72 



Proceedings of the Royal Irish Academy 







y + ay 







a + 2X-\- o? h + a/B 

 h + (3a h+2X + ft 

 ^ + ya f+y(3 







c + 2\ + y^ 



l + a{X + S) 

 n+ft{X + S) 

 »e + 7(A, + S) 



A + 8 /+a(X+S) m + )8(A + S) w + y(\+8) (? + (X + S)2-A2 



/8 



y 



A+S 



— a 

 a + 2X 



h 



g 

 I 



-/3 

 h 



h + 2\ 



f 

 m 



-y 



9 



f 

 c + 2\ 



(A + S) 

 I 



d-X^ 



Now since any first minor of the original form of A vanishes, then 

 any first minor of the latter form must vanish. Hence, 



0, 



a + 2X 



h 



9 



I 



h 



h + 2X 



f 



m 



9 



f 



C + 2X 



n 



I 



m 



n 



d-X? 



which is a quintic for X. 



ii. aZ + ygJf + yJY+ {X + 8) D = 0, 

 where Z, If, JV, D are first minors of i. 



r 



9 



f 

 e+ 2X 



- 1 a ft 



a a + 2X h 



ft h b+ 2X 



7 9/ 



Since aZ + )8J/ + yi\^+ (A + S)Z = 0, 



SJ) =- [aZ + ftM+ yiV+ AZ], 

 and x'^ -\- y^ + z' + lo \_a.x + /3y + y2 + Sm'J = ; 



.-. I){a? + y^ + s^) + a.I)wx + ftBwy + yBw% = - ZBvo^ 



= w'Ijj.L + ftM^ yi7+ AZ] ; 



.-. a.\I)wx - if;-Z] + ft{I>wy - 'w'-II~\ + y\JDw% - w^JV] 



+ Z[«^ + y2 + 2^] _ XDio'^ = 0. 



