(•(a-4»)) 



Again, its slope is given by 



dx (ia) * 



Sound Waves in the Atmosphere . 113 



The equation of the ascending portion of the ray whose 

 vertex is (f, f) is 



•- 2f -^'cSiB^' • • • (43) 



and the envelope when it exists is given parametrically by 

 this equation and 



0=f + Wa >f'^a* • . (44) 



It is clear that a value of z can actually be found to satisfy 

 (41) provided d£/dg is negative. The envelope has the same 

 general appearance as the curve of vertices. Thus if f— >oo 

 as £— >0 (last row above) then the envelope also is asym- 

 ptotic to z — Q; whilst if J has a finite limit (o)'(0)=0 > 

 o>"(0)gfcO) it is found that along the envelope 



r * 2o>" / ;0) r 01 . t v dx 



hm V2 =q -TTTkr -, lim a? = 2 lim f, hm-^ =0. 



As an example, we may consider in detail the case 



» (a) =ia(&o + &!«*)■ 



Inserting in (11) we find that the curve of vertices is 

 given by 



MS"-*)*- A, (46) 



The envelope is given by 



-^-f^ •;••■• (47) 



0= _ („-2)A?-i» + i«?»- 1 ("jJS^ji- («) 

 The last reduces to 



where A= 1 



Jo 



A 



92 



r — ^_ 



Jo (1-^' 

 so that z/f is a constant, say \, determined by 

 f* ^ _2(n-2)r i __^_ 



Jo (i-<»)»" » Jo a-< n )* 



P*tf. 21%. S. G. Vol. 12. No. 247. July 1921. 



