8 The Astrouonur Koyal un the Conditions 



11. Tliis equation dclincs the curve algebraically, and also 

 gives us the means of tracing it graphically. If we assume arbi- 

 trary values for :, and compute from the equation the corre- 

 sponding values of ^, we may then determine the numerical 

 values of x and y from these equations, 



X _ e ^ ~ 2e'- — 1 



e~^f^y e{e"-+\) ^' 



c-2{e'+\y e'^+l 

 But, before making numerical computations, the following treat- 

 ment of the symbols will give a clearer idea of the nature of the 

 curve. If \ be the angle whose cotangent = e (which angle 

 may be called " the limiting angle "), then 



. ~= ^ ^i ~'Tl-iTT^ — A'smA, + //cosX >, 



^ 2^/(e^ + l)/ c(2eHl ) , , , . ,"1 



K= ^|^^>^^^:^^+.rcosX + ysmX|. 



From these expressions it appears that z and ^ arc simjjle 

 multiples of rectangular coordinates, on a system in which the 

 ordinate corresponding to f is inclined upwards from x by the 

 angle X, and the ordinate corresponding to z is inclined from y 

 in the direction from the ship by the angle A, ; the values of the 

 new coordinates which apply to the first origin of coordinates 

 (the place where the cable touches the ground) being, for the 



ordinate of which r is a multiple, — 7—5 — :r- : and for the ordi- 



nate of which ^ is a multiple, -Vro — ,;• 



12. Remarking that the index of z in the second term of the 

 formula for f is essentially negative, it will be seen that upon 

 making ~ smaller and smaller, but always positive, f becomes 

 indefinitely great, but positive. Consequently the curve has an 

 asymptote in the dii-ection of the ordinate corresponding to f 

 positive, that is, making the angle X with the horizon, inclined 

 upwards as from points near the place of touching the ground 

 to points near the ship. The asymptote does not pass through 

 the place of touching the ground, but below it, its smallest di- 



stance being ;, // a TTx '• ^"^^ ^^ ^^oes not pass through the ship, 



but below it, its nearest distance being — y-^ — ^ x the value 



of ~ at the shij). It is unnecessary for our purpose to examine 

 the form of the branch of the curve corresponding to *r > 1. 



