280 
Peccavimus ! 
for ha — \ha\ to the third and fourth powers. But if we keep only terms of the 
two lowest orders in our results we require to ascertain the values of 
.15 
The necessary term in the latter is — — I)'', but the finding of the former is 
far more troublesome and we have not so far succeeded in determining it. But it 
is possible to obtain some idea of the deviation of the curve from normality by 
considering what its /3's are in the case of the sample being made on a population 
following the normal law. We are able to do this, if we carry a stage further the 
work outlined in a paper in Biometrika, Vol. x, p. 526. We require the values of 
2, o-s'- and ji.,, fu,^ on p. 526 carried to a higher approximation by the introduction 
of the additional term in the Stirling's Theorem expression for the factorial. 
Miss Pairman has carried this out and finds 
^'-''(i-|.+8inlf) • (™) 
which leads, having regard to (xi) on p. 525, to 
.^, = ... = ^'-S==°^:(i-i-A) (II). 
We must now turn to the equations on pp. 527 — 8 to determine and yu-j. 
We find * : _ 
2/J ~ 4?r V 2rJ [ 4n 32n- 128??/ 
3 
1 + — ) as far as our approximation is valid (KK). 
^ ■ . 1 5 If, S\fl S \ 3/1 3 V 
Again -=^']^.-2ny^- n) + 8n^) '^A^i^ 8^^) 
'=T:(i-1) (LL). 
We must now replace n by TJj, + on.^, and sum as we have frequently had 
occasion to do. 
We have: a,,^^=''-^{l+~-~] (MM), 
3 1 
"^"■^ 2^V iV; 
-IC-I) 
These values may be used to determine the nature of the distribution of S in samples of constant 
size n. We have : 
2n \ in/ n 
The term in in 5B2 could not be determined unless we went to a still higher order term in S. 
Clearly for a sample of 25 2B1 approaches close to the Gaussian and 2B2 still closer. The non-approxi- 
mate values are -0219 and 3-0014 (loc. cit., p. 529). 
