428 
Miscellanea 
and substituting for d in (21) we may then equate to (18). After considerable reduction, we have, 
to the approximation taken throughout the course of the work, 
/2 4 
405 63 
16 „ 
21 + 
656 
1575 ' 
5/64 1184 
''^-''i V5103 14175"^ 
, / 21688 
" "1 U444525 ^ 
.(23). 
§5. We have assumed that the skewness is small; this is so if be small, and as the 
coefficient of Qq-^ is approximately aj'/170 it will be seen that we have a fairly rapidly convergent 
series for the value of d expressed in terms of the coefficients of the differential equation. We 
now proceed to transform the series so as to obtain the value of djD in terms of the skewness. 
It is easily shown from the differential equation that 
«! = - £>(1 + 2a2) \ 
«(,= - P2(l + 3a2) + -D''(l+«2)) 
where is the second moment coefficient about the mean. 
■(24), 
Hence 
7)2 (1 + 2a^ 
1^2 
1 + 3o, 
Z>2 \ + a. 
.(25). 
/i., ' 1 + Sa, 
Now the differential equation is obtained on the assumption that 1 = —I— and that / (x) is 
^ y dx f (x) 
represented for all practical purposes by + a^x + a<>x^; this implies that O3, a^, ... of the 
complete expansion are neghgibly small and usually is small also. If we restrict 
— (the square of the skewness) to the cases where they are small we may expand by the binomial 
theorem all the quantities involved in (22), and after some reduction we obtain 
d /2 8 16 , 32 , \ /■ 8 64 64 
= ( 3 + 15 ~ ^ ^ " •■• I + ' " ^-^ "2 + 
21 ■ 
/ 184 
35 
9424 
V405 945"^"" 675"^'+ 
1328 
[shf + (26), 
V25515 127575 "V ' 3444525' 
where {sh.) = skewness. 
The values of the coefficients of the successive powers of the skewness are given in the 
following table: 
(,9^)" 
(sk.)^ 
(sk.)* 
(sk.)^ 
■66666667 
•01975309 
•00721144 
■00038554 
■53333333 
- ^06772486 
- ^07387027 
- -76190476 
■09481481 
0-2° 
•91428571 
§ 6. The types of curves for which the value of d/D is given by (26) are given below* with 
the values of a,: 
Type I. 
Type III. 
Type IV. 
Type V. 
Type VI. 
y = 2/ue V'- 1 + 
2/0 
- V tair 
1 + 
_,„ -ylx 
y = 2/0 X "e " , 
y = ?Ja (« - ")'"! X- 
1 
»Wj + nig ' 
0. 
_ 1^ 
2m ' 
_ 1 
P 
1 
»w, - m., ' 
* These are not in all cases referred to the mode as the origin of x. 
