M. Greenwood 
77 
Write in Euler's identity* 
1 - <Zj + ft] (1 - da) + C^OaO - Os) + ••• + <h<-h ■•• (1 - "«+i) = 1 ~ «i«-.a 3 ■•• «»+i> 
x x + pi 
a, = - . cio = — etc. 
y y +pi 
v 
Multiply by — - — and subsequently put y = 0. 
y x 
We have 1 + ± + m JZ±ti + ... . (*+&) (l±£i) ... (*±£*) (8). 
fx + \ 
\ P-2 )' 
"\ Pn J 
Pi PlP-2 
Putting n + 1 = a;, ^ = 1, p 2 = 2, ... p m = m, 
1 + + + ... J'ff f! + «±1) (9 ). 
1 2! y=1 \ y J 
Reverting to the general case, we note that for testing the divergence between 
first and second samples the formula (2) must always be employed when m and n 
are commensurable and p~q not small. This rule certainly applies to all cases of 
m and n less than 300 or 400 and p (or q) < 1. If m and n be large the best 
plan will be to fit to (2) the curve indicated by the momenta] constants, using its 
proportional areas (obtained by some convenient quadrature formula) precisely in 
the manner adopted with the tabled areas of the " normal " curve. 
Such a method is, however, not convenient for laboratory workers nor specially 
appropriate when m is a small number, since in that case the terms of the dis- 
continuous series are not closely represented by a continuous curve. 
Evidently what one needs is a tabulation of the series (2) for different values of 
m, n and p. 
Were it possible to obtain a simple formula for the sum of any assigned 
number of terms of (2), the computation of such a table would be a rapid process. 
In the particular case p = 0 or n, such a formula has been given above. In the 
general case I have not reached any resultf and more widely trained mathematicians, 
who have kindly allowed me to consult them, do not regard the problem as a 
simple one. 
I therefore fell back upon the method of direct calculation. This is a straight- 
forward but irksome task J. 
* See Chrystal's Algebra, Vol. n. p. 392, Ed. 1889. 
f Formulae are available in certain types of Hypergeometric Series. Vide M. J. M. Hill, " On a 
Formula for the Sum of a Finite Number of Terms of the Hypergeometric Series when the Fourth 
Element is equal to Unity," Proc. Loud. Math. Soc. 1907, Series 2, Vol. v. p. 335 ; and 1908, Series 2, 
Vol. vi. p. 339. The methods of these papers cannot be used in the present case. 
J Sir Konald Ross and Mr W. Stott have recently published (" Tables of Statistical Error," Annals 
of Tropical Medicine and Parasitology, Vol. v. No. 3, 1911) a set of tables for the use of laboratory 
workers. Their tables will be of great service in the cases in which p is not less say than 0T, but are 
not, I think, available for the class of problem discussed in this paper, since they appear to be based 
on the "normal" theory of errors. It must be noticed that in an immense number of examples which 
arise in medical work p will not only be less than 0 - l but less than 0-01 (the prevalence of mental defect 
in children, albinism, epilepsy, etc. are instances), and for such cases the "normal" treatment is, as 
pointed out above, inappropriate and often misleading. 
