74 
Dr. Young’s remarks on the reduction 
first factor less than unity being always £~~ = -T_ ^he 
2 . 
1+ - 
n Mean error 
2 .50OOOO 
4 - 3750°0 
6 .312500 
B .273437 
10 .246094 
12 .225586 
14 .209473 
16 .196381 
18 .185471 
20 .176196 
30 .144466 
40 .125363 
50 .112271 
60 .102574 
70 .095022 
80 .088924 
30 .083868 
too .079586 
magnitude of the mean error is exhibited in the annexed 
table. 
The general expression for this series 
being i . -J. £ . . . Hi" 1 * it is obvious that if we 
multiply it by - . * . the product will be 
T , whatever the value of n may be: and 
when that value is large, the factors of these 
two expressions will approach so near to 
each other that they may be considered as 
equal; consequently the corresponding terms 
of either, taken between any two large va- 
lues of n, will vary in the subduplicate ratio 
of n, since their product, which may be con- 
sidered as the square of either, varies in the 
simple ratio of n, so that the mean error may 
ultimately be expressed by V The va- 
lue of p evidently approximates to that of 
the quadrant of a circle, of which the radius 
is unity : thus for n = 10 it is 1.6512, and for 
n — 100, 1.5788, instead of 1.5708 ; and the 
ultimate identity of these magnitudes has 
been demonstrated by Euler and others. (See Mr. Her- 
schel’s Treatise on Series, in Lacroix, Engl. Ed. n. 410.) 
The fraction thus found, multiplied by 2 n , gives the number 
of combinations expressed by the middle term, in which the 
error vanishes, when n is even; and the whole number of 
