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



Let AB represent the surface on which the block rests, CD the general surface of the water and 

 LPPiN the wave, M the block at the time t, and P the point in which a vertical through M meets 

 the surface of the wave. 



MP and M^P^ are the sections in which the velocities are v and v^ respectively. 



Let AM = w, Al = CL = x' ; then will 



— = vel. of the block, 

 dt 



and — = v^l- of t'ls wave = V. 



dt 



Also let «2 be the velocity of a current just sufficient to move the block. Then, when the 

 velocity of the current at the point where the block is situated becomes = v.^, the block will begin to 

 move ; and as the velocity of the current increases, that of the block will always very nearly 

 = difference between the velocity of the current and that just necessary to move the block ; so that 

 we may consider the instantaneous velocity of the block as approximately =v - v.^. We shall 



d.v 

 then have "77 = ■" — "21 



or, substituting for v its value given by equation (l), (Art. 3.) 



dx h 



— = V -V.J (I). 



dt H + h 



Also %=V ■ (2); 



dx h Vi 

 •■• d^' " H + h ~ V ^ ' 



h will be a function of w - x depending on the form of the wave. This form is not known, but as 

 an approximation we may assume LP, to be a straight line ; we shall then have 



h mP L m x — x' 

 hi ttiiPi Lrrii I 



I being the length of the wave to which Z,»» is very nearly equal. Therefore 



— h = x-x (4), 



dx I dh 



and :r^ = ir*3~^ + '• 



dx A, dm 



Hence, by substitution in (3) and reduction, we obtain 



A, d^' _ _ F V V H 



I ' dh v^ V + v^' v„ ' V2 , rr ' 



— ./I + U 



F + JJa 



and integrating, 



''-l.x'=C--h + ~H.\og,. i^^.h+H). 



Let a = the original distance of the block, and I = the length of the wave; then when x'= a - I, 

 we shall have h = Aj. Therefore 



