304 Mr. F. Y. Edge worth on the Law of Error. 
The other general symbolical solution of equation (2) is 
s — - 2 " 
Here, if we may choose <f>(x) at pleasure, we may, by choosing 
it suitably, avail ourselves of the theorem given by Mr. Crofton 
(Phil. Trans. 1870, p. 187), in whose steps we are now almost 
treading. But, again, I submit that this procedure is in 
general illegitimate; that0( t i') is not to be chosen at pleasure, 
but so as to satisfy the condition 
c- doll 
and that when /is hyper-exponential then the theorem of Mr. 
Crofton does not apply. 
Subject to this limitation, the preceding proof of the law of 
error may be extended to the case in which the component 
element facility-curves are not identical (when c 2 is to be 
regarded as a function of s, and for sc 2 must be substituted 
2c 2 , approximately §c 2 ds) and to unsymmetrical curves. By 
taking account of quantities of the order -^- we may obtain a 
second approximation to the law of error. 
There is another mode of dealing with equation (1) which 
has the advantage of being applicable to those cases in which 
the mean powers of error are not finite. It begins by expand- 
ing the right-hand member in terms of x and neglecting 
those above a certain order, and then proceeds somewhat as 
follows: — Construct the form of the sought compound, partly 
by observing the initial tendency exhibited by the superim- 
position of two or three elements, and partly upon the prin- 
ciple that the compound has the same mean powers finite and 
infinite as the element, and a certain analogous theorem relating 
to mean exponentials, such as 1 e~ z2 f(z)d~. Having assumed 
the form of the sought function, determine its constants by 
the differential equations presented by the conditions 
r 
"f{z)«dz = d £ +«o=0), 
&c. = &c, 
the powers of -=- above a certain degree being neglected. 
