492 Lord Kelvin on the Production of 



Adding 3qx (46) to (45), we find 



3 2 ^) = -2^ + %) .... (47), 

 whence 



^1=2^ + 3^-^(0 . . . (48); 

 v vr 



and by this eliminating $ 2 from (46), 



[&+ 1 (u + 2v)-d+ \(u* + 2v^ Mt) = m ■ (49), 

 where ~d denotes d/dt, and 



^--^U^+^W) • (50). 



§ 18. I hope later to work out this problem for the case of 

 motion commencing from rest at £ = 0, and S(t) an arbitrary 

 function; but confining ourselves meantime to the case of S 

 having been, and being, kept perpetually vibrating to and fro 

 in simple harmonic motion, assume 



S{t) =/i sin cot (51). 



With this, (50) gives 



^0=^ 2 [(^-l)sin^+^cos^] . (52). 



To solve (49) in the manner most convenient for this form of 

 &(t), we now have 



ft (o = — i — i m 



-d 2 +-(u + 2v)-d + ~. > {u 2 -h2v 2 ) 

 q q" K y 



B 2 + \{u 2 -r2v*)--(u + 2v)-d 



[B 2 + ^(^ + 2^ 2 )] 2 - [i( tt + 2t>)3] 

 = hgu 2 X 



i-T^(" 2 + 2i> 2 )+ -Av* + dvu -n 2 ) + g) 2 ) sin^ + ^- V (^4ft) 2> jcos^ 

 I </ 4 ftr y g 2 J go) \ q l J 



. . (53). 



