228 Investigation of Determinants. (April, 
n n n |” |n— |é.|n-p 
Mp a re ( OF ie 8 he . “([p- =p)? ? 
when (n—f) is odd 
3 bs . [n  |n+|p.|n-p - (52). 
My =3- Cp. (Cp += + Cp [ap 
when (”—/) is even J 
These quantities are evidently both integers (as they 
should be). Eq. (47) is true of this class of determinants 
only when f and (m—) are both odd or both even, which 
cannot occur when 7 is odd, but always happens when » is 
even. Thus— 
"My = "Mu-», when is even...... (53). 
The following relation obtains between two successive 
values of "M,, the higher value of m being an even number; so 
that (n—p) =(n—1), an odd number, and (n—1—4) = (n—2), 
an even number. 
2. "M, =}. (n—1) =} (n—1). (n—1 +n) ="""M,, by (52), 
n being even .... (54). 
Eq. (52) shows that, 7m general, Eq. (45) is true of this 
class of determinants only when finite, 7z.¢c., only when of 
even order, thus— 
"M, =o= "M,, when is odd. | 
"M, =1= °M,,, when 1 is even. 
It will be useful to record a few values of "My, for this 
class of determinant. Thus— 
Viabuks:0f My «caja oneie (56). 
Value of p. 
im. 
fo) I 2 3 4 5 6 7 8 g. |r0 
I || o oO 
2-|| I I I 
3 || o 6 3 oO 
4 I 6 21 6 I 
5 || 0 | 15 45 55 To ° 
Osx |) 25 120 | Igo 120 15 I 
7 {|| o | 28] 210] 630 595 231 21 o 
8 || r | 28] 406 | 1540 2485 1540 406 © 28 I 
9 || O | 45 | 630 | 3570 7875 8001 3486 666 36 | o 
Io I | 45 | 1035 | 7140 | 22155 31626 22155 | 7140 | 1035 | 45 | f° 
