296 
Oti Generalised Tchebychej^ Theorems 
The process is not as long as it might seem. Indeed if we only need four decimal 
places, it is (juite adequate to integrate only through the first quadrant, the second 
contributes nothing of importance. The value given by the last Tchebycheff limit is 
P > -8375. 
This is of the same order of divergence as we found for the elliptic contour, i.e. for 
= 7, we had P = "9698, with a Tchebycheff limit P > '8600. Thus the measure 
of approach does not seem very close in this case until we reach higher values of A,. 
On the whole we must express disappointment at the results of the Tchebycheff 
process. We had found Tchebycheff's own limit based only on the second moment 
of small practical value, although it is to be found occupying a prominent position 
in many continental works on probability. By extending it to higher moments and 
product-moments we have reached results which are great improvements on the 
original Tchebycheff limit, but the method still lacks the degree of approximation 
(except for probabilities over •99, say) which would make the result of real value in 
practical statistics. It is, however, conceivable that some more ingenious application 
of Tchebycheff's idea may lead to a limit more close to the actual value of the 
probability. 
