Karl Pearson 295 
I. Generalised Tchehijcheff Limit for Schols Problem luith a Normal Distribution. 
Radius of circle = \ V a-^^ + a-^, k" = 4(1— /■-) ai"a.//(ai" + cr.rf. 
\ = 
1 
1'25 
1 •'l 
X o 
2-0 
Z 0 
Q 
a 0 
A .A 
0-0 
o(/,) 
•36 (A) 
•5556 (A) 
•8125 
(A) 
•9386 (A) 
•98400 (A = A) 
•996924 (A) 
•999528 (A) 
0-1 
0(A) 
•36 (A) 
•5556 (A) 
•81875 
(A) 
•9422 (A) 
•98574 (7,) 
•997329 (A) 
•999611 (/s) 
0-2 
0(/,) 
•36 (A) 
•5556 (A) 
•8275 
(A) 
•9459 (A) 
•98740 (A) 
■997707 (A) 
■999685 (AO 
0-3 
0(/i) 
•36 (A) 
•5556 (A) 
•83125 (A) 
•9496 (A) 
•98899 (/-) 
■998109 (A) 
■999749 (7x ?) 
0-4 
0(/i) 
•36 (A) 
•5556 ^A) 
•8375 
(A) 
•9538 (A) 
■99099 (A) 
•998478 (A) 
■999805 (A?) 
0-5 
0(/i) 
•36 (A) 
•5556 (A) 
•84375 
(^.) 
•9592 (A) 
•99193 (A) 
■998806 (A) 
■999853 (A?) 
0-6 
0(A) 
•36 (A) 
•5556 (A) 
•8500 
(A= 
h) 
•9645 (A) 
•99333 (A) 
■999096 (A) 
■999893 (A?) 
0-7 
0(A) 
•36 (A) 
•5556 (A) 
•8641 
(A) 
•9696 (A) 
•99490 (A) 
■999380 (A) 
■999927 (A?) 
0-8 
0(A) 
•36 (A) 
•5654 (A) 
•8781 
(A) 
•9746 (A) 
•99631 (A) 
■999609 (A) 
■999954 (A?) 
0-9 
0(A) 
•36 (A) 
■5852 (A) 
•8922 
(A) 
■9809 (/,) 
■99770 (A) 
•999788 (A) 
•999975 (A?) 
1-0 
0(A) 
•36 (A) 
■6049 (A) 
•90625 (A = 
A) 
•9879 (A) 
■99895 (A) 
•999920 (A?) 
•999991 (A?) 
n. Values of the functions Informing tJie denominator of the Tchebycheff Limit to 
the probability that an Individual will fall for the case of Normal Bi-variate 
Frequency within a given circle of radius X \^ai-+ai. 
/c2 
A 
A 
A 
A 
A 
A 
A 
A 
0^0 
3-0 
15^0 
105^00 
945^00 
10,395-000 
135,135^000 
2,027,025-0000 
0^1 
2-9 
14^1 
96-09 
842-25 
9,024-525 
1 14,286^725 
1,670,080-7025 
0^2 
2-8 
13^2 
87^36 
744^00 
7,747^200 
95,356^800 
1,354,429-4400 
0^3 
2^7 
12-3 
78-81 
650^25 
6,561-675 
78,279-075 
1,077,729-5025 
0^4 
2-6 
ir4 
70-44 
561^00 
5,466^600 
62,987^400 
837,665-6400 
0^5 
2-5 
10^5 
62-25 
476^25 
4,460^625 
49,415^625 
631,949-0625 
0-6 
2^4 
9^6 
54-24 
396-00 
3,542^400 
37,497 ■eOO 
458,317-4400 
0^7 
2^3 
8^7 
46-41 
320-25 
2,710-575 
27,1 67 ■! 75 
314,534-9025 
0^8 
2-2 
7^8 
38-76 
249-00 
1, 963^800 
18,358-200 
198,392-0400 
0^9 
2^1 
6^9 
31-29 
182-25 
1,300^725 
740^000 
11,004^525 
107,705-9025 
1-0 
2^0 
6^0 
24^00 
120-00 
5,040^000 
40,320-0000 
The reader may be curious to know whether the Tchebychefif limit gives 
a better result for Schols' circles than for the elliptic contours. The actual pro- 
bability of an individual falling within the circle of radius \\/ai + a.^ is given by 
5 (1 - k' cos 8) 
TT J II \ — K cos 0 
where k' = \/l — and k- — 4<(1 — ?■-) a(-a.?l{a^^ + (T-?f as before. 
I have not succeeded in finding any rapidly converging expansion for this 
expression*, and have been reduced to evaluating its argument and usinga(|uadrature 
formula. Thus for \='2, k" = "4, I find 
P = -968,3694. 
* Unfortunately Schols has not tabled P, but only gives the values of \ for ten values of which 
occur when P — 1/2, i.e. radial values for generalised "probable errors." 
