342 Theoretical Determination of the Velocity of Sound, 

 and therefore 



and the expression for^^ or ^ is 



2-'f.cos{(«+l)R-(,.H-f>.C'}. 



The velocity of the wave in the direction perpendicular to the 

 wave-plane is clearly a, as found by ordinary investigations. 



It appears from the form of this result, that though the 

 equation (4.) does not contain t, yet there is propagation of 

 motion in a direction inclined to z, with lateral spreading to an 

 indefinite extent ; and thus the inference by Professor Challis 

 on page 280 line 4 from the bottom is not supported. 



(7.) As we may combine as many of these plane waves as 

 we please, it follows that we may take an indefinitely great 

 number of plane waves, so arranged, that their normals shall 

 all be in a conical surface of which z is the axis, and shall 

 make with each other equal small angles. The former con- 

 dition, it is easily seen, is obtained by making 



b , b . , 



p=—-cosQ, o'=-'Sm9; 

 ^ a ^ a 



and the latter, by supposing the small increments of 9, in 

 passing from the normal of one wave to that of another, to be 

 equal. Also let M be the angle which the plane passing through 

 the point xyz and the axis of;:; makes with the axis of o', so 

 that .r^j'.costo, y -—r.^^\\^.'w. Then the expression for y 

 becomes 



S.A.cos<! -^cos(9— 'By) + B >, 



or, as our summation is now supposed to apply only to a series 

 of waves for which 9 varies by very small equal quantities, 



/=/;'.A.cos{^-cos(»-<.) + b}. 



The value of this integral in series, from 6 = to fl = 27r, is given 

 by me in the Philosophical Magazine, vol. xviii. p. 6 5 with 

 the proper change of notation it is 



b' , , ■ , ■ 



or, as— 2 ='ie, tlie mtegral is 



,.A X |l - — + ^j^, - ^^-^^^ + &c.|. 



