Mr. F. Y. Edgeworth on the Law of Error. 303 
Therefore ~^- and hio-her differentials may be neglected rela- 
ys 2 ° - ° 
tivelv to -X. Therefore the third and following terms of the 
ds ° 
above series may be neglected relatively to the second, for 
values of x such that ( x 2 -% -=- ) may be neglected. In that 
case the solution may be written 
A 2x 2 d\^^ 
or, if we put -j for <£, 
-ooL c 2 ih*)J' 
This is approximately equal to 
£ C 2 dr(s) . 
And, if we may regard the last written expression as the ap- 
propriate form of the sought function in x, then the condition 
of a facility-curve (that the integral between the extreme 
limits equal unity) will afford a differential equation to deter- 
mine y}r(s). The solution is -^-(s) = os/ir <t/s+A; where A is 
a constant which will be found to be zero. But I submit that 
the condition on which the differential equation for yjr is based 
is not in general valid: that, for instance, in some cases it 
would be proper to equate the truncated series of Poisson, not, 
as above, to 
but to 
1 2a? f(«) 
, . . log-i 5- T7T' 
, , s log.!— log_i— s- \ , / • 
yfr(s) ° « e 2 f(«) 
The inappropriateness of the former expression appears from 
the principle stated on page 304, which seems also to bar the 
parallel step* (from 1 — ktx 2 to e~ K2r2 ) in the ordinary method. 
The odd series is, I think, inappropriate to the case of even 
elements. 
* Mr. Glaisher suspects the safety of this step (Monthly Notices Roy. 
Astron. Soc. 1873). 
