( 657 ) 



it and freezes at tlie part B'C of the surface. The \elooity of tlie 



ice being v we tind for the quantity of water entering through E' F' 



{R -\- cl) I' cos <f d<f. 



This is also the difference between the quantities flowing across 

 FF' and EE' upwards. 



This quantity can also be determined by means of the hydro- 

 dynamical equations. Take for axis of ^ a circle with radius R -\- kd 

 and for a.xis of rj a radius of the circle. The forces acting on an 

 element KLOP are in equilibrium. Writing n: for the velocity 

 parallel to the axis of |, fi for the coefficient of viscosity, neglecting 

 the velocity Ur, and taking the intersection of the 5 circle, with 

 EE' for origin of coordinates we have: 



d' u: b sin (f' 



At the circle AB, i(i = 0, at A'B', ?/.- = v sin tf, therefore : 



b sin <p Vj^ V sin <p sin tp f b d^ 



in tp f b d^\ 



ƒ" 



and the quantity streaming across EE' is 



1 I bd' i-d) 



uc dr, rzr — — — -I [ sin w, 



' ' H (12ft ^ 2 i ^ 



the difference between the quantities of water flowing ai-ross FF' 

 and EE' will therefore be 



1 I 6 (f vd\ 



12 R^ 2 j * ^ 

 and we have, neglecting powers of '^;r: 



bd' 



V = • . (//") 



12 ft ft' 



In the experiments the wires become curved. I suppose the wire 



to be perfectly flexible and the sti-ess to have the same value .S in 



all its parts; the force per unit of length perpendicular to the wire 

 is given in each point by 



dui being the angle between two consecutive tangents to the curve. 



The curvature being not large we can nse the coefficient given by 



(/) to find the normal velocity arising from this force. This \'elocity is 



du} 

 C S—. 



