166 



Mr. HOPKINS, ON THE iMOTION OF GLACIERS. 



resultant action is a maximum or minimum. These conclusions may also be arrived at by some- 

 what different though equivalent reasoning, as follows. 



8. First to find the value of 6 which gives R a maximum or minimum, we have 

 and therefore 



which by substitution and reduction gives 



(XJ + Y,f) (sin= e - cos' 6) + (Xl - Y\) sin cos = 0, 



or 



2/" 

 tan 20 = ■' . 



And, secondly, taking (p as the angle which the resultant of X and Y makes with the axis 

 of X, we have 



Y _ r, sin0+/cos0 



*'*"'^"^"^,cos0+/sin0' 



and if we put (p = 9, we shall determine that position of the line of separation for which the direction 

 of the resultant action at any proposed point of it coincides with the normal. We thus obtain 



s)nfl{^, COS0 +/sin0} = cos {Y, sin + f cos 6] , 

 or 



{X, - F,) sin e cos 6 =/ (cos' 9 - sin' 6) ; 



.-. tan20 = ^r^^. 



This equation shews that that position of the line of separation for which (j> = 6, is that which 

 corresponds to the maximum or minimum action between the contiguous particles on opposite sides 

 of the line, as before proved. 



9. The maximum action here spoken of is the maximum tension at the proposed point, and 

 since it is perpendicular to the corresponding line of separation, there will manifestly be the 

 greatest tendency to form a fissure along that line, and a fissure will be formed along it if the 

 maximum tension be greater than the cohesive power at the proposed point. 



10. To apply the investigation to the case of a glacier, let PQ (fig. 4) be a portion of the 

 mass contained between two parallel vertical planes perpendicular 

 to the axis of the glacier and indefinitely near to each other. 

 By the more rapid motion of the central part, the element PQ 

 will be brought into the position P'Q' ; and if pqrs be an infi- 

 nitesimal rectangular portion of PQ, it will be brought into the 

 position p'q'r's'. Let the longitudinal axis of the glacier be that 

 of .r. The tangential force/ will arise from the greater velocity 

 of the central portion of the mass. It will be of the same in- 

 tensity, as above proved, for each side of the element, and will 

 manifestly act on the sides respectively in the directions q's' 

 and r'p', q'p' and rV. It is this force which distorts the element 

 from its rectangular form. The longitudinal force JST, will 

 generally be a tension arising from the greater velocity near the lower extremity. The transverse 

 force F, in actual glaciers, in which the sides have so generally some degree of convergeucy, niay be 



