PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 81 
(ii.) Now let a be a standard area and h a standard length. Let us use 
Equations ( 2 ) of Art. 4 (ii.), taking y'y' as the axis of symmetry of the probability- 
curve, and yy at a distance h to the left, then— 
jx-Jic = he. 
= (o’" + h^) c. 
= {Sh(7~ “k h^) c. 
= (3cr^ -f- 6b'^cr~ -f- h‘^) c. 
— (15cr‘^6 + 106®cr'' -b h°) c. 
Now let c/a = 2 , 0-/6 = u, and hpi = y. 
Then 2 , w, and y are purely numerical quantities, and we have for the first five 
moments round yy — 
= yzali, 
M3 = y”z (1 + a/i^. 
Mg = yh (1 + 3 ?/) ah^, 
M4. = 7^2 (1 -f 6 id + a/d, j 
Mg = y^^ (1 -f 10# -k 15 #) a/d, J 
(^)- 
( 6 .) We are now in a position to write down the equations which give the general 
solution of our problem. Let the deviation-axis of the asymmetrical frequency-curve 
be taken as axis of x, and let the axis of y be a perpendicular on this axis through 
the centroid of the frequency-curve. Let this centroid and the first five moment- 
coefficients about the axis of y of the frequency-curve, i.e., 0 , y^, y^, be found 
either analytically or graphically by the methods suggested in Art. 4 (iv.). 
Then, if the position and magnitude of the component normal curves be given 
by the quantities 64 , c^, (Xi, and 63 , C 3 , 0 - 3 , or the corresponding numerics 
71 , Uu and 73, 23, M3, 
we have, since moments round the vertical axis are clearly additive— 
MDCCCXCIV.—A. M 
