Boota—On Fresnel’s Wave Surface and Surfaces related thereto. 207 
These four planes occur also if we write the equation of the surface thus : 
(#7 +/P+2yP—(e +/P4+2)[(P+e)Pt+(C+@)P4+0C+h 2") 
+02 + CCP +02 = 
? 
or P—plP+e)P+(C+e@)m + (C+P)n7)4 00? + wm + abn? = 0, 
where /, m, and » are direction cosines. Now expressing the condition that both 
values of p* are equal, we have the locus 
[(& +c’) ae ab (e+ a’) y? + (a + 0°) EAR ih (4 +y/° + zo) (Pera? + ary? + Coe) = 0, 
and the left-hand side of this equation is the product P,P,P;P, with the sign 
changed. The singular points on V = 0 are easily determined, and also the 
equation of a singular tangent cone at a singular point. 
It is well-known that V = 0 is the first positive pedal of W=0; it is required 
to find the first positive pedal of V=0; that is, the second positive pedal of W=0; 
that is, find the envelope of the sphere 
a’ ae ee ema in che 
(2-5) +9 5) + (2 5) = aie’ 
or of P+ P+ — ave — yy — 22 =0, 
where 2’y'z’ are connected by the relation 
gy? Oe 2? 
Poa t pont poaq 
where ¢?= 27? + y"+2"3 then 
Qz2/ ape ye 2? 
Re ye | kes eee Ee Sn ae thoes 
with similar equations for y and z. Now it is known that 
i bs MES y' uy) fing eit 
Je = GR 7 Sar a (Hd Te a os 
where €, 7, ¢ and 7, on W= 0 correspond to 2’y'z't’ on V=0. Hence 
1 1 
—-2%=2d27 |=— - aa 
: # lane P| 
with two other equations. Now puttmg #=2*+y?+2, and remembering that 
ir = t?, from the doctrine of pedal surfaces, it follows that 
All 1 e 
er eee | 
ON i 2 a fo Hh 
wi w= 2a | V—@r—t | 
(3 =" a’)(¢? oe t*) t? 
