(3.) 



Mr. Hopkins on the Mechanism of Glacial Motion. 155 



and (r) the minimum tension or maximum pressure at that 

 point. To find their values we have generally, 



R = X cos 4) + Y sin (p, 

 and, substituting for X and Y, 



R = (Xi cos 9 +/sin 9) cos <i> -I- (Y, sin 9 +f, cos 9) sin <p. 

 Also, when R is a maximum or minimum, 



2 f 

 (|) = 9, and tan 2 9 = ^-^^v » 



whence we obtain, after reduction, 



(R) = 1 {X, + Y,+ Vrx;^YjM=^nl 



and (r) = -i {X, + Y,- ^/{X,-Y,r + ^Ji'} 



If Xj and Y^ be negative (/. e. pressures), the values of (R) 

 as well as of (r) may become negative, in which case (R) will 

 be the minimum pressure and {r) the maximum pressure to 

 which the mass is subjected at q. 



1 1 . If/denote the tangential tbrce acting along /> s, we have, 

 resolving X and Y along p s, 



f— X sin 9 — Y cos 9 

 = (X^ cos 9 +/ sin 9) sin 9 - (Y, sin 9 +yicos 9) cos 9 



= -l(X,-Y,)sin2 9-y;cos2 9. 



Therefore, whenyis a maximum or minimum, we have 



0= i-(X,-Yi)cos2 9 +y;sin29, 



At 



or cot 2 9 = - V -^V (*•) 



Aj— I, 



Comparing this equation with (1.), it appears that the directions 

 of the line of separation, for whichyis a maximum and mini- 

 mum respectively, differ from those for which R is a maximum 

 or minimum by 45°. To find (/'), the maximum value of^ 

 we have only to substitute from (4.) the values of sin 2 9 and 

 cos 2 9 in the above expression for/. We thus obtain 



1 



(/)=±i-^'(X-Y,f+4y;^ (5.) 



Tills expression shows that the maximum and minimum 

 values of y are the same in magnitude, and only differing in 

 sign. This ought to be the case, for I have shown that y is 

 the same for any two directions at right angles to each other, 

 and from (4.) we conclude that the maximum and minimum 



M 2 



