808 
NATURE 
[JUNE 16, 1923 

ache. Intellectually unfit to grapple with these 
subtilties, I must return to the simple, if crude and 
ignorant, physiological conception that all characters 
are products of germinal predisposition and somatic 
nurture; and to the notion that, while there is 
always blending and sometimes alternate reproduc- 
tion, there is never alternate inheritance. 
G. ARCHDALL REID. 
20 Lennox Road, South, Southsea, Hants, 
May 17. 

Martini’s Equations for the Epidemiology of 
Immunising Diseases. 
THE differential equations constructed by Dr. 
Martini and quoted by Dr. Lotka in Narure of May 
12, p. 633, namely, 5 
du : 
ap -i)-qu, . : tet) 
at = au(1 -i)-mu, . : ae 2) 
aroused my interest by the statement that they 
cannot be integrated in finite terms. I have noticed 
that in one particular case the integrals of the equa- 
tions can be expressed in a moderately simple form, 
and that in this case Dr. Martini’s second position of 
equilibrium, namely, 
gop =e? 
aq a 
is unattainable within a finite time unless it is 
permanent. 
The particular case in question which has come to 
my notice is that in which g=m, that is to say, the 
fraction of the affected population which ceases to be 
so per unit time is equal to the fraction of the immune 
population which loses immunity or dies per unit time. 
In this case it is evident by subtraction that 
du di : 
tat ag 1" ~)» 
so that u-i=Age, where A is the constant of 
integration. On substituting for 7 in (1) it is evident 
that 
du _ gt A 
di =(a -g+aAge ”)u -au*, 
an equation reducible to the linear form (and so 
soluble by quadratures) by the substitution v=1/u. 
The solution, which it will be sufficient to quote, is 
4 2 ( x efi ad sr 
U ~ n=o(¢ -9)(a- 29) . . . (amg -q)’ 
+B exp{(q-a)t+aAde}, 
where B is the second constant of integration. 
It is easy to see that the second position of equi- 
librium is given in this case by 

and if wu and 7 have this value for a finite value of ¢, 
it is readily seen successively that A=o and B=o, 
so that # and 7 have this value for all time. On the 
other hand, whatever be the values of A and B, the 
value (a—q)/a is the limit to which both u and 7 tend 
as the time tends to infinity, provided that a exceeds 
gq; if a does not exceed g, they tend to zero, unless 
B=o. ; 
I imagine from Dr. Lotka’s silence concerning these 
results that they have not been previously obtained, 
NO. 2798, VOL. II1] 

and, for all I know, the case g=m may be of no 
practical importance.. But the analysis which I have 
given seems to me to throw some light on what the 
behaviour of the solution might be expected to be in 
the general case. G. N. Watson. 
The University, Birmingham, 
May 12. = 

The Structure of Basic Beryllium Acetate. 
THE remarkable compound Be,O(C,H;O,),, the 
crystal structure of which Sir William Bragg describes 
in his letter in Nature of April 21, p. 532, can be 
given a chemical formula in complete accordance 
with its properties. Tanatar and Kurowski have 
described (Journ. Russ. Phys.-Chem. Ges. 39, 936, 
1630; 40, 787; Chem. Centr. 1908, I. 102, 1523; 
II. 1409) a series of compounds (including the formate, 
acetate, propionate, and benzoate) of the general 
formula Be,OA,. These compounds have none of 
the characteristics of salts. They are volatile, they 
have low melting-points (some are liquid), they are 
soluble in organic solvents such as benzene, and they ~ 
do not conduct electricity. 
They resemble the non-ionised members of the — 
“chelate ’’ series of compounds, to which Prof. 
Morgan has directed attention. These are derived 
from substances containing such groupings as 
HO Cites C=O, which combine with an atom of | 
a metal by replacement of the hydroxyl hydrogen, - 
and at the same time also (as he has shown) through 
the carbonyl oxygen. The simplest example is the 
volatile beryllium acetylacetonate (formula I.). The 
carbonyl oxygens becoming trivalent must each lose 
an electron. These two electrons go to the beryllium, 
which already has two valency electrons, and thus 
give it the four required to constitute the four non- 
polar links in the resulting compound. Chelate com- 
pounds of this type, in which the group is attached 
through both oxygens to the same metallic atom, 
are not formed by the carboxyl group, obviously 
because this would lead to the formation of a 4-ring, 
which is unstable. But there is no reason why the 
carboxyl should not react in this way if the attach- 
ment is to two different metallic atoms, with the © 
formation of a ring of 5 or 6 atoms. 
This must happen with basic beryllium acetate. 
We have at the centre, as Sir William Bragg suggests, 
the oxygen atom attached to 4 beryllium atoms. 
The octet of the oxygen is made up of 4 electrons 
from the four beryllium atoms, and four from the — 
oxygen. But the oxygen atom originally had six 
valency electrons, and so it must lose two. The 
attachment of the acetate group to two beryllium 
atoms is shown in formula II. It forms a 6-ring :— 
Be—O CH,—C—O ae 
i = I 
\o% \c_cH, = ee Sy AGH 
WS cts - +f, St 
a ee CH,—C=0O O= ce, 
II. I 
But in forming the ring each acetate group must 
lose an electron from its carbonyl oxygen, so that 
the six give up six electrons, in addition to the two 
given up by the central oxygen. We therefore have 
eight electrons, two of which go to each beryllium 
atom, increasing its valency electrons from 2 to 4. 
Thus each beryllium atom can form four non-polar 
links, these being (1) to the central oxygen, and 
(2, 3, 4) through three acetate groups to each of the 
other three beryllium atoms. One of these six chelate 
groups thus corresponds to each edge of the tetra- 
_hedron. 
