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 diiferential 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 '' On the Causes of Sickness and Mor- 

 tality in the Manufacturing Towns of the North- West of 

 England," by Dr. C. J. Shearman, 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 7no7'e, 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 6 per 



