Mr. R. H. M. Bosanquet on the Theory of Sound, 129 

 First, then, when r = a^ the above values give for S, 

 without flange ... 7ra^(5 4-27r) = ll'37ra^, 

 with flange 7ra^(3 + 7r) = 6*1 7ra^. 



The conditions are satisfied roughly by the following assump- 

 tion for the case without flange, 



S = 7r(a-f2r)2. 



When r is great, this represents nearly a sphere of radius r. 

 Making this substitution in the formula 



we find 



We shall see that the experimental determinations lead us to 

 a value of about *55 a ; so that experiment is thus fairly repre- 

 sented. 



The calculation of the case with flange may be derived from 

 the above by assuming S = 7r(a + \/2 r)^, which makes 

 Sa = 5*87ra^, the wave-front condition requiring 6*1 7ra^ and 

 S<^ =27rr^. Applying the above formula, we find 



r= — ^ = -707a. 



The following is another form of assumption of a more 

 elastic character, in which we employ a combination of two 

 forms of surface having a common value at r = a. Consider 

 first the hemispherical case (with flange). We can represent 

 the wave-front surface (6*1 7ra^ when r = a) by 



S = 7r(a2 + 5r2); 



then, in order to make the surface tend to that of a hemisphere, 

 we must alter the assumed form between r = a and r = oo and 

 V^^^ S = 7r(4a2 + 27'2), 



which tends to a hemisphere w^hen r is great. For the pur- 

 pose of this calculation the following formula is convenient: — 

 If 



Si = 'ir(my + ny), 



S2 = 7r(my + ny), 

 and 



m'\ + n\ — m\-\-n% 

 then 



\ ^'-^+1 ^-^ = ~-\ tan-i — + (^— tan-»— ), 



FhiL Mag, 8. 5. Vol. 4. No. 2^. Aug, 1877. K 



