510 Dr. T. Muir on the Jacobian of the 



Multiplying this columnwise by A 4 in the form 

 a 2 h 2 (f- hg ga ah 



h 2 b 2 f bf fh lib 



f r 



fi 



c 9 



9f 



2hg 2bf 2fc bc+f hc+fg hf+bg 

 2ga 2fh 2cg hc+fg ac+tf af+gh 

 2ah ±hb 2gf hf+bg af+gh ab + lt 2 

 we obtain 



aA-A «B aO dF aG 



bA bB-A bC b¥ bG 



cE cC-A cF cG 



2/B 2/C 2/F-A 2/G 

 2gB 2gC 2gF 2gG-A 2gR 

 2hB 2hC MF 2hG 2hR 

 which, it is evident, equals 



ak.bB.cC.2JF.2gG.2hH. 



i-4 i i ... 



aA 



A 



cA 



2/A 

 2gA 

 2hA 



bR 



cR 



2/H 



1- 



1 



bB 



>-£ 



This last determinant, however (§3), 



f 7 a A + bB + cC + 2 /F + 2gG + 2//H | 



2AH I 1 ~A J I 



aA.bB.cQ .2fF.2gG. 

 consequently we have 



and therefore J=— 2 A 2 . 



(5) In the case of the axisymmetric determinant of the 



