222 Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS. 



point at time t + St, and mP=m'P' = h, and MiQ = Mm = H, the depth of the canal, supposed 

 uniform. Also let V be the velocity of propagation of the wave; then will L L' = nn = mm 

 = NN" = VSt ; and let v be the velocity of the current at P, and therefore also (by the first law) 

 at every point of the vertical transverse section through P. Also let 6 be the breadth of the 

 canal; the area of the transverse section through P will = (H + h) b. 



Now it is manifest that a volume equal to that whose vertical longitudinal section is LPrL' (or, 

 in the limit LPP'L') and breadth 6, must have passed through the transverse section MP in the 

 time St. 



Let this volume = U; then if np = y, 



SU =b . area qp 

 = 6 . pp'. Sy 

 = bVSt.Sy; 

 .: U = bVhSt, 

 integrating from y = to y = mP = h. 



But by the first law we must have ' 



U = vb(H + h)k; 

 and therefore equating these values of U, we have 



Also by the second law 



H + h 



. h 



\/g(H+h,) 



(!)• 





If «, be the velocity of the current in the transverse section through the crest of the wave, 



i.V^. 



H + h, 



4. Let us now suppose the wave to diverge from a center; then assuming the breadth 

 of the wave to remain constant, and therefore the velocity of propagation {V) to be the same for 

 every part of the wave, we shall have 



SU = Znp.pp'. Sy 



= 2'7r VStpSy, 



where p = Cm, C representing the point from which the wave is diverging. U cannot be found 

 generally without knowing the relation between p and y, i.e. without knowing the form of the wave ; 

 but if we suppose the space CL (r) through which the wave has diverged to be much greater than 

 the breadth (l) of the wave, we shall have approximately p = r, and therefore 



SU=2-^VSt.rSy, 



and integrating from y = to y = m P = h, 



U =2Tr VrhSt. 



