298 Mr. Challis o?j the Forms of the Arbitrary Fimctioiis 



For simplicity I will take the case when the motion is in 

 space of one dimension, and the fluid is solicited by no ex- 

 traneous forces. Let v = the velocity of the particles at a di- 

 stance X from an arbiti'ary origin, and at a time t reckoned 

 from an arbitrary epoch ; s = the condensation at the same 

 distance, the mean density being 1, and a^ = the mean elas- 

 tic force. The usual investigation leads to the equations, 



which serve to eliminate the auxiliary quantity <$> find to de- 

 termine V and s. 



The integral of equation ( 1 ) is, 



<p = F, {x—a t) + f,(x + at) 



Hence v = F, {x—a t) ■\- f, {x -\- a t) 



a s = F, (x—a t) + J', [x + a t) 



As either of the arbitrary functions will satisfy alone the 

 equations (1), (2), (3), it will indicate a possible motion, 

 though not the most general. It will be convenient to at- 

 tend to this motion first. By taking F alone, we shall have, 

 V = a s = F (x — at), showing that for a given value of t, the 

 ordinates of the curve whose equation is i/ = F {x — a t) will 

 be proportional at once to the velocities and the condensations. 

 After an interval x the equation of this curve will become, j/ = F 

 (x — a.t + t) = F {x — a.T — at) = F {x' — a t). Its form 

 must consequently be the same as before, and the values of j/ 

 be identical for the same values of a:' and x : but as x=^ + ccry 

 the value of t/ which in the former case was at the distance x 

 from the origin, will in the latter be at the distance x -{• «t. 

 The curve, therefore, or the motion it indicates, will have been 

 propagated from the origin in the time t through a t. And as 

 this is true whatever be t, it follows that the velocity of propa- 

 gation is uniform and equal to a. By considering the func- 

 tiony by itself, we shall have v = — as =fx + a t), and shall 

 find that these equations imply a motion of propagation totaards 

 the origin of x and with the same velocity a. Hence in ge- 

 neral the motion of the particles is resolvable into those which 

 result from two simultaneous motions of propagation in oppo- 

 site directions. As these propagations must have independent 

 causes, it is allowable to suppose that in a particular case they 

 shall be exactly alike. There will then be one point at least, 

 where the particles have at every instant, in virtue of the two 

 propagations, equal velocities in opposite directions. At 

 this point, therefore, the resulting velocity is nothing, and 

 F (x — a t) +f{x + a/) — whatever be /. Let I be its 



distance 



