404 Mr. J. J. Sylvester on a remarkable Discovery in the 

 Oui' grand determinant then takes the form 



9-^^} ,f) ^> d> f^'y *; « 

 '*5 >-*g; /; ^y d] c'y b 



*; ^y ^+15' /' ^' ^' ^ 



j'y'^hf^', 9-^'y f> ^y d 



\ 

 I', k\ j; i; h; ^-gj / l 



m; /; k ; j ; i; A; <7+\. 

 H^pe it will be observed that 

 .« a and m appear only 1 time. 



b and / ... 2 times. 



c and A: ... 3 ... 



^andy ... 4 ... 



€ and ^ ... 5 ... 



/and A ... 6 ... 



ff ••• 7^ ••• 



J^t now the coefficients be called 

 : Hg H3 H4 H5 Hg H7, 



Hg and Hg manifestly are independent. 



Again, if possible, let H4 =^pYi^y then a and m would appear 

 twice in H4, contraiy to the rule. 

 Hence H4 is independent of Hg, Hg. 

 For a similar reason H5 cannot depend on Hg, Hg. 

 Agaiuj if possible, let 



H6=i>Hg3 + ^Hg.H4 + rH3^ 



H/ will contain h^ . /^', which by the rule cannot appear in 

 Hg . H4, or in Hg . Hg. 



Hence />=0. 



Also H4 will contain ^^/^ x the coefficient of \^ in 



(^+ b) (^- ft) {s+ Y^, 



which is not zero. And Hg also contains bl) hence HgH4 will 

 contain 1^. P. But Hg will evidently not contain b^ or P, or bH 

 or bP, nor can Hg contain bH^; hence ^=0. Finally, Hg^ will 



