relating to " Canonic Roots. 3 ' 459 



matrix, e. g., being 



y 4 y 8 ^ y^x 2, ycc 3 x 4 



a b c d e 



a 1 b' d d> d 



a" b" c" d" d\ 



and the other being 



au -f a'v + a"w bu + b'v + b"w cu + dv + d'w du + d'v -f d"w 

 bu + b'v + b u w cu-\- dv + d'w du -f d'v + d"w eu + dv\ + d'w 



I have ascertained, and hope shortly to publish, the method 

 of obtaining the explicit value of double determinants in the most 

 general case and under their most symmetrical form : for the 

 particular case before our eyes, this resultant will be as fol- 

 lows : — 



aba' a" 



abb' a" 



aba' b" 



abb' b" 



b c b' b" 



b c a* b" 



.bcb' d' 



b c d d' 



c d d c" 



c d d' d' 



+ c d d d" 



c d d' d" 



d e d' d" 



d e d d" 



d e d' d' 



d e d d' 



a b ! a" a 



a' b f b" a 



a' b' a" b 



a' b' b" b 



b' c 1 b" b 



b' c l d ! b 



, V d b" c 

 + d d' d' d 



V d d' c\ 



c' d' c" c 



d d' d" c 



d d' d" d 



d' e' d" d 



d' d d' d 



d' d d" e 



d 1 d d' e 



a" b" a a' 



a!'b"b a' 



a" b" a V 



a"b"b V 



b" c" b b' 



b" d' c b' 



, b" d' b d 



b" d' c c' 



c"d"c c' 



d' d" d d 



+ d' d" c d' 



d' d" d d' 



d" e" d d' 



d"d'e d' 



d" d' d d 



d"d l e e' 



And it may be noticed that if we return to the original question, 

 in which the coefficients are no longer independent, but where 

 the column a'b'dd'd is identical, term for term, with bcdef, and 

 a"b"d'd"d' with cdefg, the above determinant becomes 

















a 



b 



c 



d 





* 





b 



c 



d 



a 



b 

 c 

 d 



c 

 d 

 e 



d 

 e 

 f 



e 



/ 



u 





* 





c 

 d 

 e 



d 



e 

 f 



e 



f 

 9 



b 

 c 

 d 







* 





c 



d a 



b 



















d 



e b 



c 







* 









* 





e 



f c 



d 



















f 



g d 



e 



















