212 Boota—On Fresne?'s Wave Surface and Surfaces related thereto. 
equation (9). Also following Mr. W. Roberts (13) enables us to obtain the nega- 
tive pedal with reference to any origin of the “surface of elasticity,” and also 
gives us with reference to any origin, the negative pedal of its parallel. If we 
apply this method of finding envelopes to the determination of the envelope of the 
plane f ? : 
OF zZ 
/ / — /2 anh} Be ] re * = 
wetyy +ee=ao+y + wher Po@tPopt Poe 0, 
t? being #?+y7"+2", we find two equations A? + B= 0, and C6 + D=0, where 
A = 0 and B = 0 are the two well-known Cartesian Equations of the Wave 
Surface, and C= 0 and D= 0 are 
ip y i Zuegh, ft ae 
(? as ay (7? us Ry ae 
and 
r being tbe radius vector of the wave surface, and p the perpendicular; in short, 
the process is substantially the same as the method of Archibald Smith. 
The solution of the following problem illustrates the method of finding 
envelopes, some account of which has been attempted in this Paper. 
The cone 2’ cota + 7’ cot’?B — 2?=0 intersects the sphere 2+ y7?+2=qa@ in 
a sphero-conic. Show that the equation of the tubular surface, which is the 
envelope of a sphere of constant radius /, whose centre moves along the sphero- 
conic, is had by equating to zero the discriminant of the following cubic in \:— 
2 
4a? 2’ sin’a " 4a’? sin?B Zz 
P? + 4a’? cosa =P? + 4a? cos*B Xd 
—1=0, 
where PHHP4+ 74240 —-Fh’. 
[See Unsolved Questions, ‘‘ Educational Times,” Reprint, No. 5447, vol. 55, 
page 136.] I leave it as an exercise to the reader. 
I ought to add that equation (9) is due to the late Rev. W. Roberts, M.A., 
sometime Fellow of Trinity College, Dublin, but responsibility for the solution 
above given rests on myself. 
