210 
Shew Variation, a Rejoinder 
whole determiuantal theory of inheritance. A complete theory of asymmetrical 
frequency must describe in the manner of Laplace's probability integral either 
continuous or discontinuous variation. 
(iv) The apparent or practical continuity of many variation data may be due 
either to real continuity or to discontinuity effectually masked by: (a) the relative 
paucity of material and roughness of our measurements, which compels us to 
divide it into groups of the same order of number as the number of determinants, 
{h) the influence of age, nurture and environment superposed upon the pure 
hereditary results, or (c) the fact that many of the characters measured by us 
are built up of a larger or smaller, but not necessarily an infinite, number of 
simple organs or characters, which may possibly individually have discontinuous 
variation*. 
(v) The assertion of Ranke that a finite number of fundamental cause-groups 
must lead to discontinuity is disposed of by illustration. It is quite possible to 
invent a great variety of deterniinantal systems — the number of the determinants 
being finite — which lead to continuous variation. In our present ignorance of the 
sources of variation, especially of the mechanism of inheritance, it would be idle 
to lay weight on any special interpretation of this kind. It is important, however, 
to observe that continuity or discontinuity of variation are not, as Ranke asserts 
them to be, associated with the finite or infinite number of the cause-groups. 
(vi) The absurdity which Ranke finds in the values taken by some of the 
constants of my curves, exists only when a very narrow view is taken of the sources 
of organic variation. A binomial series with negative power or with negative 
]) or q \& capable, as is shown in this paper, of perfectly rational interpretation. 
But in the present state of our knowledge it would be idle to specify any particular 
interpretation as the correct one"f". 
(vii) The problem of variation can be looked at in the following manner 
without the least loss of generality. Modify Gauss by replacing his third axiom, 
the independence of contributory increments to the variate, by the postulate that 
the increments are correlated with previous increments |. Start with any binomial 
and we reach the generalised probability curve for an infinite number of cause- 
groups : 
\ dy _ —X 
ydx~ aff {xlaa) ' 
where / is an arbitrary function. This theory covers Galton, Edgeworth, Kapteyn 
and Fechner. Expanding f {xja„) in a .series of ascending powers of x/ao we have 
* Kanke has quite overlooked the work by Galton and myself on the discontinuity of the series 
of individuals even when the population obeys the Gaussian law. See Biometrika, Vol. i. pp. 289 — 299. 
t Ranke apparently considers that (p + q)"' with ^j, q and n positive is interpretable. A little 
philosophical consideration will show that it is merely "familiar," not really intelligible. There is no 
physiological meaning in p, q, n, and we cannot as yet associate them with any true organic 
mechanism. 
+ This postulate of course abrogates the first two axioms of the Gaussian theory as well. 
