152 Mr. Hopkins on the Mechanism of Glacial Motion. 



which, since — = sin L and — = cos L prove the above 

 ps ps ' 



formulas*. 



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 p q, the intensity of which will not differ from 

 y' by any finite quantity; and similarl}', a force {/') will act 

 on the side p j- in the direction opposite to that on q s. The 

 moments of these forces with respect to the middle point of 

 the rectangular element, will be 



—frV^l-qh ^^^^fl'Pq-qs. 



The direction of the resultant of the normal forces on q s 

 will pass at a distance from the middle point of the element 

 small compared with qs\ that distance will therefore not ex- 

 ceed a quantity of the second order; and consequently the 

 moment of the force X.^.qs about the middle point of the ele- 

 ment will not exceed a quantity of the third order, and may 

 be neglected in comparison with the moments of the tangential 

 forcesy'andy, which are of the second order. Hence the 

 equilibrium of the element requires that we should have 



-^fl 'Pq-qs = ~f,.pq.qs, 



or ^ fl=fr 



With this condition we have 



X = Xj cos 9 + f^ sin 5, 

 Y = Yj sin 6 + /, cos 9. 



If a line be drawn through q parallel to 77.9, 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 be 

 considered to have for its resolved parts the forces X and Y, 

 from which they cannot differ by quantities exceeding infini- 

 tesimals of the first order. 



Let the length of^ 5, or of an equal and parallel line through 

 <7, = A; the resolved parts of the forces upon it will be aX 

 and A Y. Let X R be the force on A estimated in a direction 

 making an angle 1:^ with the axis of ,2?, then shall we have 



' A R = A X . cos <:f> + A Y . sin $>, 

 or R = X cos <$ + Y sin 4) ; 



* See Poisson's memoir, " Sur le Mouvement des Corps Elastiques," in 

 the Memoires dc Plnstitut, vol. iii. p. 383. 



