706 



SCIENCE. 



[N. S. Vol. XIX. No. 487. 



w the weight in water of a unit length of the 

 wire, and, T the tension at the upper end, 

 = vlw; it is assumed that the wire is nearly 

 vertical throughout its length. From recti- 

 fying the parabola, we obtain 



Height error = J ^, P = i -f-i L 



Assume /^= 0.001 lb., y = 5, iy = 0.003 lb.; 

 the position error will be Z/30 and the height 

 error Z/1350. As will be noted below, p. = 

 0.001 lb. and w = 0.003 lb., imply, for a steel 

 wire, a velocity of about 0.6 foot per second. 

 For smaller velocities p./w will be much less. 

 In deep water, with v = 5, we should not ex- 

 pect to generally find velocities such that the 

 error in height due to the tidal current could 

 exceed one part in 100,000, unless the law of 

 resistance given below does not hold good for 

 feeble currents. See velocities given near the 

 end of this paper. 



Problem Jf. — Taking into account the weight 

 of the wire and assuming that the horizontal 

 impulse of the current is the same for each 

 vertical unit, required the error in height or 

 depth and in position when the upper end of 

 the wire is exactly vertical. 



The forces acting upon a length Z extend- 

 ing downward from the surface, give 



dx fj. 



dy~ Tt — ioV 

 From this we obtain, to the third power of 

 the small quantities ij-V/T^, wy/T^, 



" r, 2 Ts^ 3 K' 4 ' 

 as the equation of the curve, the origin being 

 the point where the wire crosses the surface. 

 This rectified gives 



.-. Position error = Y,\}^^T,^^^?'~^ WX 



ments showed that the force is well repre- 

 sented by the expression 



t^' 



I '"i" 



Height error = i^,Z»-Kijf^i*. 



I have made numerous experiments for 

 determining the force of the impulse of water 

 upon slender cylindrical rods. The rods used 

 were of steel and varied in diameter from 

 0.036 to 0.5 of an inch. The velocity ranged 

 from 1 to IJ feet per second. The experi- 



25 



yld. 



where v denotes the velocity of the water; 

 y, the weight of a cubic unit of water at the 

 temperature of the stream; Z, the length of 

 the rod; d, the diameter; and C an empirical 

 abstract number found to be 0.95, approxi- 

 mately, or about one half of the value (1.86) 

 obtained from the experiments of du Buat 

 and Thibault for the case of a plane perpen- 

 dicular to the flow of the stream. In all cases 

 where length is involved in the above expres- 

 sion, the same unit of length must be em- 

 ployed. 



For sea water y may be taken as 64 lbs., 

 and the above expression gives as the force 

 for each foot of rod or wire 



0.995 f i;=(i =: 0.945 v'^d 

 pounds, =//.. 



In a steel wire suppose d^ 0.003 foot; the 

 force of impulse per foot i^=iJ-) is 



0.945 v-d = 0.002835 v"; 



while the weight per foot for wire immersed 

 in sea water is 0.003 lb. (^w). Hence, for 

 a wire of this diameter the force of the im- 

 pulse of the water will equal the weight of 

 the wire in water (i. e., fx will be equal to w) 

 when the velocity is a trifle more than 1 foot 

 per second. 



For a simple progressive long wave of am- 

 plitude A in a body of water whose depth is 

 h, the maximum velocity of the water par- 

 ticles is 



Let A=l foot; then the "velocities, expressed 

 in feet per second, corresponding to various 

 depths, are as f olows : 



5 10 25 50 100 fathoms, 

 30 60 150 300 600 feet. 



Depth = j 



Velocity = 1.035 0.732 0.463 0.327 0.232 feet, 



tl _ J ^00 1000 2000 3000 4000 fal 

 P {3000 6000 12000 18000 24000 fei 



Velocity = 0.104 0.073 0.052 0.042 0.037 feet. 



