a 
Mav 12, 1923] 





rf 
_ positions of the electrons in certain atoms The 
conclusion must therefore be drawn that the principle 
_ has not been established. On the other hand, I am 
not disposed to attach too much weight to the evi- 
_ dence against the principle furnished by tartaric acid 
for the following reason. The object of Astbury’s 
investigation was to explore the connexion between 
optical activity and enantiomorphism, and it was 
therefore necessary to choose a substance of relatively 
complicated composition. The crystals of tartaric 
acid are much too involved for any effective test of 
the symmetry principle. It is, for example, by no 
means certain that a slight deformation of the sym- 
metrical crystal molecule into an asymmetric form 
could be detected by the X-ray method, and yet this 
deformation would be enough to substantiate the 
‘principle so far as tartaric acid is concerned. 
_ It seems to me, therefore, that the whole question 
is still open, and that the suitable choice of material 
for an eventual test is worthy of a careful considera- 
tion. Such aromatic compounds as those under 
investigation for other purposes by Sir W. H. Bragg 
would seem to be unsuitable, for they are so compli- 
_ cated that the positions of the individual atoms can- 
not at present be deduced from the measurements ; 
consequently, the shape and symmetry of the mole- 
cule have to be assumed. The compounds should 
rather belong to the simplest order of molecular 
Structure ; a molecule containing one atom of carbon 
is much better than one containing two. Hydrogen 
should be avoided, as it cannot be placed by the 
X-ray method. There should be as few kinds of 
atom as possible, for the quantitative connexion 
_ between atomic weight or number and reflection 
intensity is, perhaps, not too well known. The 
symmetry of the crystal should be beyond reproach, 
_ and it should be part of the investigation to assure 
it, since as a rule the crystallographer does not take 
the necessary trouble to determine the class of 
symmetry. Perhaps a suitable commencement might 
_ be made on carbon tetrabromide, CBr,, the corre- 
sponding iodide and possibly hexachloro- and hexa- 
bromo-ethane (if these are not already too compli- 
cated). Such compounds have the advantage that 
the carbon content is almost negligible (being much 
less than the average percentage of hydrogen in 
organic compounds), and the X-ray effect of the 
carbon atom might therefore be neglected as a first 
approximation, the investigation being, as it were, 
simplified to that of a solidified bromine (or iodine) 
in which the halogen atoms are limited stereochemi- 
cally by the insignificant carbon atom. Moreover, 
the dimorphism of the tetrabromide (monoclinic at 
_ ordinary temperatures and cubic above 49°) might 
afford information on the extent to which the mole- 
_ cular configuration changes with change of crystal 
structure. 
Closely related with the above compounds are 
others of the same simple chemical type. Tin tetra- 
iodide, SnI,, for example, might give interesting 
results, since from the X-ray point of view. the in- 
vestigation might be regarded as that of an element 
(by virtue of the approximate equality in atomic 
number of tin and iodine), while from the chemical 
oint of view one can be quite certain it is a compound 
(though whether the grouping of iodine atoms is 
tetrahedral is not so well grounded as in the case of 
a carbon compound). Work on such simple com- 
ides as these might possibly establish the Fedorov- 
hearer principle, and so be of assistance in the study 
of more highly developed carbon compounds. 
T. V. BARKER. 
” 
University Museum, Oxford, 
April 10. 
NO. 2793, VOL. III] 
Ce —————— 
NATURE 
633 

Martini’s Equations for the Epidemiology of 
Immunising Diseases. 
E. Martin, in his ‘‘ Berechnungen und Beobacht- 
ungen zur Epidemiologie und Bekampfung der 
Malaria’’ (Gente, Hamburg, 1921), sets up asystem of 
differential equations to represent the presumptive 
course of events in the development of an endemic in 
which recovery is accompanied by acquired immunity. 
He adopts the notation : 
u =fraction of population affected with the disease, 
and infective. 
i=fraction of population not available for new in- 
fection (immune or already affected). 
(1-1) =fraction of population available for new in- 
fection: 
p=fraction of population newly aftected, per unit 
of time. 
q =fraction of affected population that ceases to be 
so, per unit of time, by recovery or by death. 
_ m=fraction of immune population which loses 
immunity or dies, per unit of time. 
a =infectivity (a proportionality factor). 
Martini puts the new infections, p per unit of time, 
per head of population, proportional both to the 
infective fraction wu of the population, and also to the 
fraction (1-7) of the population available for new 
infection, so that =au(1—7), and accordingly writes 
his equations : 
d . 
op = au(l—i)—qu=(a—g)u-aui, . . (x) 
di - 2 : : 
3 =au(l—1)—mi=au-—mi-aui. . . (2) 
Martini remarks that these equations cannot be 
integrated in finite terms. They are of a type dis- 
cussed by the writer elsewhere (American Journal of 
Hygiene, January Supplement, 1923). Their solution 
in series is 
u=Pye(a—9)'-+ Poe—mt + Py ,e%(a—9)t-+ Pyge—2mt+ . . . (3) 
4 =Qye—9+ Qae—m + Oy e29—-Mt+ Qone—2mt+ (4) 
From this it is seen that : 
(1) The equilibrium at the origin (% =i =o) is stable 
if, and only if, a<g. When this condition is satisfied, 
the disease will die out. 
(2) The solution near the origin cannot take on 
oscillatory form, since («—g) and m are necessarily 
real quantities. 
There is, however, another equilibrium (as pointed 
out by Dr. Martini), namely, at 
FU SAY, a koe oe fa) 

PPUBAY 3) fo ~ shh 4 (6) 
This has a real meaning if and only if a>g, that is to 
say, just in that case in which the equilibrium at the 
origin is unstable; at the same time, in the neigh- 
bourhood of «=U, i=I, we have again a solution— 

(u—U) =P’ yet + P’ge\t+ P’e7At+ 2. (7) 
(i- I) =Q’yeM#+ O%,er# +O’ et + 2. (8) 
where eHAgd 
am a®*m? 
n= - afte / Z ~4(a-q)m}. Bra) 
We need here give no further consideration to the 
case a<q, since the second equilibrium has no real 
existence in this case, and the first equilibrum was 
found to be stable, the disease dying out. 
