532 
0)1 the Constants of Index- Distributions 
If they be uncorrelated we have 
^yl I V' y 
, = fiV(l+.,^-^ + ^-etc.l (i)*- 
if, = = |v 4- V - 1^ + 3^.^ V + '^^V/^'^' + etc] (ii)-, 
^ /|y _ ^ _^ 6^' _^ - ^.-) ^ I 
Formulae (i) to (iv) are extensions of formulae given by me in a paper on 
spurious correlation*. Formulae (ij^'^ to (iv)'''^ are due to Dr M. Greenwood, Jun., 
who obtained them in dealing with the problem of the distribution of the opsonic 
index. They show at once two noteworthy but not yet noted points, namely, 
(a) if the distribution of both x and y be symmetrical, i.e. /Xg and fi^ = 0, M3 will 
not be zero or the distribution of indices must be skew; (b) the mean of the ratio 
of two numbers picked out of the same series is certainly greater than unity if the 
series be symmetrical, and will probably be always greater than unity even if it be 
not, i.e. i>xly, which is unity for the same x and y series f. Dr Greenwood 
found, however, that these formulae did not give with sufficient accuracy the 
constants of the index distribution. This was probably due to two causes : 
(a) clearly we ought only to keep to the square order in and the cubic order in 
M3 if we retain only to the 4th order in M^; or if we keep to the higher terms 
in ilfa and il/3, we must go further with i/4; and (h) the values of Wa; or Vy are not 
so small, that the convergency is sufficient when we take these lowest terms of the 
expansions. It seemed accordingly desirable to find some other way of attacking 
the problem, and Dr Greenwood asked me for suggestions. The problem he had 
in view was the distribution of the opsonic index when the blood of the same 
individual taken in the same manner at the same time was treated as test and as 
normal. If a Avide range of values of the opsonic index could thus be obtained, it 
would cast some light on what deviations from unity must be looked upon as 
significant, when test and normal were different individuals. 
(3) The idea that suggested itself to me was a fairly simple one, namely to 
tabulate the 7/-frequencies to a variable z=l/y. The units of the ^^-frequency 
groups will not be equal, but they will all be sufficiently small for us to concentrate 
their frequencies at their mid-points. We can then calculate their moments easily. 
Let i/j, V2, V3, be the moments of x about the zero value of x, and i//, v^, v^, vl be 
the moments of z about the zero value of z. Then 
i = z -K X , 
* R. S. Proc. Vol. LX. p. 492. 
t Given two dice, it would be advantageous to bet that the ratio of the number of pips on the two at 
a cast will exceed unity. 
