Mr. Faraday on Ventilating the Coal-Mine Goaf, 169 



conclusions have, in fact, been drawn from the conditions of 

 equilibrium of an element of the mass, conditions, the analytical 

 expression of which is only equivalent to the differential equa- 

 tions of the problem. The quantities involved in the prece- 

 ding results are Xj, Yj andy^. The directions in which the 

 couples depending onyjact is determined by the observed fact 

 of the central being greater than the lateral motion ; the facts 

 of the forces Xj and Yj being pressures or tensions are in- 

 ferred, and without any risk, as I conceive, of essential error, 

 from the form and inclination of the glacial valley. And these 

 are the points on which our conclusions depend; they are in 

 a great degree independent of the actual values of the above 

 forces. It would seem impossible to draw any accurate con- 

 clusions depending on less ambiguous circumstances. It may, 

 however, be remarked that the whole investigation is tacitly 

 based on the assumption of the continuity of the mass being 

 preserved in the same sense as in the ordinary investigations of 

 the motion of fluids. In the case before us the assumptionhas 

 been, that, while a continuous mass having elasticity changes 

 its form by the application of external forces, the particles 

 which constitute an element of the mass, such 'ds p'q'y^s' (fig. 6), 

 in its original state of no constraint, continue to constitute a 

 separate element, j)/ git', Sp during the whole change of form 

 up to the state bordering on fracture. It is the common as- 

 sumption on which all mathematical investigations of this 

 nature are founded, and it is one which appears to me to pos- 

 sess the strongest a priori claims to our confidence, except 

 under particular conditions with respect to the constitution of 

 the mass. But I must reserve any further observations on 

 this point for my next communication, in which, after this de- 

 tailed exposition of my own views of the mechanics of the 

 problem, 1 shall have some comments to make on those of 

 Prof. Forbes. I am. Gentlemen, 



Your obedient Servant, 

 Cambridge, January 6, 1845. W. HoPKINS. 



Erratum in Mr. Hopkins's Fh-si Paper. 

 Page 15, line 2, for u sin « read a (sin » — sin /3) = u sin x. 



XVI. On the Ventilation of the Coal- Mine Goaf 

 By Michael Faraday, Esq., D.C.L,, F.R.S. 



To Richard Taylor, Esq. 



Dear Sir, 



'VT'OU have honoured the Report by Mr. Lyell and myself 



-^ with a place in the Philosophical Magazine, p. 16, and 



this induces me to send for insertion also, certain considera- 



Phil. Mag. S. 3. Vol. 26. No. 1 7 1 . Feb. 1845. N 



