68 Mr. HOPKINS ON THE MOTION OF GLACIERS. 



If we suppose an upward expansion of the mass to take place in consequence of the freezing and 

 consequent expansion of infiltrated water, according to the theory of dilatation, this expansion will 

 also increase the thickness of the glacier above A. Let e denote this increase for a unit of thick- 

 ness, while P moves through the horizontal space PM ; then will eA be the whole increase, h being 

 the thickness AQ of the glacier. On the contrary, the thickness will be diminished by the melting 

 of the superficial ice during summer, occasioned by external influences, and of the ice in contact 

 with the bed of the glacier, as the effect of internal heat and subglacial currents. Let A and ^ 

 denote the depressions of the surface below the point C, due to these causes respectively, in the 

 time (/) of moving through PM. Then if D denote the whole depression of the surface in the 

 vertical through C in the same time, we shall have 



D=i^ + l-eh-a (tan a - tan /3). 



Of the quantities involved in this equation D, A, a and a may be easily observed. For this 

 purpose conceive two vertical poles fixed firmly in the ice at P and Q in the same ];ne of motion, 

 their upper extremities coinciding as nearly as possible with the mean level of the glacier at the 

 time. The inclination to tile horizon of the line joining them would give the value of a ; and the 

 height to which the poles should project above the surface of the glacier after the time {t) 

 would o'ive the value of A for that time. To determine the corresponding value of D, we 

 miffht observe the vertical distance of the surface of the glacier from the fixed point C when the 

 poles should be first fixed, and after the time t of moving through PM, repeat the observation. The 

 difference between the observed vertical distances below C would give the required value of D. 



The only attempts at the independent determination of 6 have been made, I believe, by observ- 

 ing the distances at different times of fixed points on the surface of the ice. Such determinations 

 I consider entirely valueless, on account of the impossibility of separating the effects of dilatation 

 from those of pressures and tensions depending on other and independent causes. If, however, 

 instead of horizontal we should make vertical admeasurements, the value of e for a given depth 

 of ice might, I conceive, be determined with great accuracy. If two short horizontal poles were 

 firmly fixed in a vertical line in the vertical wall of a crevasse, and an inextensible line or chain 

 were fixed to the lower one, any variation of the known distance between the two poles might 

 be ascertained with great accuracy by observations made at the upper one, and thence the 

 value of € might be accurately determined*. 



Supposing the quantities D, A, a, a and e to be determined, our equation will still contain 

 three unknown quantities, /3, ^, and h, which cannot be determined by direct observation. I think 

 it probable, however, that e might be found to be inappreciable, or, at least, extremely small, so that 

 the term eh might either be neglected or expressed approximately by means of an assumed value 

 of h. We might also eliminate ^ from the above equation by making one of the observations for the 

 determination of D as late as jwssible in the autumn, and the other as early as possible in the follow- 

 ing spring, since the corresponding value of ^ would doubtless be very small on account of the absence 

 of subglacial currents during the winter. The value of D in this case would probably indicate 

 an elevation of the surface. Let this value therefore be denoted by - D^. We should thus have 



D^ = eh + a (tan a - tan /3) - A, 



a 



nearly, the value of eh for the winter being small enough to be neglected. If tan /3 were thus deter- 

 mined, the value of S corresponding to any observed values of D and A, would be given by our 

 previous equation. 



Also if /3 were known we should have immediately the difference of thickness at P and Q : 

 for this difference = Qp = a (tan a — tan /3). 



• It appears singular that those who itlsist so much on glacial dilatation should never have subjected their views to this simple test. 



