1905 - 6 .] Studies in Immunity : Theory of an Epidemic. 511 
case in the large epidemic of smallpox in London during 1902. 
It is unfortunately impossible to consider the whole of this 
epidemic, as the centre of the outburst was in that part of London 
adjacent to West Ham. As the cases occurring in the latter are 
not included in the spot-map of the epidemic issued by the 
Metropolitan Asylums Board, the complete distribution is not 
known. If, however, London be divided into squares by a series 
of lines, of which one set is parallel to the boundary between 
London and West Ham, the distribution of the cases as these lie 
to the right and left of the centre of the epidemic may be studied. 
The summing has been taken along those parallels which are at 
right angles to the boundary between the two areas.* 
The space distribution of the epidemic estimated in this way 
is seen to be of type IV., and the constants are as given in the 
annexed table (diagram XXVI., Table B, No. 1). The correspond- 
ing distribution of the epidemic of relapsing fever in Glasgow in 
December 1871 (Table B, No. 4), given for comparison, shows the 
same form. It does not seem necessary to elaborate evidence on this 
point, as the general theory is quite obvious, and the instances 
given sufficiently accord with it. 
Two other examples will be referred to later (Table B, Nos. 3 
and 4), namely, the north and south and the east and west dis- 
tributions of the epidemic of smallpox in Liverpool in 1902. The 
latter of these is again a curve of type IV., which is nearly 
symmetrical, while the former is exceedingly asymmetrical, due 
apparently to the fact that the centre of the epidemic was adjacent 
to the docks, and in consequence the spread thereby so limited on 
one side as to prevent the development of the usual form. In this 
case also, however, the criterion 2/? 2 - 3/5j — 6 is positive. The 
* If the distribution were normal, it is easily seen that the sum taken in 
this way will also partake of the same distribution. 
Thus the equation to a normal distribution is of the form 
_»2 __ 
y = Tce “ 2 & 
of which the integral with respect to y is 
_*2 r y _y* 
y=-ke a2 e 62 dy 
— a 
the curve of normal frequency. 
The same holds with regard to the symmetrical form of type IV. 
