Mr. Hopkins on the Mechanism of Glacial Motion. 153 



R is therefore a function of the two independent variables 5 



and <+i; and I shall now proceed to find the values of fl and 4), 



which render R a maximum or a minimum. Differentiating 



with respect to 4), we have 



= X sin ^ — Y cos i^, 



which shows that for any assigned value of 3, or position of the 



line of separation, the maximum value of R will be that of the 



resultant of X and Y, and the corresponding value of ($», that 



of the angle which the direction of that resultant makes with 



the axis of ^. Differentiating with respect to 9, we have 



^ rfX c/Y . 



= -^cos<^+-^sm^. 



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

 (Xj cos 6 +/^sin5) sin <$> — (Y^ sind +/yCos6)cos <$> = o? 

 (Xj sin fl —ff cos fl) cos 4> — (Yj cos 9 — /y sin fl) sin <f) = 0. 

 Eliminating 4>, we have 



(Xi cos 9 +/, sin 6) (Xi sin 6 —f,cos 5) 



— (Yi sin S +/) cos Q) (Y^ cos 3 —f, sin fl) = 0, 

 .-. {XJ, + YJi) (sin2 Q - cos^ fi) + {X\ - Y\) sin 9 cos fl = ; 



.• tan 2 5 = ^ — . . (I ) 



■ ^ ..ian-t;_^^_Y^ U-J 



Again, from the two preceding equations containing Q and <$, 

 we have 



(Xi 4-/y tan fl) tan <^ — (Y^ tan 9 +/) = 0, 

 (Xi tan 9 — /y) — (Yi — / tan 9) tan cfs = 0, 

 or 



Xj tan 4> — Yj tan 9 4-// tan 9 tan cf> — / = 0, 

 X, tan 9 — Yj tan <^ +/ tan 9 tan 45 — y; = 0. 

 9 and <$> enter exactly in the same manner in these two equa- 

 tions, and must therefore be equal. Hence 



tan 2^ = ^{^ (2.) 



Equation (1.) shows that there are two positions of the line 

 of separation through any proposed point, at right angles to 

 each other, for one of which the resultant action between the 

 particles on opposite sides of the line at the proposed point is 

 a maximum, and for the other a minimum ; and since ^ deter- 

 mines the direction of the resultant action, equation (2.) proves 

 that direction to coincide with the normal to the line of sepa- 

 ration, whenever that line is in a position for which the result- 

 ant action is a maximum or minimum. These conclusions 

 may also be arrived at by somewhat different though equiva- 

 lent reasoning, as follows. 



FUh Mag. S. 3. Vol. 26. No. 171. Feb. 1845. M 



