164 Dr. E. H. Barton on the 



from (5), (G), and (9). Thus, dropping subscripts, we have 

 cosec0 = cosec# o — oy'l 



or cosecu = z = o — ay J 



where 2 = cosec0 o — a #> an( ^ 6 = cosec0 o ; 



also tan <j>=-~r = tan 0-f- (c + a?/) sec 0, . . (11) 



whence f tfy C{c + ay)zdy (%i) \ 



X = 1 — ' + 1 , . • . . ( I &) 



The first integral here represents the catenary found by 

 Lord Bayleigh for the path of the ray when taken at right 

 angles to the wave-front. The second integral is a small 

 correction for the obliquity of the ray, and arises from the 

 second term on the right side of equation (6). On evaluation 

 (12) yields 



2ax= \b + 2c) >JW^l- (6 + 2c + ay)>/i^l + log e — ^^i. (13) 



z + sJz^ — \ 



Thus, for any given ordinate, the abscissa is the sum of 

 three terms, the first being a constant, the second forming 

 with y a curve of the fourth degree, while the third is the 

 abscissa of a catenary. This is the general expression ex- 

 hibiting the relation between x and y, and therefore completes 

 the required solution. In considering various special cases 

 it will, however, often be simpler to go back to equations 

 (10) and (11). 



Hay with Horizontal Wave-Front. — We now have 6 =0, 

 £> = cosec # =G© j and (13) becomes indeterminate ; so, either 

 by evaluating it or by use of (10) and (11), we obtain 



2x = 2cy + ay' 2 , (34) 



i. e., the path of the ray is the parabola 



If the ray starts in still air, put c = 0, and we obtain 



y*=2*/a, (16) 



a parabola with vertex at the origin. 



Numerical Illustrations. — To exhibit the two phenomena of 

 total reflexion and the parabolic path of rays with total 

 reflexion impossible, take the following numerical data. 

 Let the region be specified by equation (9), where c = 002 

 and o=0-0001, i.e., 



speed of wind 



speed or sound * 7 ' v ' 



y being expressed in feet. Then, if the temperature is such 

 as to make the speed of sound 1100 feet/second, the wind- 

 speed at the origin is 22 ft./sec, = 15 miles per hour. 

 \Vhereas at 1000 feet high the speed of the wind would be 

 90 miles per hour. In this region let sound at the origin of 



