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Miscellanea 
numbering the deformed. The criticism was justified in that the three famihes had been 
repeatedly made use of to support the view criticised. Another family of " brachydactyly " 
is now instanced by Puu nett (Proc. Roy. Soc. of Med. Feb. 28, 1908), who states that "no member 
of a brachydactylous family who is free from the defect can transmit it to his or her offspring." 
For certain families this conclusion might be justified from a consideration of the facts, and 
apart from the question of Mendelism. But that it is of general applicability is disproved by 
the family recorded by Hasselwander {Zeitschr. f. Morph. v. Anthrop. Bd. 6, 1903, S. 510 — 526), 
in which such transmission actually occurs. Drinkwater in the article referred to by Punnett 
{Proc. Roy. Soc. Edin. Vol. 28, 1908, p. 38), states, that of 75 descendants of abnormal parents 
in the family he records, 39 were abnormal, " a result corresponding with what we should expect 
from Mendel's law." The statement is misleading, for it implies that 36 of the offspring were 
normal. As a matter of fact four of these were uncertain, and there is as much reason to 
consider the ratio 43 to 32, as 39 to 36. The mean, 41 to 34, is the fairest expression of the 
ratio, and it shows the hypei'dominance of the deformed, common to so many malformations. 
In regard to the possibility of stability of a deformity, the Nettleship family is of interest 
{Ophthcdm. Soc. Trans. Vol. 27, p. 269), in that "night-blindness" has been traced through nine 
generations and shows no marked tendency to abate. A single instance of this character, 
though suggestive, does not constitute proof ; and in any case the conclusion could be applied 
solely to "night-blindness." 
The point which requires emphasis is that a statement to the effect that a family of brachy- 
dactyly is an example of Mendelian heredity (Punnett, loc. cit.), or the inclusion of that deformity 
in lists which purport to show examples of instances in which Mendelism has been witnessed 
(Bateson, Progress. Rei Botan. 1906), is not justified by present evidence*. It may be, and very 
probably is, the case, that Mendelism applies to certain hereditary human deformities; but the 
conclusions which are being drawn, or implied, conclusions having a serious sociological aspect, are 
at present ahead of the facts at our disposal. 
THOMAS LEWIS. 
IX. On a Formula for Determining V {x + 1). 
EDITORIAL. 
The statistician so often requires the value of the r-function, or more usually the value of 
T{x+\)j{x^e-^), that a suggestion as to a ready means of determining it may not be out of 
place. Every biometrician may be supposed to have at hand Barlow's tables and an arith- 
mometer of some form. The usual series, Stirling's or the Bernoulli number formula, involve 
inverse powers of x, and thus do not lend themselves to rapid calculation in cases where 
X may involve three or four figui-es. Forsyth's formula, i.e. 
r(,.-n)=V2^{^^^^±^}"' (i), 
is very accurate. It may be read for oiu* purposes as 
log^^^|^^=0-181,9427 + i(.r4--5)log {{x + -bf-^}-x\ogx (ii). 
* Those interested should refer to Joachimsthal (Virch. Archiv, 1898, Bd. 6, S. 429). It is obvious 
from his paper that so-called hypophalangia or brachydactylia is a by no means simple malformation, and 
is possibly not an entity at all. The number of digits affected and the grade of affection is open to wide 
variation. There is an instance in the paper (Fall 1), of a symmetrical affection of two fingers, trans- 
mitted through a normal individual. Ammon (loc. cit. S. 96), quotes the Kellie family, in which hypo- 
phalangia is said to have passed through ten generations, and shown sex limitation in transmission. 
