Notices and Bibliography 
207 
The general form reached is written 
I i d \\ I (d^\* .w 1 id 1 ^ 
>i~'"'-d\\dx) ^''U\Ux) " [t+2)\\dx] -r==r^e- -la 
where = ; ^2 = ^'-^^ ; etc. 
If this form be rewritten as 
i^(,r) = /io(^(x) + .430'"W + '44'/'''(.^-)+ - 
where = g - - * ) •-72<r 
v27rcr 
it becomes the same as that called Type A by Dr Charlier in (3), (5) and (6) and it is also the 
same as that given by Dr Thiele in "Theory of Observations" (London, C. and E. Lay ton, 1903) 
p. 35. Charlier's method of reaching his form is by following Hagen's development of Laplace. 
The same writer also gives in (4), and considers more minutely in (.5) and (6), the form (Type B) 
which he writes 
F {.v) = B^f (x) + BiAyj, (.i-) + 52A> (*•) + . . . 
, , , , sin TT.r ri X . ~1 
where \fr (a-) = - — +^-r-, ... . 
^ TT \_x l!(.r — 1) 2!(.r-2) J 
This curve with a range limited in one direction is, we believe, new though Thiele has given a 
form very closely allied to it (loc. cit. p. 21). 
Charlier uses the method of moments for fitting his curves, but though both Edgeworth 
and he do this, and their series finally take the same form, different graduation results will 
be reached owing to the index form being used in the one case and not in the other ; the diflference 
rnay, in some cases, be negligible but in others it becomes of more importance and we shall 
therefore refer to it later. 
It will be noticed that in all cases it is proposed to use a series to describe the frequency 
distribution and there seem to us so many objections to this course in practice that it is well to 
take this opportunity of examining it. The objections to it are as follows : 
(i) If one of the later coefficients has a large value the neglect of later terms of the series 
may involve a considerable error, while their inclusion demands the use of the higher moments 
which are untrustworthy owing to their large probable errors. 
(ii) In some cases the series lead to negative frequencies, which is objectionable. This 
can often occur with Type A and is noticeable with Thiele's example {loc. cit. p. 50). 
(iii) It is necessary to make successive graduations using an increasing number of terms 
in order to find how many terms of the series are required to give a satisfactory graduation. 
(iv) As we cannot tell at the first how many terms to use, it is necessary to base the 
solution of the equations for finding the constants on integrations over the whole series from — oo 
to + 00 and then neglect terms which may or may not be significant, or else to make successive 
trials with an increasing number of terms from equations formed from the actual number of 
terms used. The latter method would be better if the position of negative terms could be 
decided at the outset and if integration could be effected between any limits that might be 
indicated. This would however .seem to be impossible and Charlier uses the former method ; 
the objection does not apply to Edgeworth's series. 
The effect of these objections in the case of Charlier's work is interesting as it is quite 
impossible to reproduce one of his frequency curves (the bi-modal curve, fig. 5 of (6)) statistically 
because the negative frequencies play so important a part in the series that if positive frequency 
only be taken (which is what would happen in practice) an entirely different curve is obtained. 
We are by no means satisfied that in such cases the integration for moments from — oo to + oj is 
