Mr. HOPKINS, ON THE MOTION OF GLACIERS. 165 



be considered to have for its resolved parts the forces X and F, from which they cannot difFer 

 by quantities exceeding infinitesimals of the first order. 



7- Let the length of jts, or of an equal and parallel line through 5, = A ; the resolved 

 parts of the forces upon it will be \X and \Y. Let \R be the force on X estimated in a 

 direction making an angle (p with the axis of oc , then shall we have 



\R = X^.cos ip + \Y. sin (p, 

 or R = X cos + F sin d) ; 



R is therefore a function of the two independent variables 9 and <p ; and I shall now proceed 

 to find the values of 9 and (p which render R a maximum or a minimum. Differentiating with 

 respect to (p, wc have 



= X iia <p — Y cos (p, 



which shews that for any assigned value of Q, or position of the line of separation, the max- 

 imum value of R will be that of the resultant of X and Y, and the corresponding value 

 of cp, that of the angle which the direction of that resultant makes with the axis of x. 

 Differentiating with respect to 6, we have 



dX dY 



= — cos<^ + _sm0. 



Substituting for X and Y in these two equations, we obtain 



{X^ cos Q +/sin Q) sin (p — (Fj sin Q +fcos0) cos^ = 0, 

 (Xi sin 6 -fcosO) cos(p - (YiCosO —/sin 6) sin cp = 0. 

 Eliminating (p, we have 



(X, cos d+f sin 6) (X, sin - / cos 6) - ( F, sin +/ cos 9) ( }' cosG -f sin 0) = 0, 

 .-. (XJ+ YJ) (sin-' e - cos^ 0) + {X\ - Y\) sin cos = ; 



••• tan2g= ^^[^ (1). 



Again, from the two preceding equations containing Q and (b, we have 



(X, + / tan 6) tan (^ - (F, tan + /) = 0, 



{X, tan e-f)-{Yi-f tan 9) tan <p = 0, 

 or 



X^ tan - Fi tan + / tan tan ^ - / = 0, 



^, tan e - F, tan (p + f i&nd tan (p - f = 0. 



and enter exactly in the same manner in these two equations, and must therefore be equal. 



Hence 



2/" 

 tan2^=^^p~f-j^ (2)- 



Equation (1) shews that there arc two positions of the line of separation through any proposed 

 point, at right angles to cacli otlier, for one of wliich the resultant action between tlie particles on 

 opposite sides of the line at the ])roposed point is a maximum, and for tlic other ;i minimum ; and 

 since (p determines the direction of tlic resultant action, equation (2) proves that direction to coincide 

 with the normal to the line of separation, whenever that line is in a position for which the 



y2 



