Eleanor Pairiman and Karl Pearson 
251 
Maclaurin Theorem ftxils theoretically, (b) our auxiliary curve is unreasonable, for 
cannot be expanded at the origin terminal in powers of a; — are found to give 
results within ^ °/. of the true values for both mean and standard deviation. 
The variety of illustrations we have taken seems to suggt'st that for most 
practical statistical problems — even with J- or U-shaped distributions — we shall 
obtain reasonable results from the system developed in the first part of this paper. 
At the same time the method adopted indicates that for the best possible results in 
asymptotic frequency curves it may be needful to use a more suitable auxiliary curve 
for the asymptotic terminal. This loads us directly to the second part of our paper. 
Fm^t II. Cases of Asy inptotic Frequency. 
(11) In selecting our auxiliary curve to give the first five frequencies we must 
remember that it has (i) to give an infinite ordinate but a finite frequency, (ii) it 
must be of such a character that its constants can be readily determined. 
If we adopt 
Z=N{1+ x'i {A + Bx + Cx' + Dx^ + Ex')), 
where q is chosen less than unity, we have the adequate number of constants and 
y = — dZjdx is infinite when a; = 0. 
If we leave q undetermined, however, we should have six not five constants and 
might then omit E. But the process of determining A, B, G, D and q would be 
very laborious and involve a troublesome series of approximations. We are ac- 
cordingly thrown back on the retention of E and an arbitrary choice of q. CHearly 
to give an infinite ordinate and finite area we may give q any value from slightly 
over zero to slightly under unity, and the. size of q measures so to speak the intensity 
of the asymptoting. This is probably rather an important feature of the frequency 
curve, but as we see no way of determining it accurately without very great labour, 
we give q its mean value i. Accordingly our problem becomes that of determining 
A, B, C, D and E so as to give the first five frequencies or the values of iV/;/, Nn.!, 
Nvs, Nn^', Nn.' as before. After a good deal of work they are found to be 
' A=- 1-64964,8475571/ + 3-35035,15245»./ - 3-72071,62874h/ 
+ 2-05278,64045»; - •44721,35955/(.,', 
B = + ■91328,76419h/ - 5-50337,90247«,,' + 7-10669,19065«3' 
- 4-15163,83427«/ + •93169,49906h/, 
G ^- •31317,72759h/ + 2-6451 5,60574»; - 4-30806,06243»;/ 
(xxviii) A . , 
^ ' +2-7G448,01733n/- •65218,64934m/, 
i) = + •05299,17797/i/- •5 3 0 3 4,15 5 367;/ + l-00l72,3139Uu/ 
- -73032,76686/1/+ •18633,89981h/, 
E = - •00345,36703h/+ -03821 ,29964h/ - •07963,81338/;/ 
+ -06469,94335?//- -01863,38998?//. 
The large number of decimals is requisite owing to the high coefficients they have 
to be multiplied by in ascertaining the values of the abruptness coefficients. 
