6 The Astronomer lloyal on the Conditions 



the same equation which gives the tension in the common cate- 

 nary. Hence we obtain this singular result : that the form of 

 the travelling curve is exactly the same as the common catenary ; 

 and that the tension at every point is exactly the same as at the 

 corresponding point of a similar fixed catenary^ increased only 

 by an absolute constant a, which depends only on the velocity 

 with which the curve travels. In the instance of the Atlantic 

 cable, this constant a is insignificantly small in comparison with 

 the other tensions. 



8. In the second place, I shall investigate the form of the 

 curve when the velocity of delivery of the cable is equal to the 

 ship's velocity, and when the frictional resistance is fully taken 

 into account. 



9. The accelerating force produced by the frictional resistance 

 in any direction z= — bx velocity in that direction. Attaching 

 the proper terms therefore to our former equations, wc now have 



gd'^x , d /^dx\ , dx' 



ds^^ ds\ dsJ ^ dt 



Remarking that -7- =n—n-,-, -y- =— n-p, and that — r =e, 

 ° dt ds dt ds (/' 



these equations become 



Integrating, with the same attention £0 the constants as before, 

 (T— fl) -Y- = c—ex + es; 



from which -f- = — , the differential equation to the curve. 



dx c — ex + es 



10. The treatment of this repulsive equation is much simpler 

 than might have been anticipated. Expressing 5 in terms of the 

 other quantities, 



^Ay .,Ay 



c~ — ex— + ey 

 dx dx 



dx 



