of Deposit of a Subniarine Cubic. 9 



13. Combining the consideration of the. asymptote with the 

 previously estabhshed characteristics of the cvu-ve (that when 



x = 0, i/ = 0, and -j-—.0), its general form will be seen to be this. 



In its lower part it resembles a circular arc, or the lower part of 

 a common catenary. But in its upper part, the curve does not 

 tend to become vertical, but tends to approach to an asymptote, 

 making the angle X with the horizon : which angle, when the 

 ship's speed greatly exceeds the terminal falling velocity of the 

 cable, will be very small ; implying that the asymptote and the 

 upper part of the curve will be little inclined to the horizon. 



14. Other formulae, necessary for a complete knowledge of the 

 properties of the curve and the tension at different points, will 

 be obtained thus : — 



Differentiating the equations at the beginning of (11), 



d fx\ e d^ 1 



dz\c)~2{e'^ + \) dz eie' + l)' 

 dz\cJ : 



1 -^^?+. i 



dz\cJ~2{/- + \) dz ' e^+r 

 Squaring these, adding, and extracting the square root, 



(the negative sign being taken because, as z diminishes, s in- 



-1 



^|{^(eHl) + e}.^-'^('-^73) + (^(e2+l)-.}-^0-^70\ 



~2ev/(e2 + l) 



Integrating, and making s vanish when x and y = 0, 



15. Again, to find -j- ; putting for abbreviation 



we have 



But 



{^(e2 + l) + e}^ ^(' + f) = , 

 ds ds dz —c r .11 dz 



tt("%-} 



dx dz dx 2^^/(6^ + 1) L qj dx' 



dz^_\ ^ 2e{f+ 1) _ -1 _2e{e^±ll . 

 dx c a d^ 



di 



-2 



<,-!)-.■ 



