A 



542 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



We will suppose that the initial conditions are now, 

 u = 3> (x, y, z), 



(10) du , 



Z- + b*& 



Then the solution of (9) is, by 247 (22), inserting explicitly 

 the rectangular coordinates and the direction cosines cos a, cos f3, 

 cosy, of the radius r in the functions <& at , <f> at , 



(i i) u(x,y y z) j-g- I \t&(x+at cos a, y + a* cos fi,z + a cos 7) dco 

 + 7 I lt<f> (x + a cos a,y+at cos fi, z + at cosy)d<w. 

 The equation (8) holds for all values of t Solutions of it are 



<I> 0, y, z) = F(x, y)cos~z + H (x, y) sin - s, 



(12) 



f / \ / \ ib 7 / ^ . ib 



<f> (x, y,z) = g (x, y)cos- z + h (x, y) sin ^, 



Ov CL 



and for ^ = these reduce to <& = F, =/, so that for t = we 

 have the proper values of u t ^- . We therefore insert the values 



(12) in the integral (n). Now as we integrate over the whole 

 sphere, the sine terms, being odd functions of z, disappear, while 

 the cosine terms, being even functions, give us double the value 

 that we should get by integrating over the hemisphere for which 

 z>0. 



Accordingly, giving z the constant value zero, 



(13) u ~- - JTT 1 1 tF(x + at cos a, y + at cos ft) cos (ibt cos 7) dco 



~*~ 9~ \\tg (x + at cos a, y + at cos /3) cos (ibt cos 7) c?a>. 



In the telegraphic equation F and g are independent of y, and 

 are therefore constant on all small circles of the sphere normal to 

 the X-axis. We will therefore employ polar coordinates, a the 

 angle made by r with the X-axis, and % the angle that the plane 

 of r and the X-axis makes with the XZ-plane. Then 



, v cos 7 = sin a cos ^, 



dco = sin a 



