Mr. CAYLEY, on THE THEORY OF DETERMINANTS. 87 



From the equation (20), it follows that when (n) is even, the values of a symbol of the 

 form 



(^a,/3„ (n)] (22), 



is the same, over whichever of the columns a, /3 . . the mark (j-) is placed. To denote this 

 indifference, the preceding quantity is better represented by 



t 

 \Aa„ /3, ..(n) (23), 



this last form being never employed when (ti) is odd, in which case the same property does 

 not hold. Hence also an ordinary determinant is represented by 



t ft, 

 Mai/3i , Mill (24), 



the latter form being obviously equally general with the former one. 



It is obvious from the equations (17), (18), that the expression (22) vanishes, in the case 

 of (n) even whenever any two of the symbols (a) are equivalent, or any two of the symbols 

 {(i), he; but if (n) be odd, this property holds for the symbols (/3), &c., but not for the 

 marked ones (a). In fact, the interchange of the two equal symbols, in each case, changes 

 the sign of the expression (22), but they evidently leave it unaltered, i. e. the quantity in 

 question must be zero. 



Consider now the symbol 



t 

 Mil — (2p) (25), 



kk 



which, for shortness, may be denoted by 



t 

 {A.k.2p\ (26). 



I proceed to prove a theorem, which may be expressed as follows : 



t t 



\J.k.2p\ . {B .k.2q} = \JB]k.2p + ^q.2] (27), 



where 



^r.,.....y...=S.J,,,,„iS,,,,„, (28). 



The number of the symbols r,s ... being obviously 2p — 1, and that of x, y ... 2q—l. The 

 summatory sign S refers to I, and denotes the sum of several terms corresponding to values of 

 I from 1=1 to 1 = k. Also the theorem would be equally true if I had been placed in any 

 position whatever in the series r, s ... li and again, in any position whatever in the series at,y ... /, 

 instead of at the end of each of these. With a very slight modification this may be made to 

 suit the case of an odd number instead of one of the numbers 2p, 2q-, (in fact, it is only to 

 place the mark (f) in |y<ij|.. | over the column corresponding to the marked column in |yi..|, 

 {ji . . ^ being the one for which the number is odd), but it is inap|)Iicable where the two numbers 

 are odd. Consider the second side of (27). This may be expanded in the form 



2 +±.... ±.±,...Zb1, .._..,,,,.. . Zb1,.,,,.,^,^....Zb',. ,,,... (29), 



where 2 refers to the different quantities s, . . , w, y, . . as in (II). 



