H. E. SoPER, A. W. Young, B. M. Cave, A. Lee, K. Pearson 343 
or, if ^ = e-^ == — r = r , n x = pr. 
Jo cosn z — X Jo cosh z — pr 
But . (l-P^r^ n ,.)";^' ^"-^ /cos-M-pr)\ 
2\ -2 
11-4 
77- (n — 3)! d{pr)'^ - Jo cosh z — pr 
/( - 1 
«-4 
2) ^ (l-r^) ^ ; ^ --^ ...(xxxvii), 
' 7T ^ Jo (cosh z - pr)"-i ^ ' 
(n 
II. - 1 
u -4 
(1 — r^) - say (xxxvii)'^'^ 
Similarly, if I'n-i = 
77 
Substituting in Eqn. (viii) we find 
n{l- p^r^) /„+i = (2n - 1) pr Z„ + (w - 1) (xxxviii), 
as the reduction formula for the /„'s. 
dz 
Jo (cosh z - po^)"~^ ' 
then n (1 - po*) = (2^ - 1) Po^I'n + [n - 1) ...(xxxviii)'"^. 
Now using value (xxxvii)'^''^ for i/^, the equation for the mode is 
{n-'^) rr„_, = p{l- P) {n-l)I„, 
or, if as before, = pr, we have : 
{n - 4) p24_i = (p-^ - p*) (« - 1) /„ (xxxix). 
This combined with 
n (1 - p*) = (27^ - 1) p^/„ + (n - 1) (xl), 
should determine the mode. 
Now assume p^ = p^^ + e, where po^ is some first approximation to p, then we 
find 
(n-4)po'^/'„_,-(»-l)(p'-^-po-^) /,/ 
(n - 4) + (n - 1) (n - 2) po^// - (n - 1) (p^ - p^") /',^^^ 
(xli). 
If we had obtained an approximation po^ to p-, we could start with 
, _ cos-i {-p/ ) , _ 1 po^ cos-i (- po^) 
vr^ ^ ~ Wo*+ " (i-Po¥ 
and by aid of (xxxviii)''!^ determine the /"s in succession. If = ^nl^n-x we can 
put our results in the forms (xliii) and (xUv) below, and calculate successive £"s : 
j<^' -(»-!)(,= -,..)[ 
e = ; i ^ 1 . . . (xliii), 
n — 4 ^ ' 
En 
+ {n-l){n- 2) po^ - n {n - 1) (p^ - po*) E, 
n+i 
