416 Prof. Kowland on Spherical Waves of Light 



-£> = V 2 A*F, 



^F 



V indicates the velocity of a plane wave in the medium. 



Each one of these equations is the same as that of sound, 

 and can be solved in a similar manner. If we have solutions 

 of these equations F 0> G , H , which do not satisfy the equation 

 of continuity, then we can get other solutions F m , Gt m , K m from 

 these which shall satisfy this condition. Denote s/ —1 by t, 

 and let V_(„+i) be a solid harmonic. Then, as V_ (n+ i) is a 

 homogeneous function of a, y, and z of the degree — -(w+1), 

 we have 



Make -^ n » TT ' (a-«6)( P -vo 



where C is a function of p of a complex form, 



This form has the advantage that, by the addition of another 

 term of the same form in which i has a negative value, the 

 sum will reduce to a real form with circular arcs instead of 

 powers of e ; a is introduced for generality. Substituting 

 this form, we have for the determination of * the equation 



dp dp p 



* 



Making 



Cn + Dnp2 e-p(«-^), 



we obtain #D_ 1 ^_ D / (a _foy + («W\ =0 



dp 2 p dp \ p 2 J 



If we replace p by a new variable equal to ^(a — $)p ; this reduces to the 

 equation of Bessel's functions. Hence we can write 

 D»=J(»+$)(&+wf)p, 



F =\J(nU)(b+ia)p}pn+5Y- (n+ l)e-Vt(a-ib) 7 



with similar terms for G and II . 



Hence Cn = C 'ph-P^-ib)J (n+h) (b+m)p, 



where C ' is a function of n to be determined, and one must add a corre- 

 sponding term with — i in place of -\-i. 



