338 Prof. W. M°F. Orr on the Radiation from 
Hence the time-mean value of a+ 6?+97 is | 
= 97? O7K2a2R-?{cos? @ sin? od . 0? J5(xa sin 0)/x?a? sin? 0 
+008? o¢ . J (Ka sin 0)}, 
if @ is even, and an expression obtainable from this by inter- 
changing sin od, cosad, ifo is odd. And the mean rate of 
radiation across the sphere of radius R, being the mean 
value of (4ar)- 1V\(@? +2? +y°)d8, necuednnel’ becomes 
moreary * sin 0{07J2 (ka sin 6) cos? 6/x2a? sin? 0+ J. (xu sin 0) } dO, 
0 
except that if o=0 the above must be doubled, since in that 
case the mean value of cos*o@ is 1 instead of $. By means 
of the relations 
ot J (v7) =${J,_1(2) +3 541(a) } 
J,(0) =43{3,_1(«)—Io4i(2)} 
this may be written in the form 
Ta 
yecreat | sin O{J?_,(«a sin @)+J?_,(«a sin @) 
0 
2-272 . 
— 20° "a "J (Ka sin 6) $d. 
Using the relation 

J5(«) =N(o)[72 "a Fhe +4; o +1, 2o+1; =a 
where I a > Pir P23 y) denotes the hypergeometric series 
sail 
er ae emg 
Pip2 - 1 7 P1(p1 + 1)po(p2+1)1. on 
and integrating each term separately, we see that 
17/2 29 
ve 
sin 0.J-(w sin 0)d0= 7. F(o+4; 043, 2041; — 2’) 
\ hoon EO aeea ly, ie 
=a | Jog(2u)du. 
“0 
The expression for the rate of radiation may thus be 
written in the co 
dar? C2K? “ide —. F(a—4; ¢ +4, 20—1; —x’a’), 
Ka) fhe 
: Oe 
tases Pet 5 +4, 20643; —K*a’),. 
Da (ea) 2752 
ee aie BRA 1. 2 Bree? 
orn o +3, 2041; wa) b 
* Proc. Camb. Phil. Soc. May 1899. 
