 

 x 1 y 1 x,y 2 x z y % x 2 y 3 + y 2 a? 3 a?gy x + 3^ ^z/ 2 + y x oc 2 



occurs in a paper by Brill* published in 1870. The proof of 

 the general theorem may stand over for the present. 



(4) Beginning with the axisymmetric determinant of the 

 third order 



a h g 



h b f or A, 



9 f c. ; 



and agreeing to take the six independent variables in the 

 order a, b. c, /, g, h, the Jacobian sought is 

 B(A, B, C, F, G, H) orJ 



where A, B ? ... denote the signed complementary minors of 

 a, b y ... . As first obtained it takes the form 

 e b -if . 



c . a — 2g 



b a . 



j 



-9 



-h 



— a 

 I, 

 a 



h 

 f 



-2h 



9 



f 



* Brill A., " Uober diejenigen Curven eines Biiscliels," Math. Annalen, 

 iii. pp. 459-468. 



