302 Mr. Woolhouse on the Deposit of Submarine Cables. 
the asymptote, e times the length of this tangent will be equal 
to T—a. 
II. We propose now to renew the investigation, assuming the 
resistance and friction to each vary as the square of the velocity, 
which supposition is more nearly in accordance with the actual 
resistances as determined by experiment. Let 
B be the coefficient of lateral resistance, 
B! that of longitudinal friction, 
ack Bn* _ lateral resistance to velocity 
I: diminished gravity 
isos 
Then with respect to an element 5s of the cable, we shall have 
Normal resistance = Bn? sin? a, 
Tangential friction = B!(m—n cos w)?; 
and, proceeding as before, we obtain 
i 5 =g! = (Tcosw) — Bn?sin*w + B!(m—ncosw)*cosw, 
tne — £ (Psino) —g! + Bn?sin2wcosw + BiG —>nciines) Sa 
or, transposing and dividing by g’, 
! 2 
0=£,(T—a)coso} —e*sin?w + B (™ cose) cosa, 
5 Pi btis , . DF 
O=7, {(T—a)sinw} —1+esin*wcosw +R (“—cosa) sino, 
which, in substance, are the equations finally arrived at by 
Mr. Airy. 
Multiply equations (1) respectively by cos @, sinw, and add; 
and afterwards multiply them by sinw, cos@, and subtract ; 
then 
Oi ‘cx Bm rs 
ee Be (2 —cos 0 . 
b cise co: al 
(Pa) @ = cos o—e*sin®a, | 
which agree with the equations obtained by Messrs. Longridge 
and Brooks*. 
* Tn the investigation of Problem III., Messrs. Longridge and Brooks 
have disregarded the effective forces as inconsiderable. ‘To supply these, 
we have, at the point C, the horizontal velocity =v(1— cos @), and the ver- 
tical velocity =—vsinz. Now with any variable velocity V, the effective 
accelerative force = ea and multiplying by ae and in- 
