548 
[549 
549. 
NOTE ON THE MAXIMA OF CERTAIN FACTORIAL FUNCTIONS. 
[From the Messenger of Mathematics, vol. II. (1873), pp. 129, 130.] 
I CONSIDER the functions 
y 2 -x{x- 1), 
y 2 = x(x-\) {x- 1), 
y 3 = x{x - f) (x - (x - 
(*-Î)(*!) (a-i). 
Attending only to the absolute values, disregarding the signs, y n has n maxima, 
viz. if n be odd, = 2p -f1 suppose, these are 
V V V V V V 
* 1> J 2> ••'> J- p> -* p+li -Ip,'" J. 1, 
where Y p+1 corresponds to the value x = \, and Y lt Y. 2) ..., Y p to values of x between 
0 and —-—-=r, ^ ^ and ¿r — , ..., £—i- and —^ -- . 
2p + 1 2p + 1 2p + 1 2p + 1 2p + l 
But if n be even, = 2p suppose, then the maxima are 
Y Y V V V 
where F 1} F 2 ,..., Y p correspond to values of x between 
0 and -xr- , ¿r— and ~, ..., — and 4. 
2p 2p 2p 2p 1