April 29, 1904.] 



SCIENCE. 



705 



passes over the weight. The aim of the ob- 

 servers on board is to' so manoeuvre the boat 

 that the wire shall become apparently ver- 

 tical as many times as possible throughout the 

 period of observation, and at each such time 

 to note the depth of the water and positions 

 of the floats. For small depths, verticality 

 can be estimated sufficiently well by aid of a 

 plumb line held alongside of the sounding 

 wire. Por greater depths, more accurate 

 means must be provided, such, for instance, 

 as a small telescope supplied with vertical 

 sight wires and hung in gimbals, together 

 with a mirror placed alongside of the sound- 

 ing wire and likewise hung in gimbals. If a 

 boat were to be used exclusively for such work, 

 it should have, at the point of least motion, a 

 well through which sounding operations could 

 be carried on. Of course, means must be pro- 

 vided for securing nearly uniform tension at 

 the time of taking a reading. In small depths 

 the approximation to uniformity need not be 

 very close. 



Problem 1. — Ignoring the impulse of the 

 current upon the wire, also the sagging due 

 to the wire's weight, required the amount of 

 error in height or depth and in position due 

 to the want of verticality of the wire. 



Obviously, 



Height error ^l — I cos ij)s=l versed sine (Jj, 

 Position error = Z sin (pg, 



where I denotes the length of wire extending 

 from the bottom to the surface, and <p , the 

 small angle which it makes with the vertical 

 at the time of reading. Giving to <p^ several 

 values, we have 



'Ps= Vi" %° 1° 



Height error =0.000010 0.000038 0.000152 



Position error =0.00436.3 0.008726 0.017452 



<!,,= 2° 5° 



Height error =0.000609 0.003805 



Position error =0.034900 0.087156 



In depths not exceeding 100 fathoms, an 

 error of 1 degree in verticality can not cause 

 an error of more than 0.1 foot in height or 

 depth. 



ProUem 2. — Ignoring the impulse of the 

 current upon the wire, and supposing the 

 (vertical component of the) tension at its 



upper end to be v times as great as its weight 

 in water, required the error in height or 

 depth and in position when the wire is not 

 exactly vertical. 



Let ^^ denote the want of verticality at the 

 surface and <p, that at the bottom; then 



where 



True depth = a (coseo (l>s — cosee ^j) 



v — 1 



cot <pi = cot (ts 



and 



a = vl tan fs- 

 Height error = I — true depth, 

 Position error = a [log (cosec 0s + cot (jis ) 



— log {oosec<pb-\- oot<i 



0). 



If the wire is nearly vertical, we have the 

 following approximate equations : 



True depth = a cot <ps — ^^~Y) 9s) , 



Position error = a log ~ + — 1 — ( | <pl''. 



^v—1 4L V>'— 1/J2 



By aid of these equations doubtful interpo- 

 lations can be avoided. 



Assuming >' = 5 and an apparent depth (I) 

 of 3,000 units, we have as errors corresponding 

 to a few values of <p^: 



<!>,— r i" 1° 



Height error = 0.035 0.143 0.571 

 Position error = 14.60 29.21 58.41 



This shows that in a depth as great as 3,000 

 fathoms the upper end of the sounding wire 

 should not deviate more than one half degree 

 from the vertical if a tide of ordinary ampli- 

 tude is to be determined. 



Problem S. — Ignoring the sag due to the 

 wire's weight and assuming that the hori- 

 zontal 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 wire forms the arc of a parabola with 

 vertex at the surface and whose equation is 



This gives 



Position error =; 



;a(depth)g _ III 

 271- ~^ 2vw 



where p. denotes the impulse per unit length, 



