Formulation of the Principle of Huygens. 267 



surface integral pertaining to the sphere tends to the 



value — ~^<f)lt ), as p approaches zero, and we have, 



therefore 



M *( t ~q) _ M (T[Y_L dr, 1 drA / r, + n,\ 



R 47rJJ L\r?r dn r Y r* dnj^\ a J 



ar^QXdn an Jot T \ a /J 



M being any constant. If we suppose A to be a source 

 of disturbance, the vibrational velocity or displacement 

 due to which at any point at a distance r can be expressed 



by — (pit — ), t being the time and a the velocity of 



propagation, the above equation expresses the equivalence 

 of the direct effect at B due to A at any instant to that due 

 to a source distribution on the surface S, the secondary 

 disturbance being sent out from each element of surface at 



a time — after it was sent out from A and at a time — 

 a a 



previous to its arrival at B. 



It is usual to write the surface integral in the form 



dS. 



47rJJ ion r x r r o0 n r x ( t - 9 ~ 9 )^ 



In the first term the differentiation with respect to the 



normal operates on r only and in the second term t~^-^ 



a 



is written f or t after differentiation with respect to the 



normal ; but although there may be a gain in conciseness in 

 writing the expression in this way, there is perhaps some 

 loss in clearness. 



Remembering now that at every point in space we have 



where V denotes the vibrational displacement or velocity, 

 and (f> is a function of the co-ordinates of a point and the' 



