K. Pearson and A. Lep: 
65 
deviation, and li the distance from the mean at which the distribution is truncated. 
Then the frequencies of the shaded part are known, but N, h and a are unknown. 
Let n be the known total frequency of the tail, Vi the distance of its centroid, or 
mean, from the 'stump,' the second moment of the tail about the stump, and E 
the standard deviation of the tail, of course about its mean ; then Vi and v^' (or 
2 = \^Vo — Vi'-), are known from the observations. 
Now nvp is the ^;th moment of the tail about its stump and 
iV - },^, 
' \/27raJh 
I e""^'^^ {x - li'yp dx , x' = xja, h' = h/a, 
J w 
= Nai' [^ji, (W) - ph'f,,_, (h') + (h') - etc.) , 
where iXpQif) is the incomplete normal moment function of the pl\\ order. Repre- 
senting it by fXp for brevity, we have 
11 = JS^fMu, iiVi = iW (/u./ — h' fji^), 
711'.,' = Na" (yu./ — 2/;'/^/ + /t'-yu-,,'). 
Hence we find 
S"/vr = -v/^i = (v.: - 2//')/ 2^/- = (fi^fi,,' - /U,/-)/(/x/ - /;>„')- (1). 
Now i/tj is a known quantity, being the ratio of the squared standard deviation 
of the tail to the squared distance of its mean from the stump. Accordingly if the 
value of i/tj be tabled to each h', then, — /.to' being known from a table of the 
probability integral, and /i./ and //,./ from tables of the incomplete normal moment 
functions — we shall be able to find h' from the known value of -v/tj. 
In the next place: N—nj^o (2) 
will be known from the probability integral table of /.to' soon as h' is known. 
Lastly : a = T'l'/io'/CA*-/ — /''a'-o') = Vi'^lr., (8). 
Or, if -v/tj = /io'/(Mi' — ^(^'/io') be tabled for values of h', a will be given at once in 
terms of v/. Then h = h'a is determined, and the constants of the complete 
normal distribution found from the constants for the truncated tail. 
We require accordingly tables of i/r, and -v/r.^. It was found impossible to 
calculate these with sufficient accuracy from the usual 7-figure tables of the 
probability integral and the new tables of the normal moment functions. 
When h' gets at all considerable -v/tj and yfr., depend on the differences of very 
small quantities. Accordingly /z/ and /a./ were calculated de novo, and owing to 
the kindness of Mr Sheppard an unpublished 9-figure table of the probability 
integral calculated by him was used. Even thus, it has not been considered 
desirable to give the table to more than three decimal figures. 
Biometrika vi 9 
