302 Mr. F. Y. Edge worth on the Law of Error. 
pandecl (the odd terms vanishing by hypothesis), 
Cf(z)dzu„+ £/0£«fc**» +&C.J 
provided that we take for granted what DeMorgan has postu- 
lated {Eneycl. Melr. § 88) and Mr. Crofton calls the " usual 
assumption " *, that the mean powers of the elemental errors 
are not infinite. In this case an approximate solution of (1) 
is afforded by a solution of the partial differential equation 
du_(? d 2 u 
ds~4 da? { ' Z) 
c 2 r x 
(where -~ — 1 z 2 f(z)dz). For let to be a solution of (2); 
then the right-hand member of equation (1) 
= j f(z)dru+ j~§+ terms of the order of (^ -^) 
raised to the second, third, &c. powers, 
= " + I % + termS ° f the ° rderS (s)'' (I)'' &C '' 
= u+ j y^ + terms of orders which may by hypothesis 
be neglected, 
=u + -j-= the left-hand member of equation (1). Q. E. D. 
Equation (2) has two general symbolical solutions in terms 
respectively of — and -r-. The former is resolvable into two 
series given by Poissonf, involving respectively even and odd 
powers of x. The even series is 
Now the rough general experience, or intuition, which allows 
us to assume that -^ and higher differentials may be neglected 
for some values at least of a-, allows us to assume that such values 
occur about the centre of the compound curve. The assump- 
tion holds accordingly for x = 0. But when x = 0, u = <f>(s). 
* Phil. Trans. 1870, p. 183. 
t Mecanique, p. 808. Of. Fourier, Chakur, ch. 9. 
