( '41 ) 

 OP 



cos d-„ 



OP' 



Tlie infinitely small cone of normals will intersect R in a curve 



lying in a plane, normal to OP; the plane touching R at the point 



Q is also normal to OP. Let these last two planes, which are 



therefore parallel, cut OP in S' and S. Then 



0SX0P=\ OS' X OP' = 1 , 



and 



u= OS' .OP cos &„ 



du, = — SS' . OP cos d-„ . 



Further 



cos d'. 



d£i, = —— ° da , 



if do is the infinitely small surface of the just mentioned plane curve. 

 But we have also 



da =^ 2,T l/pd q'o SS' 

 if Q^ and q'„ are the two principal radii of curvature of R at the 

 point Q. Combining the obtained equations we find therefore 



,duju = u,~ OQ'OP ' 



1 



or since OP = r and --— = OP cos &^ ^= r cos 9^„ 



/'d£i\ 



-— = — 2jir 1/ Q, q', cos' », . 



so that 



g T'V^' cos &„ 2,iir 



T\ pj 



or by (13) and (14) 



fj J % 



if ^j„ is the velocity of propagation of the ray OP, as it is defined 

 for ^>/ane waves. Thus tlie electric force appears to have the same 

 direction as it has for plane waves whose corresponding rays coincide 

 with OP. Its magnitude is given by 



u IV, r t/g;:^'' CO. &, ^^^ 2a r_ ^\ 



1- r„' T V Po J 



^11. We must add to this a second vibration which may be 



obtained by the composition of all wave systems due to the y'-com- 



ponents of the infinitely small vet-tors into which the original 



E. M. F. has been divided. It is this action we lia\'e left aside in 



