12 CHARLES CARPMAEL ON THE LAW OF FACILITY OF ERROR. 
results have been verified by Cauchy. From their investigations it appears that the series 
(iv) is approximately equal to 
L RL ONE eS —— -" 1 
s+ / | He dy (0 = WD \ (vi) 
where 7 is the ratio of the circumference to the diameter of a circle, and eis the base of 
Napier’s logarithms. 
The series (v) is also shewn to be approximately equal to 

SEE ay oe 72 _ 62 a 
ie FL n ae 24 + ss :" | dy (vil) 
In practise it is generally sufficient to use the first term of vii, so that we may say that 
the chance of an error between y and y-+ dy is approximately 
tr ay (viii) 
To take a numerical example suppose x —6, the chance of the error in the sum of six 
quantities lying between 0 and dy is by the true formula (v) 0°55 dy, by the approximate 
formula (vii) it is 0°550085 dy, and by (viii) 0°564 dy. For the chance of an error between 1 
and 1 + dy we have in the three cases 0°216! dy, 0°21447 dy and 0:20755 dy respectively. 
