12 The Astronomer Royal un the Conditions 



value of the other side of the equation. Therefore D' must = 0. 

 Again, the index of the second exponential term is positive; 

 and therefore if at any other point of the curve x—ey have a 

 value, while D' = 0, that second term will be infinitely great. 

 Consequently, x—cij must always = 0. This equation, it will be 

 seen, with its derivatives, 



satisfies the original equations from which it was deduced. 



20. In this case then (when the cable is delivered exactly as 

 fast as the ship sails, but when the tension at the bottom = 0), 

 the form of the cable is a straight line, inclined to the horizon 



at the angle whose tangent = -, that is, at the angle X,. Its 



tension will be found from the original formula 



T-»=*-)S=V('+?)-4\/0+?)-^}- 

 °-V 04)V04 )-^}- 



21. The result for the last case of our investigation, showing 

 that the cable may assume the form of a straight line (a form 

 which simplifies the mathematical conditions of the problem so 

 much as to enable us to dispense with some of the limitations 

 of supposition formerly made), suggests the following as au 

 investigation to be made in the third place. Assuming (as a 

 hypothesis to be confirmed by the investigation) that the cable, 

 from the point of quitting the ship to the point of touching the 

 ground, has the form of a straight line ; assuming that the 

 resistance in moving through the water is proportional to the 

 square of velocity ; assuming that the coefficient of resistance for 

 longitudinal motion is diff'erent from that for lateral motion ; and 

 assuming that the rate of delivery of the cable may be greater 

 than the ship^s speed; to find the position and tension of the cable. 



22. The notation of Article 2 may be retained, with the 

 following addition only : — 



A the coefficient of the square of cable's lateral velocity in 

 the formula. Ax (lateral velocity)^=lateral resistance. 



B the coefficient of the square of cable's longitudinal velocity 

 in the formula, B x (longitudinal velocity) - = longitudinal 

 resistance. 



If we limit the meaning of e to the proportion of the ship's 

 speed to the terminal velocity of the cable when falling laterally 



