Miscellanea 
427 
we have the following equations for the determination of B^, ... 
Ca + 2flo = 0 C'2 = 
3C3 + 4ai£i = 0 C'3 = 2£iJ52 
46'4 + 402(72 + 4aiB2 = 0 C4 = 2£i£3 + B^^ 
5C5 + 4a2t'3 + 4ajS3 ^- 0 C'5 = 251^4 + 2^253 
.(Ifi). 
The solution of these equations gives the following values of B^, JSj, 
2 
3' 
2ao „ 2 
±io = - ^ a. 
i-(l< + aoa.2) 2 V 2 
^^^^ / , ^*=-W5t^-5"-''^- 
5, V540 a; 30 12 """^ ^ ^« = " 8505 ^ 945 " "^"^^ 
^1 V340200 a.o^ 12600 «o 4200 8* ) ~ 25515 2835 ' + 2835 63 
(17). 
The area from the mode to the median is then 
f /2 4 88 2 16 . \ fliV 4 16 16 , 
i V3 5 " 105"- 21 " -J ^ VT35 " 315"^ ~ 945"- 
/' 8 16 \ fli' / 16 \ ] 
§ 4. Now this area can be found in another way in terms of d, the distance from the mode to 
the median, by assuming that (p (x) can be expanded in powers of x by Maclaurin's Theorem. 
Professor Pearson* has shown that a high order parabola does not adequately fit these curves 
throughout the range, but if we assume that the skewness is small and therefore d small, we may 
use Maclaurin's expansion legitimately within the range x = 0 to .r = rf. Since ^ is zero at the 
mode we have 
2/=2/o(l +A^^ + A,x^+ ...) (19), 
[cl 
and substituting in | ydx we have the required area equal to 
yjd+i'd^ + i'd*+...) (20). 
3 4 
The values of the coefficients A2, A^, ... may be obtained by using the differential equation (13) 
and the expansion (19); whence we obtain 
As = (s + 2) ^, + 2 a„ + (.s + 1) ^s^i «i + sA, a.^, 
and therefore A, = 1, ^3 = " = 4^ ^o^'"'' ^'^ " 6a> ^ - " 
Thus the area required is 
/, _ d^ai_ d^(l - 2a^) d^af _ d^^ ] 
^"t 6ff„ 12ao^"^ 40rto^ ^ 20a„^ 36ao-'"^"'/ '* 
This expression is in descending powers of and so is the one obtained in (18). Hence we may 
assume 
'^ = ''»+a' + ^-^^^+ (22), 
"0 "0 "(I 
* Biomeirika, Vol. 11. p. 19. 
28—2 
