88 Mr. CAYLEY, ON THE THEORY OF DETERMINANTS. 



Substituting from (28), this becomes 



Effecting the summation with respect lo w, y . . this becomes 



2. 5',, . . ^,^ . . + ±. • . ^,.„..,, • . A.,...^ {B1.1..1A (31), 



( kk..l^\ 

 2 now referring to s,...only. The quantity under the sign 2 vanishes if any two of the 

 quantities I are equal, and in the contrary case, we have 



(fill../,] =±,|S.fe.2g}, (32), 



which reduces the above to 



{S.fc.29}.2+±...±,^,.„..,,..^j.,,..,., {S3), 



2 referring to the quantities s ... , and also to the quantities /. And this is evidently equivalent 



to 



t t 

 {A.k.'2p}{B.k.'2q}, (34), 



the theorem to be proved. It is obvious that when p = 1, 9 = 1, the equation (27), coincides 

 with the theorem (O)) quoted in the introduction to this paper. 



