234 Lord Kelvin on the Production of 



or, substituting the values of K and L from (65), 

 D 4wp q-hco f r a /9v 4 3u 2 ,\ /9w 4 3u 2 A"l 



' -^) 2 +i(^-^)]sin W ?} (72'), 



+ ^[ 9 ^(" 2 + 2. 2 + ^ 



^ 4 o) 4 



where D denotes the denominator in equations (65). 

 Finally for to we have, denoting the period by t, 



w= X -{ T dt?S(t) 



= ^y-q*h*o> fo(u- v)aK + 2(w 2 - 1' 2 ) - + 3va \~)dt cos 2 o>* 

 = ?|^ ? 2 A 2 (w r2( M _ t ,) &) K + 2(w 2 -t; 2 )-+3^1 . . (73), 



)](73'). 



2tt P q*h*a>* f 9 / 9o 4 dv* \ ( 



9w 4 3m 2 

 i f- 1 



4. 4. 9 9 ^ - 1 - 



\q (o q*a> 



§ 30. To verify the agreement of this direct formula for the 

 work done, with (69) which expresses the effect produced in 

 waves travelling outwards at great distances from the centre, 

 is a very long algebraical process, with K and L in (69) given 

 by (65). But it becomes very simple by the aid of the following 

 modified formulas for $\(t) and $i{t), which are also useful 

 for other purposes. From (48), (49), and (50), by eliminating 

 ^\i we get an equation for <^ 2 similar to (49), viz., 



[b 2 + - (« + 2v)-d +\(u* + 2v*) ] ? 9 =W{t) , 

 where 



^(0 = ^(i+fB- 1 + ^B- 2 )^(0. 



We may now write (50) in the form 



§{t) =hqu' 2 G sin (wt + a), 



and similarly 



#f(f) =hqv*K sin {cot-/3) J 



where the values of G, H, a, /3, are given by the following 

 equations : — 



y (74), 



I 



J 



(75), 



Sv 2 

 G-cosa=-5— g — 1 ; Gsin« = 



q'co' 



Hcos/3=l- 



3m 2 



•1 9 J 



</ or 



Hsin/3= 



3v 



3m 

 17®. 



(76). 



