Karl Pearson 
293 
round the origin. This corresponds to Schols' Problem*. It may be useful to 
have a Tchebycheft' limit for this case, although we have yet to meet the particular 
instance in practical statistics where it would be of marked advantagef. 
We can best investigate this problem de novo. 
Let = Jf {x- + y-)- (f) {xy) dxdy. 
Then if R be any radius round the origin, 
/,/i^- = // i^~f)\(xy)dxdy, 
the integral being taken to include the whole volume of the probability surface 
2 = ^ {x, y). Now pick out those elements of the integral for which x- + y" is 
> R\ then 
/,./i»:->// f'g'^)<^{xy)dxdy, 
where the integration extends over the above-mentioned elements only, and is 
therefore 
■ Jl {xy)dxdy, 
but this integral is 1 — P, where P is the j^robability that the individual talis within 
the distance R of the origin. Thus the Tchebycheff' limit is given by 
R'' 
Now clearly we have 
^6 = |j + y-f <f> dxdy 
s(s-l) 
Now write 7^ = A, Vo-i" + a.f, and further take tan 6 = a.^jo'i. Then 
^ = jcos--"-' 6^r/os,„ + s cos-''-" 6 sin- Oq^^.,^., + cos-"'-'' 0 sin^ ^'/■j,-4,4 
+ ' \ cos="-" 0 sni" ^73s_„,,; + . . . 
For the particular case in which 5 = 1, 
^ = l(cos^^+sin^^) = l 
Fors = 2, ^ =;;^ rcos^^?/5., + 2cos-^^sin-6'(/,„, 4-sin^6'y8.,'V 
* Over de Theoiie Jer Fouten in Kuimte en in het platte Vlak, VcrhamUintjcn dcr K. Akademii' vmi 
Wetenschapen, Deel xv, pp. 1—68, Amsterdam, 1875. Translated into French in the Aiinulf!; de VKcole 
polytechnique de Dcl/t, Tome ii, pp. 123—178. Leide, 1886. 
+ It is conceivable tliat the solutions given might be serviceable in the case of testing machine guns 
against a target. 
