4 
208 
Notices and Bibliography 
sound because of the terms which must be omitted in practice, and we think the point deserves 
more consideration in the mathematical treatment of (5) than it receives. It will perhaps be 
advisable to give the details of the curve given by Charlier to which our objections refer, and 
show our failure to reproduce it. The equation to the curve of fig. 5 is 
F{x)=N[<t>,{x)--lct>,{x)l 
where (/>„ = o-" + 1 (.f), 
and the ordinates corresponding are given in the first row of the following statement in which, 
as the curves are symmetrical, the last few terms are omitted. 
- -0021 
- -0060 
- -0089 
-f- -0095 
+ •0810 
+ 
1999 
+ 
•2904 
+ •2971 
+ •2792 
+ •2971 
- -0012 
- -0016 
+ -0035 
+ -0269 
+ -0832 
•1695 
+ 
•2572 
+ •3155 
+ •3333 
+ •3155 
The moments were calculated about the mean from the figures given but the negative frequencies 
which Charlier does not give in his diagram and which are meaningless in practical work, 
were neglected. The values were as follows : 
Second moment = 4^7089 
Third 
Fourth 
= zero 
= 46^987 
0- = 2^1700 
and the equation is 
/'(.i') = iV'[</>,(*')--03671(/.4 (.*■)]■ 
The resulting ordinates are given and will bo seen to be very far from the original figures. 
While of course we know we can reproduce the curve in Charlier's figure by using the negative 
frequencies we cannot help thinking that there are strong practical objections to the use of the 
curve in the form in which he writes it so long as such results as that just given can be obtained. 
If integration had been effected only over the positive area of the curve instead of from - co 
to + 00 , the difficulty would not have arisen — but how is such integration to be effected '? 
The objecti(jns here I'aised to negative frequencies have been surmounted (as is, we think, 
theoretically necessary) in Edgeworth's work by leaving the equation in the form already given 
from which it can be seen that negatiA'e frequencies are impossible. There are however other 
difficulties that may arise and one of them can be seen in the example given by Edgeworth 
on pp. 522 and 523 of (2). This examp)le deals with statistics of fecundity and the total 
frequency in the series of observations is 1000 while the totals in the first, second and third 
approximations in Table III, p. 523, are 947, 977 and 960 respectively. These differences 
between the calculated and observed frequencies are due to the area of part of the curve being 
neglected in reading oft" the gi-aduation figures ; in other words the frequency curve (Third 
Approximation) gives 40 cases out of 1000 as having less than no members in a family and the 
effect of this is that the frequency is on the average understated for the remainder of the curve. 
The application of Charlier's Type A would have given the graduation shown in the following 
table and a comparison of this graduation and Edgeworth's brings out the difference between 
the two methods to which reference has already been made. 
For families of from 2 to 9 members, Edgeworth's graduation is close but both tails in his 
graduation and the start in Charlier's are quite unsatisfactory, while Charlier's curve gives 
a distorted graduation prior to 7 members, from which point however it agrees admirably. It 
seems probable however that Charlier would use his Type B for such a distribution and we have 
added a graduation by the third of his methods of fitting ; the agreement is poor in comparison 
with that shown by Pearson's Type I. An attempt with Charlier's first method of fitting led to 
