4?64- Prof. Challis on the Nature of Aerial Vibrations. 



As the integration, for reasons I have elsewhere insisted 

 upon, must generally be taken along a line of motion, the 

 arbitrary function added contains the co-ordinates a v 6„ c l of 

 a given point of this line. Supposing ndx + vdy + wdz to be 



an exact differential (tty), we have V==~. Hence 



d& V 2 

 a 2 .Nap.log P + g + — =F(a l) b li c l ,t). . (1.) 



In the instance of motion before us, the axis of z is a line 

 of motion. Also 



v Jf> dt J dt y 



and for a point on that line 

 Consequently, 



V= — =f& 

 dz J dz 



« 2 Nap.lo g ^+/.f +^-^=F(« I ,6 1 ,c 1 ,0. • (2.) 



I shall now proceed to investigate a general equation which 

 must be satisfied whenever a given state of density is propa- 

 gated with a constant velocity, for the purpose of ascertaining 

 whether the propagation along the axis of the ray-vibrations 

 obeys this law. If a x = the constant rate of propagation, and 

 ds be the increment of a line 5 drawn in the direction of pro- 

 pagation, it is clear that the following equation must be satisfied, 



f # j =°> 



because the integral of this equation is p = F(s—a 1 t). Now 



dp d.pu d.pv d.pia __ 



dt dx dy dz 



Hence 



^£H + ( Lfl + i^-- a 4 ^o. . . (3.) 

 dx dy dz ' ds " . * 



This is the general equation sought. For a first applica- 

 tion, let us suppose the motion to be that of plane-waves pro- 

 pagated in the direction of z. The equation for this case 

 becomes 



d.piso dp 



-t -"' ah - 



or 



dw . .dp 



