Mr. R. D. Oldham on the Modulus of Cohesion of Ice. 243 



It will be necessary also to determine what is the greatest 

 pressure which could be transmitted by the prism he hi. This 

 is governed partly by the modulus of cohesion of the ice, and 

 also by the extra support given by the excess of thickness of 

 the ice over b compared with that over c. Giving this its 

 utmost power, 



Q = m + ff-y, (II.) 



where 



Q = the ultimate strength of, or the maximum pressure that 

 could be transmitted by, the prism bcki, 



m = the modulus of cohesion, 



x = the thickness over b, 



y = the thickness over c. 



But it is evident that if P is greater than Q, or, in other 

 words, if the pressure required to overcome the resistance is 

 greater than the utmost pressure that can be transmitted, no 

 motion as a whole can ensue, and consequently no abrasion of 

 the bed b c can take place ; so that for given values of 0, x, y, 

 and fju, D attains its maximum when 



P=Q, 



or 



m + x —y = D(I + /ub cot 0), 

 or 



m +x-y .... (HI) 

 U l+/xcot<9 V 1,; 



In the above equations the pressure has been supposed to 

 be limited, in the first case, only by the resistance to be over- 

 come, and in the second by the ultimate strength of the ice. 

 But if the slope of the bed of the lake-basin from a to b is less 

 than the angle of repose — that is (taking {3 as the angle of 

 slope), if tan/3 is less than fi } the ice cannot slide down of its 

 own weight, but must be pushed down ; and consequently the 

 actual pressure exerted at b could not equal Q, but would be 

 diminished by a quantity equal to that due to the resistance 

 to motion down ab. Here the same equations as before hold 

 good, substituting — /3 for 0, 



S = 2# sin — /3+^ycos — /3; 

 or 



S = fiw cos /3 — w sin /3 ; 

 or 



S = D0*cot/3-l) (IV.) 



But the total pressure which could be transmitted at a only 



