1 54? Mr. Hopkins on the Mechanism of Glacial Motion. 



8. First, to find the value of 5 which gives R a maximum 

 or minimum, we have 



R2 = X2 + Y^, 

 and therefore 



which by substitution and reduction gives 

 (Xi/+ Yi/) (sin'-^fl - cos^d) + (X2, - Y^) sin S cos 5 = 0, 



or tan 2 fl = „ ^^„ . 



And, secondly, taking <$> as the angle which the resultant of X 

 and Y makes with the axis o^ x, we have 



Y Y.sinfi +/;cosfl 



tan <t) = -^7 = —^ . . 1— — i ; 



^ X XjCosS +y^smfl' 



and if we put 4> = 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 



sin 9 {Xi cos 9+/^ sin 6} = cos 9 {YisinS +yjcos9}, 

 or (X, ~ Yi) sin 9 cos 9 =/ (cos^ 9 - sin^ 9) ; 



.-. tan 2 9 = ^ -^'^ . 



Aj— Yi 



This equation shows that that position of the line of separation 

 for which 4> = 9, is that which corresponds to the maximum or 

 minimum action between the contiguous particles on opposite 

 sides of the line, as before proved. 



9. If the forces acting parallel to the axes of x and y be 

 pressures, Xj and Yj will be negative. 



If X^— Yi= 0, 9 = 45°. This accords with the result pre- 

 viously obtained in article 5, where Xj and Yj were both 

 s=0*. 



10. If (R) and (r) be the maximum and minimum values 

 of R at g, they will act respectively in directions perpendicular 

 nnd parallel to p s, the position of that line being determined 

 by equation (1.). (R) will therefore be the maximum tension, 



* It should be observed that in the investigation of the article above re- 

 ferred to, the condition of the transverse force Yi being ^ after the point 

 N {fig. 1.) has moved to n, is only secured by supposing /3 so small a quan- 

 tity of the first order that all small quantities of higher orders may be neg- 

 lected. In that case the compression along M L becomes a small quantity 

 of the second order, and therefore such as may be neglected, if /3 be very 

 small, and not otherwise. If /3 were more considerable, we should obtain 

 by the same mode of investigation a different result. In this case, however, 

 the second mode of investigation is much the best. 



