554 PROFESSOR TAIT ON MIRAGE. 
To draw the corresponding curve we may construct, for different values of 0, 
the set of curves 
; b ee. 
é =tee( 24/5 i ) Se ke ee 
2bi ees 
| 
or 
The ordinates of these curves are proportional to the reciprocals of those of 
a common catenary. 
Next construct, for the given value of a, the equilateral hyperbola whose 
equation is 
aie Jat+7?. 
Then we have, at once, for any given value of n, 
ee 
For the purpose of carrying out this process we have tabulated as below, a few 
rough numerical values :—and by the help of these the curve (4) has been 
drawn, along with (8), in three forms ; forb=2a, b=4a,and b=6a. See fig. 
2. In each case (4) is represented by a dotted curve, (3) by the corresponding 
full curve. 
soho Bgl JOPTC Ye Ok age a aya 
> log (+ 73 1: id 73 atR (ratio). 
0:0 oe) 1-0 0°5 05 
0:05 3°69 0:99 0-51 0:51 
0-1 2°99 0:99 0°51 051 
0:2 2°29 0:98 0:54 0:55 
0°3 1:87 0:95 0°58 0°61 
0:4 157 0-92 0:64 0°70 
05 1:32 0°87 071 0°82 
0°6 1:10 0:8 0°78 0:97 
07 0°89 0-71 0°86 1:20 
0°8 0°69 ° 06 0:94 1:56 
0-9 0:47 0:44 1:03 2°36 
10 0:0 0:0 112 ~ 
4. Let us digress to consider what we learn, in any case, from the form of 
the Locus of Vertices. 
It is obvious that if, instead of the special law of refractive index assumed 
in the preceding section, we had written quite generally 
w=fy) » 
(3) would have become 
i eee: ae : 
E=sonf, Fiaia ° : - (ey 
