120 UR. \y. F. SHEPPARD ON THE APPLICATION- OF THE THEORY OF 
Let the mean ^’th power of this last expression be denoted by so that 
p ,2 = {ci'cc -|- 6'/3 -f- c'y . ..) — {ci^i -{- d" cy d" • • 
By expanding the expression at the end of § 15, and writing nd for 0, we see that 
fji/. = n~-^'\kx coefficient of 6’' in {1 + -h 
where C 3 , C 4 , . . . are functions oi a,h, c, . . ., /3, y,. . . Denote -fg 2 ^+C 3 ^®+C 4 ^+. . . 
by ©, and expand (1 + ©)" by the binomial theorem. Then the highest power of n 
contained in comes from the term involving 0 "^' when k is even, or from the term 
involving when k is odd. Hence, when n is made indefinitely great, 
p..,; = R " X ( 2 /^ 2 )" = 
M 2 .+ 1 =W 25 + 1 X ^ .5 (ipo)'"' C 3 
[1 
and therefore the distribution is ultimately normal. 
It follows that the distribution of values of a {a — a)h — /S) c {y' — y )... 
is also normal. 
It will be noticed that, when n is finite, the number of terms in pjs or P 2.-+1 increases 
with 5 , and becomes infinite when s is infinite. Thus the approximation of the 
actual distribution to the ultimate normal distribution is close as regards the low 
moments, n being supposed to be moderately gTeat, but is not close as regards very 
high moments. The difterence between the two distributions is therefore due 
mainly to the values of 7i {a (a' — a) b c (y' — y) + • • •} which are 
great in comparison with \/But these are values Avhich only occur very rarely ; 
and therefore, for practical purposes, Ave may regard the two distributions as 
identical. 
( 2 .) To obtain the same result from the geometrical definition of the curve, we 
must use § 14. 
(i.) To find the distribution of values of \/;? (a — a), we take a series of points 
Mo, Ml, . . . M,„ at equal distances ijy/n along a straight line X'X ; and then druAv 
ordinates MqPo, MiPj, . , . M^P,^ equal to the coefficients in the expansion of 
-v/ri {l^x + oLij)'\ where a + ^8 = 1. Thus 
M,,P^ = v/h.a^/3'‘-^C;, 
where Cp stands for - j- ^ . Then, if n is increased indefinitely, the locus of the 
points Pq, Pi, . . . P,t will be a curve, which will be the curve of frequency of A’alues of 
v/?i (a' — a). 
