of the Motion of Fluids, Sf-c. 403 



Also we can now shew why a disturbance, independently of its 

 particular nature, produces propagations in opposite directions 

 from the place of its application. For either of the equations 



w = a hyp. log p, w = — a hyp. log. p, 



proves that where the fluid is condensed, the velocity is in the 

 direction of propagation ; where it is rarefied, in the opposite 

 direction. Now every disturbance will condense the fluid in one 

 part, and rarefy it as much at another, but will impress motion 

 at all parts in the direction in which the impression is given; 

 therefore the same disturbance will cause propagations in oppo- 

 site directions. 



6. Recurring now to the complete integral, we may infer 

 from it, that two propagations will obtain simultaneously in oppo- 

 site directions. It is not possible, as it seems, to obtain this 

 integral in exact terms: let us suppose that, 



2a(D = f{s — at — w^t) — F{s + at — w„t), 

 2a- hyp. \og.p = f{s — at - to^t) + F(s + at — oi„t); 



in which a>i, Wi, are unknown functions of s and t. If the two 

 opposite propagations be exactly alike, there will be one point 

 at least where the velocities due to them respectively will be 

 equal and opposite, and where consequently the resulting velocity 

 will be nothing whatever be the time. Now at this point w, 



will be equal to - cu,, whatever these functions may be, becau.se 



we may presume that the value of-yr, obtained by differentiating 



s--at — u>it-c, on the supposition that (u, is constant, will differ 

 only in sign from that which is obtained from s + at - w„t = c^, 

 on a like supposition with respect to w„. Let therefore m = the 

 distance of the point of rest from the origin of s, 



and (a + a),)<, or {a ~ w.^t = z. 



