164 



Mr. HOPKINS, ON THE MOTION OF GLACIERS. 



particles immediately above pq in the annexed figure, Fig. 



on those immediately below it, estimated in the direction 

 qB; and let f.pq represent the tangential action on pq. 

 Again, let the line of separation coincide with B'B, paral- 

 lel to the axis of y, and perpendicular to A' A ; and let 

 X, . qs denote the normal force exerted by the particles 

 immediately on the right of qs on the contiguous particles 



immediately on tlie left of it, and /'. qs the tangential — 



action. Join p and s, and let a perpendicular to ps 

 make an angle with A'A or the axis of a. Then if 

 X.ps and Y.ps be the resolved parts of the forces 

 which the particles on one side of ps exert on those on 

 the opposite side, estimated in the direction qA and qB 

 respectively, we shall have 



X = X^co%9 + /sin 9, 



Y= Fjsin0+/cos0. 



To prove these formulas we have only to observe that the forces acting on the sides pq and 

 qs of the triangular element pqs must be in equilibrium with the forces -X and - Y acting 

 externally on the side ps, neglecting small quantities of the third order. Hence we have 



- X .ps + Xi.qs +f.pq = 0, 



- Y.ps + Yi.pq +f.qs = 0, 



which, since — = sin 6, and — = cos 0, prove the above formulae*. 



-— ps 



B' 



ps 

 We have also the relation 



*/' = /• 



To prove this equation, complete the rectangular element pqsr. A tangential force will 

 act on the element along the side rs in a direction opposite to that of the tangential force (/) 

 acting along pq, the intensity of which will not differ from / by any finite quantity ; and 

 similarly, a force (/') will act on the side pr in the direction opposite to that on qs. The 

 moments of these forces with respect to the middle point of the rectangular element, will be 



If.pq.qs, and \f'.pq.qs. 



The direction of the resultant of the normal forces on qs will pass at a distance from the 

 middle point of the element small compared witli qs; that distance will therefore not exceed 

 a quantity of the second order; and consequently the moment of the force X,.qs about the 

 middle point of the element will not exceed a quantity of the third order, and may be neglected 

 in comparison with the moments of the tangential forces / and /', which are of the second order. 

 Hence, the equilibrium of the element requires that we should have 



\f.pq.qs = lf.pq.qs, 



f'-f- 

 With this condition we have 



X — X^ cos + / sin 0, 

 r= r, sin +fcos0. 



If a line be drawn through q pai-allel and equal to ps, the distance between the two lines 

 will be a small quantity of the first order, and therefore the action on the line through q may 



" See Poisson's memoir '* Sur le Mouvement des Corps ^astiques,'* in the Mtmoires d€ I'ImtUut. Vol. iii. p. 3R3. 



