192 
SIR G. H. DARWIN ON THE 
Now in 8 5 we defined 
E* = 
(2i+ I) 
Hence 
47ra ! X/(l + X) 
&'M ®<Wi> <*<>•• 
(e + S s + W(if+z 2 ))l = - A(^p a >)»—A- 3 sfAT {A.' + series} 5T/ . (19) 
the summation being for all harmonics. 
In determining the lost energy ±(ll) we may treat the layer as surface density. 
A typical term in the surface density is and the surface value of its 
potential is 
l <^ a3 ) t^x W©■'(>'•) WM W- 
a.' = f/M ©,*(»„), 
Then, since 
a typical term of \{ll) is 
ff (Up* 3 ) ~ (/?)’&■ {[»/W &W* d 
T 
1US 
1 (U)- 9 (4 a 3\2 A * {fif (xtor* 
2 w 2 /i- l37rpaj (l+xy^&n^®*- • 
(20), 
the summation being made for all harmonics. 
1 he value of jr(LL) may be written down by symmetry 
9. Final Expression for the Lost Energy of the System. 
We have 
V = (eE) 1 + (vv) + ( VV) + (eE) 2 . 
The several parts are to be collected from (14) or (15), (1G), (17), (18), (19), (20), 
and we have 
(vv) = {*** + ! */?©.' (»/*) ?<* M W (I) ««* (4») («>3, s even) 
_l_ yA v(./*) E-i cn s /\i ^ cos" y ( i . 3 (3+ «")/■" 5 (5 + 2/c“’+/c 4 )& 4 
+ L^~ Al ~3WWM l + - 14K 2 ~r 2+ 56A r* 
(all 
harm.) J 5 
( VV)— symmetrical expression with 1 /a in place of A. 
