4 The Prohable Error of a Mean 
In a similar tedious way I find : 
(/I - l)(« + l)(n + 3) 
and = 
(H l)(/i +3)(?! +5) 
The law of formation of these moment coefficients appears to be a simple one, 
but I have not seen my way to a general proof. 
If now il/j. be the ii"' moment coefficient of s- about its mean, we have 
M, = ^- (ii^> j(„ + 1 , _ („ _ l)> „ .V , 
Jl/, = ,<.= I <''-1)('' + 1 )(" _ 3 ) 2( n-l) _ (it-iy 
il/, = ^'^^ [(» -l){n + 1) (h + 8) (/; + 5) - 32 (// - l)'^ - 12 (« - 1)^ - («. - 1)^} 
n: 
^ — "a W+ 9"'+ 23h + 15 -32h + 32 - 12?/H24/i - 12— »H3ji— 3?? + ! ! 
^ 12^M."_- IK" ±3) 
Hence P-^^-JL . _ _ 3(»±3 ) 
Hence " " « - 1 ' ^'^"il//- n-l ' 
.-. 2/3, - 3/9, - 6 = [6 (h + 3) - 24 - 6 (n - 1)1 = 0. 
Consequently a curve of Professor Pearson's type III. may be expected to fit 
the distribution of s". 
The equation referred to an origin at the zero end of the curve will be 
y = GxPe-y", 
Mo Vo-(«-l)«-' n 
where j = 2 
and ^ = = _ 1 = ^-- 
Consequently the equation becomes 
n - .1 n.v 
y = Ox e '^''2 , 
which will give the distribution of s'-. 
n-S nx 
The area of this curve is Cj x e '^f^- dx = I {say). 
Jo 
