226 



DYNAMICAL THEORY OP SOUND 



on the other, and is therefore more adequately represented, in 



the simplest cases, by a combination of two simple sources near 



together but in opposite phases. Idealizing this a little further 



we are led to the mathematical conception of a "double source." 



We begin with a simple source of strength m at a point 0, 



and a simple source of strength -f m at an adjacent point 0', 



the signs indicating the oppo- 



sition of phase. If we next 



imagine m to become infinitely 



great, whilst the distance 00' 



becomes infinitely small, in such 



a way that the product m.OO' 



remains finite, we have the ideal 



" double source " of theory. The 



direction 00' is called the 



" axis," and the limit of m.OO' 



is called the " strength." The resulting motion is evidently 



symmetrical about the axis. 



If the direction 00' be that of the axis of x t and be 

 taken as origin, the velocity-potential at P due to simple 

 sources + m at ' and 0, respectively, will be given by 



p -ikr\ 



- V)' 



where r= OP, r' = O'P. If we draw PP' equal and parallel to 

 O'O, we have /= OP', and the expression in brackets is equal 

 to the change of value of the function e~ ikr /r caused by a 

 displacement of P to P'. Hence, ultimately, if P'P = 8x, 



Putting m&x=I, we deduce the formula for a unit double 

 source at 0, having its axis along Ox, viz. 



(20) 



this is a particular case of 73 (3). When x alone is varied, 

 whilst y and z are constant, it appears from the figure that 



