Mr. HOPKINS, ON THE TRANSPORT OP ERRATIC BLOCKS. 223 



Again, since U is now the volume which passes through the cylindrical surface whose radius 

 = Cm (p) and height = MP{H + h), in time ^t, we must have by the first law 



U=2irp'{H + h)vit 



= Z-n-r (H + h) vSt nearly. 



Equating these values of U we obtain 



h 



v=V. 



H + h' 



u, = F. 

 ' /f + A, 



These approximate expressions for u and v^ are of the same form as the accurate expressions 

 obtained in the preceding case, but h and h^ are not here independent of the distance through which 

 the wave has travelled ; they are functions of r. To determine them let us assume the vis viva of 

 each wave to remain constant during its motion. The element (Sm) of the mass in motion at the 

 time t, will be the portion of the fluid included between the two cylindrical surfaces whose radii are 

 p' and p + Sp' and height H+h (MP). Therefore 



Sm = 27rp' (H + h) 3p' 



= ZTrrffSp' nearly, 



if r be much greater than I, and H than h. Also 



v = r.~^ 



H+h 



= V —- nearly. 

 H 



Hence 



^v'Sm^2^^rr*'h'Sp\ 

 H J, 



Now let 



A = (jo' - r) 



be the equation to the curve LPN when CL = a, a particular value of ;■; then assuming the 

 form of the curve so to change that each ordinate shall be diminished in the same ratio, we 

 shall have generally when CL = r, 



and 



or putting p - r - p, and y^' = {<j>f, 



which will be independent of r, if 



r|\M-j> = c = a constant, 



