288 On Generalised Tchehycheff Theorems 
owr ellipse that of the normal contours, although of course for the general case this 
will not be a contour of equal probability, although it may roughly approximate to it. 
-Thus we find for this case, 
/: = 2, 
= (]7r^3 f^™ + + ^ + ^^^^ ~ ^^^'1 + + ^'^^.-O + 12r2 ((742 + ^04) - 87-^^33}, 
^4 = ^^.,-^4 {^80 + ?08 + 4 (r^,a + 7„,;) + 6^44 " ^'^ iqvi + Q^v + ^Qs, + S^^r,) 
+ 24r^- ((/,3 + q,, + 2^44) - 32r^ (g,, + + IGr^q,,} (vii), 
and the general value of /, will be 
1 H-s m = s—n ( „ I 
Is = 7T^-i;v, 'S' S j(-l)«2"r'- 
(1 - r-y „=o ( (« ~u- m.) ! m ! m ! ^ ^ j 
For the case of a normal distribution the qs, are all given in terms of r 
{Biometrika, Vol. xil, p. 87) and on substitution we find 
I, = 2, 1,^ 8, /3 = 48, /4 = 384; 
generally = '2s (2s - 2) {2s — 4) ... 2, which can be shown directly, thus : 
•'0 . 
if we integrate by parts, 
= 2s {2s - 2) (2,9 - 4) ... 2 X e~ ^x^X 
= 2.S (2s - 2) (2s - 4) ... 2. ' . . 
Accordingly our generalised TchebychefF's limit becomes 
„ ^ 2s (2s -2) (2s- 4) ... 2 . ^ 
P > 1 — (vni)*, 
and our best value of s will be determinable from 2s < xi^, or s must be the greatest 
integer less than or the integer equal to l^o'- 
Now the actual volume of the frequency surfece inside the contour 
^ 1 — 7- \«Tj- a^a., cTg-y 
is known to be 1— e" "^"^"", and it is thus easy to test the present generalised 
Tchebycheff limit as applied to this case. 
* This result is almost at once extensible to any number of variates following the normal distri- 
bution, but as the actual value of the probability is known Ihcvo is no value in writing down this 
limiting value. 
