34 
PBOFESSOR A. E. H. LOVE ON THE INTEGEATION OP THE 
the functions of r and t, that are to replace the expressions in { ], being factors in 
certain particular solutions of the diflPerential equations, and even functions of t, 
could be arrived at by changing, in {},t into — t, and taking half the sum; for 
example we should rejdace 
by 
cos K [ct — r) — KV sin k (ct — r) 
5 [cos K [ct — r) — Ki- sin k [ct — ?•) + cos k [ct + '?’) + xr sin k [ct + r)] , 
or by 
cos K ct (cos KT + KI' sin kv). 
This comes to the same thing as picking out from each expression in { } the terms 
that contain cos k Ct, and rejecting those that contain sin k ct. 
We accordingly take for ii_, . . . forms given by such equations as 
4:Ttu_ = cos K CM dx'dy' 
4 TTiv_ = cos K CM dx'dy' 
r 0/-1 
— i COS KV + KV sin K!' 
-[cos/Cl’ + fcc sill Krj 
— ^ Vi — kV') cos Kr + 3/cr sin/cr] 
iy - y'f + (2 - / 
--q-(/i{cos K/q- 
/. , _ a/-i\, , • 1 
zq + iq j [cos kv + kv sin kt] 
-1 
(-2)= 
02,-1 _ 02,, 
— f\Tr^]\['?> — K'l’-) COS K-r + S/c^’sin/cr] . . (73). 
oxoz " oycz MV / 
29. According to explanations already given, we shall have 
477 lim [u^ (P) — ■u_ (P')[ = cos KCt Ij dx'dy' 
u, 
dr /d?’ 
a 7 L + 
0^ 
= 4771'? j (Q) COS K .("•!)• 
Again 
477 lim [7C+ (P) — w_ (P')} is the limit of 
3/c' J J 
dx'dy' 
9i 
drdz 
-/ 
er r ^ 
\dydz 
{(3 — /cV“)cos K[ct — r) — S/cr sin k [ct — r)] 
9i 
/0'-/’-l 
. /ah-1 
^/_ ~ + -SK)'sm Kr] cos k Ct 
