399 
1911-12.] Point Binomials and Mendelian Distributions. 
be M, the distribution of the new expression will be equivalent to 
M (a + b + c + . . .). We have therefore to obtain the moments of the 
compound expression round the new centre of gravity. 
Let h v h 2 , h B , etc., be the distance of a, b, c, etc., from the centre of gravity 
then by the preceding formulae (A) 
or 
likewise 
M (a + b + c + . . > 2 = M£ 2 2a + M2^ 2 , 
A4 = £ 2 "t £ 2 i 
M'(a + 6 + c+ . . .)^ B = M^a + Ml^ah + M^ah 3 = + C's > 
since '2ah = 0 by definition. 
And in general 
So also 
/*4 = £4 + f 4 + efaf'a • 
H = 3£'o 5 
/*3 = ^£ 3 5 
/ , 4 =^ 4 +^r 2 r2- 
Lemma II. — The even moments /x 2 and of (1 + g)™ (g-fl) n (l + l) 9 
(l+_p) fc (p + iy etc., are constant if m-\-n — const, and /<? + £ = const., and the 
odd moment /u B depends on the difference m — n, k — l, etc. This follows at 
once from the preceding since the second moment of g + 1 is the same 
as those of 1 + g, while the third moment of these is equal but of opposite 
sign. 
Lemma III. — If (l + g) m (g + l) n (1 + 1)p (l+_p) r (p + l) s etc., be 
a distribution, then the first moments round an horizontal axis are the 
same if m-\-n = c, n-\-s = c', etc. For, consider the distribution given by 
a + &-fc-f . . ., where a, b, and c are posited at equal distances m. 
The first moment is 
a 2 b 2 c 2 
— 9 9 H 9 4 - 
m l m l m z 
a 4- b + c + ... 
Multiplying the distribution by 1 + q and q + 1 respectively, and limiting 
to three terms which is sufficient, we have the two expressions 
a + (b + qa) + (c + qb) + qc } 
qa + (qb + a) + (qc + b) + c. 
The first moments are then 
ct 2 + (b + qa) 2 + (c + qb) 2 + q 2 c 2 ' 
m 2 (a + b + c) 
and 
q 2 a 2 + (qb + a) + (qc + b) 2 + c 2 
m 2 (a + b + c) 
which are obviously identical. 
