K. Pearson 
285 
where of course = 0, x^ = /(, x.^ = 2h, etc., if h be the base element. Then we 
easily find 
Zdx = CJi 
J 0 
720 ' 30240 
\ 
Zxdx = CV' + ^JSI — + TTTv^Q ^4 — 
Jo 
Zx-dx = GJi + -— - 
in lol2 
12 240 ' 6048 
28800 
as + 
I Zx 
J 0 
^^^^ = ^^^^ - 1440 + 15840^^ 
0 252 480 - 3168 ' / 
(X), 
where the values of the Bernoulli numbers have been substituted. Now let us use 
(viii) and write UsjN = cts, then we find 
IJ-2 
f^3 
720 ^ 30240 ~ 
^ N '^'6 l-20"' ^3024"^ 
SGJi 
IT"*" 504" 
9600 
4C,/i //^ 
" ~ 30 + 126 ~ 1440 + 
5 CVi _ / 
^ 288 
as + 
3168 
, _ 6CJi k' , ' _ 
^'~ 'N 42 ~ 80 ^ 528 ~ 
(xi). 
If the base element be as usual unity, then we have simply to put /; = 1 throughout. 
If there be high contact at a; = 0, then we have simply 
where B.^,. = 0, fijr in this case — a/ — a/ = etc. = 0. 
We can modify (xi) in the following manner. Since Zx^, s > 0, vanishes at both 
ends of the range we have, substituting the value of Z (see p. 282), and putting the 
base element unity : 
