280 On Generalised Tcliehiichef Theorems 
(2) Two Variates ; Limit to the Frequency loitiiin an Elliptic Area round the 
Mean as Centre. 
■Let the law of frequency he z = cf) (w, y) and let the standard deviations of x 
and 2/ be cTi, o-.,, and r be the coefficient of correlation between x and y. Let us take 
as our ellipse, 
I — r-\ o-j- CT] (To an- J ^ 
X and y being measured as deviations from the mean. 
Then by giving special values to ^n, 6.^ and %^ we can get any ellipse we 
please. Further since the curve is to be an ellipse r-By? < 0n^22* and we shall take 
^„ and always positive. Thus and all its powers will invariably be positive. 
Now consider ; if N = jj cf) (xy) dxdy, 
the integration extending all over space covered by the frequency surface. Divide 
both sides by 
Take out all the' values for which x gi'eater than ^o, then 
X^^^ Njj ■ ■ - \Xo 
when the integral extends over the area for which % is > Xo- Hence 
> chance . of an observation falling outside the 
ellipse Xo- 
Let P be the chance of an observation falling inside this ellipse, then Ave have 
at once 
^>1-^ (i")- 
Now we define 
= ^[J (})(xy)x'y''dxdy 
in our case, as the .s, s'th product moment-coefficient about the mean. And it is 
very convenient to write 
qss'=Pss'/(<^i'o-/) (iv) 
and term q^^' a reduced j)roduct moment-coefficient. 
* We shall generally wish to have symmetry of expression between a; and y, and in this case we take 
B.n = 0u = (> say and write di-n-jd^p and we shall have as necessary condition for the ellipse /xl. 
