PROFESSOR TAIT ON THE LAW OF FREQUENCY OF ERROR. 145 



If we erect these as ordinates at successive distances, each equal to unit, along a 

 line, we may graphically represent their relative values by a curve drawn, libera 

 mann, through their extremities. The area of this curve will evidently approxi- 

 mate to unity, which is the exact value of the sum of the areas of the rectangles 

 of unit breadth, each of which is bisected by one of the ordinates laid down from 

 the expansion. 



To find the corresponding curve of error, notice that the maximum ordinate is 



20 . 19 11 1 184756 



2 10"2 20 "1048576 



= 0-1762. 



Taking this as the value of ■,,— we have for (12) the expression 



1 



v = x € 10 ' 253 (17). 



y 5-675 . v ; 



The following table shows a few of the values of y from this formula, compared 

 with the corresponding terms in the binomial : it is sufficient for our purpose, as 

 it would not be worth while to take the trouble of calculating the areas of the 

 curve of error corresponding respectively to the rectangles above mentioned. 



X. 



y from (17). 



y from Binomial. 



Difference. 







0-1762 



0-1762 



o-oooo 



1 



0-1598 



0-1602 



-0-0004 



2 



0-1193 



0-1201 



-0-0008 



3 



0-0733 



00739 



-0-0006 



4 



0-0370 



0-0369 



+ 0-0001 



5 



0-0154 



0-0148 



+ 0-0006 



6 



0-0053 



0-0046 



+ 0-0007 



15. Nothing is better calculated to show the general soundness of the method 

 we have adopted in this paper, than the fact of the excessive closeness of the 

 above approximation : the case having been specially chosen as one in which we 

 could hardly have expected more than a rude resemblance to the law of error. 



