of Deposit of a Submarine Cable, \7 



Thus the tension maybe diminished with smaller loss of cable 

 when e is large (that is, when the ship is running quickly) than 

 when e is small (or when the ship is running slowly). 



31. The results of the preceding investigations will be ap- 

 plicablej I conceive^ with great accuracy in tlie circumstances 

 supposed ; namely, uniform depth of water, and uniform motion 

 of the ship ; requiring only for their complete numerical inter- 

 pretation an accurate evaluation of e (which, it is believed, is 



T) 



pretty well known), and an acciirate measure of -j (on which, 



apparently, little is known). It is almost unnecessary to re- 

 mark that the investigation takes no account of the unevennesa 

 of the bottom or the disturbance of waves. 



32. Before quitting the subject, I will revert to the second 

 investigation, founded upon assumptions which make the equa- 

 tions linear and resolvable in the general case. On referring to 

 the results in Article 11 and following articles, it will be seen 

 that they do not give facilities for ascertaining the form and 

 tension of the curve when e is very small, that is, when the 

 resistance of the water to the cable's motion is very small. I 

 will here indicate the form and the results of a process applicable 

 to this case. Assume y = Y + SY, s = S + SS, where Y and S are 



the values corresponding toe =0. The equation -7^ = , 



dV S c—ex + es 



of Art. 9 gives, when e = 0, -j- = -, from which, as is wellknown, 



Y is found = J (e^ + e"^ - 2), S = I (e^ - e~7) . Now let Y and 



S be increased by the quantities 8Y and SS of the same order as 

 e, and let the squares, &c. of BY and SS be neglected. Since 



•.r„ .t- . ^S '^•SS dY d.SY ^ d.BS dY 



it follows that -7- • — ; — = -3- -J — . -Let —j — =e .n -j— ; 



dx dx dx dx dx dx 



then — ^ — will = e.»-7-: also 8S = f e.o -5-. Substituting 

 dx ' dx J ^ dx 



these in the equation (c—ea? + es) — =s — ey, differentiating, and 



putting W for y-f S -T-;- — Xy^ +Y), it is found that 



«^__2 W 



dx c f _- ' 



Phil. Mug. S. 4. Vol. 16. No. 104. Juhj 1858. C 



