H. E. SoPBR, A. W. Young, B. M. Cave, A. Lee, K. Pearson 349 
Table C in Appendix gives the values of log ^^.^z^^- , log (1 — p'^)^, log Xi , 
Vn — 1 
^^SX2> 4'!' 4'%' 4'i' ^"^^ enables the ordinates of the frequency curves to be 
calculated with considerable rapidity for w = 25 and upwards*. 
Approximate Expressions for the Mode. 
d.z 
Writing /,/ = 
" Jo (cosh z — prY' 
we find /„' = \ T^^—^s^ — fl + ~ + ^ + ^\ + ' 
2(1— prf- V n \ n nr J 
where stands for (j)^ [pr). We now use Eqn. (xxxix) and find 
r — pp _ n — I /n — 1 ^ 
p — pr^ n — 4: 
where 
V n "'// \ {n — 1) [n — l)^ (to — 1)^ ""/ 
(liii). 
If we expand this in inverse powers of , we deduce 
P = 1 _ 'A/ , -^4>2+<f> l _ 4>/' + </'/^ + <f>l - Hl<f>2 - + I 
(w-l)2^ (w-l)3 (n~l)4 
Again 
n-1 / n^ _ 5 ^3 373 898^ 
rT^ V '~ir ~ 2 (w - 1) SXn^l? ^ {n - 1)^ ^ I28 {n^- i)^ 
Thus we have 
r - pP _ 5 63 - (/./ 3 73 - 240/ + 16<^/2 _ 32,^^^ 
^ + 2 {n-1) 8{n- 1)2 + 16 {n - 
8987 - 816(^/ + 192(/>/2 _ 1280/ ^ + 384(^/</.2' - 256(^/ - 384^3' 
^ ~128(M^lj* ^ "" 
Bringing the first term on the right to the left we reach after substituting for 
the 0"s 
{} I 5(l-r2) ( 61-p r)(l-r2) (367 - Spr - 2p2;.2) (i - f.2) 
* "^2(to-1) 8 (to -1)2 ^ 16 (to -1)3 
(17606 - 195pr- - 81p^P - bOp^P) (1 - P) \ 
^ 256 (TO -1)* "7 
* The ordinates calculated by the rising difference formula were tested in this manner. For n - 25 
the accordance was excellent, and quite good enough for piactical purposes at n = 10. Below this 
(Hi) becomes less reliable and needs more terms. 
