Mr. F. Y. Edgeworth on the Law of Error. 305 
Example 1. (Mean powers all finite.) 
2/3 2 \ 1 * 
iW-(. + £0 
s/ttc 
(where « + /3=l, /3<i). > 
Assume as the appropriate final form 
Changing x to s in this expression, and prefixing f(z) as a 
factor, and equating the integral between extreme limits to 
(1 + -7-1—7= — ^otm„- -fc — 7=-^, we find C = sc 2 . Which so- 
lutionis such that the superimposition of the element/^) trans- 
forms u into n s+l , not only as regards the absolute term of u, 
but also for the coefficient of x 2 -j -j- &c. being neglected > . 
Example 2. (Mean powers above second infinite.) 
Assume, upon the principles above stated, as the final form, 
2P 3 1 
7T (# 2 + P 2 ) 2 ' 
and, proceeding as before, construct a differential equation 
for P; from which it will be found that P = 2 v /s + A^> — a 
solution true for the second power of x, the first power of — . 
ds 
A similar result will be found when the denominator is not 
square ; in which case pq takes the place of p 2 in the com- 
pound. 
It may be observed that if the average curve be formed 
(which is done by changing x in the compound to si; and 
multiplying the result by s), the absolute term of the average 
is approximately equal to the absolute term of the element 
multiplied by \y/ s, when the denominator is square, other- 
wise an additional factor appears. When the denominator 
is higher than a biquadratic, the elemental absolute term is 
still to be multiplied by \A; but for the factor \ must be 
substituted a larger fraction — on to the case when all the 
mean powers become finite and the fraction becomes unity. 
Since now the fundamental equation (1)* is satisfied in the 
same sense in example 2 as in example 1, I submit, in the 
* Above, page 301. 
Phil. Mag. S. 5. Yol. 16. No. 100. Oct. 1883, Z 
