38 Mr. 0. Heaviside on Maxwell's Electromagnetic 



where the U^s are functions of (i?VO~^ given by 



T] —pvtv TT —cwtvM- 1_J TTo = e''Ml — -I - 1 



Uo-6 , Ui-6 yi ^^^J, U2-e ^i ^^y + (,,^^^2;^ 



TT -.^^vfl _ ^(^ + ^) 4. r(r^-l')(r+2) I (24) 



^'~ L 2vtV 2A.{vtS7Y 



2.4.6 (t;^V)' +-"J^ 



being in fact identically the same functions of vf^ as those 

 of r which occur in the investigation of spherical waves. 

 Arranged in powers of s^ajv^, we have 



q vS7^ ^' ^' ^'^ . . . (25) 



where 



_o-^ _ 1 . M' _ 3°-^ , M' 1 



^i-y, .72- -2+ 2.4' ^^~~^¥"^ 27476' 

 ^4-8 2.4 2.4.6.8' ^'~^2 * 2 .4. 6 ^ 2 .4. 6 .8 . 10' '^^ 



i/6- 16^-162.4 22.4.6.8 2. 4. ...12' 



7 _^« 7 _(^0' 7 _ 1 / .o.^^'O^ 

 /ii--2J "2--^, '*3-22|2\^-°"*+Y73"j' 



7 _ W «^ N2 (o-0' \ 7 _ 1 / , 6(0^3 (g-Q^ \ 

 ''^~ ¥\S\'^^''^^ + T:TJ' ^''~ 2^V TTI "^ 2^475 j' 



^«~2*|4r^^ ^^ 2.5 ^22.5.67' 



1 / 45M^_15(^ _M^\ 



^~2^14\ ^^^^^ 2.5 2^5.6^22.5.6.7/' 



The following properties of the g's and h's are useful. 

 Understanding that g^ and Aq are unity, we have 



(at)^ 

 gy+a-tgr+i+ .^ gr+2+ ...=0 when r is odd, 



. Jr (o-fi) 



and when r is even, =1 .3.5 ...{r — 1) ( — 1)^ — —' 



{atiy^ 

 except r=0, when =Jf^(^ati). 



}-{27) 



W28) 



J 



