184 
the theory of quadratics, a new and simple method of 
deducing the arbitrary constants that occur in the integration 
of the resolvents, together with the actual calculation of the 
foregoing differential equations and some interesting formulae 
of verification, by which the correctness of those equations 
may be ascertained. Such formulae play a somewhat con- 
spicuous part in the general scheme, for, if the differential 
resolvents be employed as tests of existing theories of differ- 
ential equations, it is of the utmost importance that their 
accuracy should be placed beyond dispute. Notwithstanding 
the complexity of the calculations by which I have arrived 
at them, the differential resolvents for quartics and quintics 
are, it will be noticed, of a remarkably simple form. It is 
proper to mention that a large portion of the calculation of 
that for the quintic was performed independently by Mr. 
Cockle, and that on comparing results and making one or 
two corrections, I found that his calculations coincided (so 
far as they went, for they only extended to the determination 
of the differential coefficients) with my own. 
A Paper was read (e On the Causes of Sickness and Mor- 
tality in the Manufacturing Towns of the North-West of 
England,” by Dr. C. J. Sheaiiman, of Sheffield, communi- 
cated by Dr. R. Angus Smith, F.R.S. 
Previously to any examination of the disturbing causes of 
vitality, the proportion of age to the whole population should 
be ascertained for town and country districts ; for under all 
circumstances, favourable or otherwise, age of an individual 
determines to a great extent the kind of sickness and the 
consequent mortality. In the towns of this part of England 
there are up to 15 years of age less, from 15 to 55 more , from 
55 upwards less than in the adjacent country. Hence one 
of two evils results — either that towns kill young people 
and old fast — or middle age is imported. 
As to the feeding of town population, an average of (J per 
