400 Mr. Challis on the general Equations 



Hence, 2a<o = 2a= hyp. log. p=/(s-a^- -^) =/(^^ - at) (I). 

 So by supposing the function / to disappear, we .should find that, 



2aw= -2a'' hyp. log. p=~f(-^^ + at J (2). 



These two particular integrals will be found by trial to satisfy 



iq) : consequently each indicates .separately a motion which is 



possible. Let us suj)pose in (i) s and t to vary in such a man- 



ds 

 ner that w does not alter. The consequent value, a + w, of ^, 



is the velocity of propagation together with the velocity at the 

 point whose abscissa is s. Hence the velocity of propagation is 

 «, the same quantity for every point in motion. By considering 

 in the same manner the equations (2), the velocity of propaga- 

 tion will be found to be —a, the negative sign indicating the 

 contrary direction. It thus appears that, when an impression 

 is made on the fluid in a single direction, whatever be the 

 magnitude of the motions, the velocity of propagation at every 

 point in motion is ])recisely a. This remarkable result, which 

 discontinuous functions have hitherto concealed from the eyes 

 of mathematicians, may be confirmed by the following reasoning. 

 Suppose the line of fluid we are considering, to be divided into 

 an unlimited number of equal masses, indefinitely small, and 

 call three of them taken in succession, a, (i, y, a being that 

 which is nearest the origin of abscisste. Let z = the distance 

 between the centres of gravity of a and /3; s, that between the 

 centres of gravity of /3 and 7 ; <«', u>, the velocities respectively of 

 the centres of gravity of /3 and 7 : then w - « is their relative 

 velocity. Suppose at the end of an interval t reckoned from the 

 instant at which the distances were z' and z, that the latter 

 becomes z': the relative velocity w'-w will be constant during 



