590 



binary and ternary quantics. But the theory of binary quantics is 

 considered for its own sake ; the geometry of one dimension is so 

 immediate an interpretation of the theory of binary quantics, that 

 for its own sake there is no necessity to consider it at all ; it is con- 

 sidered with a view to the geometry of two dimensions. A chief 

 object of the present memoir is the establishment upon purely 

 descriptive principles of the notion of distance. 



III. " On the Mathematical Theory of Sound." By the Rev. 

 S. EARNSHAW. Communicated by Professor W. H. MILLER, 

 For. Sec. U.S. Received November 20, 1858. 



(Abstract.) 



The principal feature of this communication is the discovery of an 

 integral of a certain class of differential equations. This class in- 

 cludes, as a particular case, the differential equation of motion when 

 a disturbance is transmitted through a uniform elastic medium con- 



fined in a horizontal tube. If the equation =F/ be differen- 



dt \dx) 



tiated with regard to t, it produces the equation 



dx 



which, by means of the general function F', can be made to coincide 

 with any proposed differential equation in which the ratio between 



^ and |- is dependent on -^- only. The integral obtained in this 

 dt dx dx 



manner is that which arises from the elimination of (a) between the 

 two following equations, 



y = a# + F (a) . t + <(> (a), 

 0= s + F(a).* + 0'(a). 



This integral, though not found by the direct integration of the 

 differential equation, and though evidently not the general sym- 

 bolical integral of it, is proved to be the general integral for wave- 

 motion, from its affording the means of satisfying all the necessary 

 equations of initial disturbance and wave-motion. 



The author first discusses wave-motion when temperature is sup- 

 posed to be unaffected by the passage of a wave ; and then when the 



