K. Pearson and A. Lee 
61 
We see accordingly that the chance of an outlying observation being reasonable 
or not, — the observation consisting of a complex of n variates, — can be readily 
found, if the incomplete normal moment functions have once been tabled, and the 
constants of the correlation surface be known. These incomplete normal moment 
functions serve a variety of purposes which will be developed in later papers. 
The present paper merely refers to the means they provide of determining the 
probability of any observation lying outside a given contour ellipsoid, i.e. there 
being any " fringe " beyond this value of x- 
In Table I. pp. 66, 67 the values of m,i{x) 
= fin i^)/ {{n — 1) {n — 3) — 5) ... 1|, if n be even, 
and = - 1) (n - 3) (m - 5) ... 2}, if n be odd, 
are tabled for values of a; from 0 to 5 proceeding by tenths. They were calculated 
in the following manner. Let 
if 11 be even, 
9n-e-i^' + T + 2^ + - +2 . 4. 6 ...^^ - i j ' " 
Then: ™,i= ^(1 — qn), if n be even, 
= 2^ (!-?«'), if " be odd. 
The even series involve the probability integral which was taken from 
Sheppard's Tables (Biometrika, Vol. i. p. 182). Consequently the even series 
may have an error of unity in the seventh figure. It is hoped that arithmetical 
blunders have been avoided, but the labour has been very considerable and some 
blunders may have escaped notice. 
If we make IJN = \, we find the contour ellipsoid within which half the 
frequency lies. In other words an observation is as likely to lie inside as outside 
this contour ellipsoid. The corresponding value of -)(, ^^^Y be spoken of as the 
"generalised probable error." Since the first 10 incomplete moment functions 
have been calculated it is possible to determine the generalised probable errors for 
1 up to 11 variables. These are given in the table on the next page. 
Of these, the first is very familiar. The second was first given in Bravais' 
classical memoir {Mem. pres. par divers Savmis Etrangers, T. ix. 1846, pp. 255 — 
332). The third professes to be given by Czuber, Theorie der Beobachtungsfehler, 
S. 404, but I do not agree with his results. The value 1"53817 was given by me 
in College Lectures in 1896. The remainder have, as far as I know, not hitherto 
been published, although they have been for a considerable time in use in my 
Biometrical Laboratory. 
