122 Mr. K. Moon on the Transmission, in one 



view the problem is simple, not to say trivial, and possesses but 

 slight interest to a mathematician; and to the naturalist it is 

 too mathematical to be pleasing. A mathematician cannot 

 sometimes help feeling some regret that the great discoveries 

 and advances in his science can be only known to the very few 

 who have learnt to read the language that can alone express 

 them ; but from the contents of this communication more than 

 one argument might be drawn to show that it is any thing but 

 an unmixed gain to a science to admit of popularization. 

 Cambridge, June 10, 1873. 



XII. On the Integration of the Accurate Equation representing 

 the Transmission in one Direction of Sound through Air, deduced 

 on the Ordinary Theory. By Robert Moon, M.A., Honorary 

 Fellow of Queen's College. Cambridge*. 



THE following investigation, independently of its bearing on 

 the problem to which it ostensibly relates, I offer as an 

 essay towards a complete theory of the solution of linear partial 

 differential equations of the second order. 



The transmission of sound through air in one direction may 

 be represented on the ordinary theory by the single equation 



n __ d 2 ct a 2 d^a ... 



dF~"^d^' (j 



where a and % respectively denote the ordinate of the particle at 



the time t and in the state of rest, and where «„~ -% — -or by the 



lv dx J 



pair of equations, 



dt u{ dx 



n __dv da x 

 dx"~~oJ' 



where v and a* 'are no longer to be regarded as the partial dif^ 

 ferential coefficients of a, but as independent functions of oo and t. 

 The solution of this last pair of equations must consist of a 

 pair of relations between the variables vct x % and t, such as 



F 1 (^/r« ir )=0, F 2 (^« T )=0; .... (3) 



from which it is evident that (1) must be derivable from a pair 

 of first integrals of the form indicated by equations (3) when we 



substitute in them — > — for v and u x respectively. 

 * Communicated by the Author. 



