10 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
§ (3.) To show that the Series for r is Convergent if r < 1 , whatever be the Values 
of h and k. 
Write the series in the form of p. 6 , i.e. :— 
Now 
ft rpn _ _ 
c — ^ r n _ x w a—\- 
v,i+i = hvn — nv 
w n+1 = kw n — nw„_ x 
(xii.) and (xiii.). 
From these we deduce 
v»+i = {^ 3 — (2n — 1)} v n _ x — (n — 1) (n — 2)v n _ s 
w n +i — [W — (2 n — 1)} w„_ x — (n — 1) (n — 2) w n _ z 
Now let s n = v H _ x r i>, j {jn}-, t„ = w )l _ 1 r h> j 
Then we find 
5 „ = Id ~{2n- 1)_ _ / (re - 1) (n - 2) 2 g r3 
" +3 + 1) ( n + 2) 11 * n(n + 1) (n + 2) “ 3 
l? — (2 n — 1) / (n — l)(?i — 2) . 3 
,i+3 ~ v/(»+T) ("% + 2) “ V «(» + l)(n + 2) < *-® r * 
Thus, when n is large, we find the ratio of successive terms s /l+ Js n or t n+i /t n is given 
by p, where 
p = — 2 r — r 3 /p or, p — — r. 
The ultimate ratio of s /l+2 t n+ 3 to s n f n is accordingly given by r 3 , hut this is the 
ratio of alternate terms of the original series. The original series thus breaks up 
into two series, one of odd and one of even powers of r. Both these series are 
absolutely convergent whatever h and k be, having an ultimate convergence ratio of r 2 
(4.) To find the Probable Error of the Correlation Coefficient as Determined by the 
Method of this Memoir. 
Given a division of the total frequency N into a, b, c, d groups, where 
a-}-& + c + c? = N, then the probable error of any one of them, say a, is ’67449 or,,, 
where* 
«■.= \/" ,x r 3 • ■ ..(«'!•). 
Let b -f d = n lt c + d = n 3 , then 
* The standard deviation of an event which happens np times and fails nq times in n trials is well 
known to be Jnpq. The probable errors here dealt with are throughout, of course, those arising from 
different samples of the same general population, 
