Equations in a Homogeneous Isotropic Medium. 39 



A. + c.M.,.x+^^%.+.+ ... = (-l)'-cri.^, . (29) 

 except when r=0, which case is not wanted. Now if 



€2^(l+-)=6"'Ml + ^/i + «y2+.-.),. . (30) 

 the/^s* will be given by (25), viz. 



/o=l, fi=go+K f2=gi + h,&C'; • • (31) 



and the properties of ihef's corresponding to (28), (29) are 



/, + <+! + ^V+2 + . . . = 6'^' when r=0, \ ^^^^ 



= when r is even, except ; J 



and 



Jr-i{ati) — iJr+i(a-ti) 



-.+1.3.5...0'-2) -^ ^ ; . . (33) 



(a-ti) 2 



when r is odd, with the + sign for r=l, 5 9, . . . , and the 

 — sign for the rest. The first case in (32), of r=0, is very 

 important. But in case r=l, the coefficient in (33) is +1 ; 

 thus, 



= {Jo—iJi){ati). 



12. Special Initial States. — Now let there be an initial 

 distribution of Hq only, so that, by (17), 



II = 6-''Ycosh + -sinh]^«. Ho,") 



TT, , smh qt V TT 



E = — e-p* ^ — Hn 



q G 



by (17). Let Hq be zero on the right side and constant on 

 the left side of the origin, and let us find H and E at a point 

 on the right side. The operator e"'^ is inoperative, so that, 

 by (30), 



^=Le-pie-^tv(^l-sg, + s%-s'g,+ ...)BoXfiv,S ' 

 the immediate integration of which gives 



* These /'s are the same as in my paper " On Electromagnetic Waves,' 

 8, Phil. Mag, February 1888 ; but s there is a- here. 



