Axisymmetric Determinants. 



455 



2 3 

 2 3 



n-3 



71-3 



(1, n) vanishes : why it should vanish when 



vanishes is not so apparent, and the reason when found is of 

 considerable interest. 



As a preliminary let us consider an axisymmetric determinant 

 of the form 



/ 



j k in n 



d 



e 



f h o 



h m q 



i n r 



b c 



h i 



on the supposition that the minor 



d 



* f 



e 



j k 



/ 



k o 



q r 



V 10 

 10 X 



= 0. 



cl 



* f 



i 



+ 2bc 



e 



J k 



n 





/ 



k o 



r 





i 



n r 



X 





d 



e 



f 



i 



- c 2 



d 



e 



f 



h 



e 



J 



k 



n 





e 



j 



k 



m 



f 



k 







r 





f 



k 







<1 



h 



m 



q 



10 





h 



m 



2 



V 



It is at once evident that the determinant is equal to 



- V 



— — 6 2 1 + 2bcr) - c 2 £, say, 



where £, rj, r m ay be looked on as the principal minors correspond- 

 ing to the elements v, w, x of 



d e 



* j 



f h 



h m 



n 



f 



h 



i 



k 



m 



n 







2 



r 



q 



V 



w\ 



r 



10 



x 1 



and where therefore 



r) 



d e 



e j 



f h 



h m 



i n 



f 



h 



i 





d 



e 



f 



k 



m 



n 



. 



e 



J 



k 







<L 



r 





/ 



k 







9 



V 



ID 









T 



w 



X 











= 0. 



From this it follows that 



- b 2 t, + Zbcri - c 2 £ 



(b& - C4*) 2 ; 



so that we have the theorem : Any axisymmetric determinant icJiicJi 

 has (l,r) equal to zero for all values of r except n - 1 and n is cxpres- 



2 3 ... 7£-2 



sible as a square if the minor 



2 3 



n-2 



vanishes. 



(XII. 



