Formulation of the Principle of Huygens. 263 



where U 1? U 2 , U 3 , . . . , Y lf V 2 , V 3 , . . . are such that the 

 volume integral 



JjJCCUiV'Vx-VxV'UO 



+ (U 2 V 2 V 2 ~ V 2 V 2 U 2 ) + . . .] dx dy dz 



taken throughout the space S vanishes. This is merely 

 a simple and obvious generalization of Green's theorem. 



t J, where t and a are 



for the present merely algebraic symbols, but which, subse- 

 quently, will be identified with the time and the velocity 

 of propagation. We proceed to show that the surface 

 integral 



J J L\rfr an r°r l dn] T ar x r§ dt \dn dnjj 



vanishes when taken over the surface S. 



Expanding the integrand by Taylor's theorem, we obtain 

 for the surface integral the expression : — 



JJV^o dn r*r % dn J 



_ 1 d(f>(t) CC 1 r/1 ch\__l drA \_ dr i ^ol jo 



a dt Jjrir \_\ri dn r dn/ °' dn dn\ 



i_ fm ccj_ r/i dn_i dr,\ 



+ 2a 2 dt 2 J J vo LV, dn r dn) [ ' ' + o) 



_ j_ £*(o rrj_ r/ 1 *j_ i drA 



3! a 2 rf« s J J r in [An dn r dn l Kl+ 0> 



+ + 



f-i)»*»(Q ( j J_ r/i ^_i * \ ( )n 



n\a n dt n J J r^o LVn dn r dnJ K 0J 



+ + 



