of Deposit of a Submarine Cable. 13 



in water; since A x (terminal velocity)^ = y, we have, tenninal 

 velocity = \/ -T) and therefore e-=n\/\. 



23. Taking the first equations of Article 9, we must make 



dx' dx da dy 



dt'''^"' ds^' dt^~ '"' ds^' 



whence 



and 



m^ -rr =9' -r ( T -^ ) — (resistance of water to motion in direc- 

 os'^ ds\ ds/ ^ 4.- e I ■ • \ 



tion 01 x' increasnig), 



or + (resistance of water to motion in direc- 

 tion of x' diminishing) ; 



wi^-rf =y-7-( T-^j — y+ (resistance of water to motion in 

 ds ds\ ds/ directionofy diminishing). 



These equations apply to any form of the cable-curve. 



24. To express the last terms of these equations by means of 



dx ds 



-J-, -J-, &c., we must express the lateral motion and the longitu- 

 dinal motion by those quantities ; then express the resistances 

 to lateral and longitudinal motion, and then, from these, find 

 the resistances in the direction of x and y. And we must be 

 careful to remark that both the lateral motion and the longitu- 

 dinal motion are always downwards. Now the lateral motion 

 downwards 



_dx' dij dy dx _/' dx\ dy dy dx _ dy 



^ dt ' ds dt ' ds ~\ ds) ' ds ds' ds ds ' 



and the lateral resistance, normal to the curve and acting upwards, 

 = X71H -j ) . The longitudinal motion downwards 



_ dx' dx dy dy _f dx\ dx dy dy _ dx ^ 



~ dt ' ds dt ' ds~\ ds) ' ds ds ' ds ds' 



and the longitudinal resistance, in the direction of the tangent 



/ dx\'^ 

 and acting upwards, = Tiym—n-j-j , From these we find. 



Last term in the equation for x = 



Vw ds) ' ds \ds) ' ds' 



