4 The Astronomer Royal on the Conditions 



the proportion of the cable's real weight to its apparent weight 



in water. 



fl = "= twice the height due to velocity n, with diminished 



. S 

 gx'avity. 



b the coefficient of the simple power of velocity in the for- 

 mula, 6 x velocity =frictional resistance. 



e the proportion of the ship's speed to the terminal velocity 



of the cable when falling freely in water, = -^. 



Additional symbols will be introduced as they are required. 



3. In i\\e first place, the form of the curve will be investigated 

 when the velocity of delivery of the cable is equal to the ship's 

 velocity, and when the frictional resistance is neglected. 



4. The equations of motion of the point, whose coordinates 

 are a' and y, are 



-j-^— accelerating force in the direction of a'; 



d^ii 



— = accelerating force in the direction of y ; 



(if- 



or, considering the small part of the cable included between the 



points « and s + h, 



fPx^ 8 (real value of the horizontal part of tension) 



7//~ ~ real mass of Ss ^ ' 



d'^ij 8 (real value of the vertical upward part of tension) , 



lifi ~ i-cal mass of hs J J ' 



Now, considering the tension as expressed by a length of 

 weight of the cable as weighed in water, the real value of the 



horizontal part of tension =T-^, and therefore 8 (real value of 



the horizontal part of tension) =8f T— j= -r^f T-^ j x Si\ But 



the real mass of 85 in the denominator, on which the inertia 

 depends, is not expressed by the length 8s as weighed in water, 



but by the length 8s as weighed in air, or by the length Zs x -^ 



as weighed in water. Thus we obtain 



'df~^ds\ds)' 

 and similarly, j/^y _ , ^ 



dt^~ ~'^ Is 



(4f)-*'- 



