651 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
Hence we find without difficulty the tangent to be given by the equation 
x cos 3 <p-\-y sin 3 <p=z ; (H?) 
and this is therefore the equation of a tangent to a nodal line of A. 
168. If from any point of the curve four tangents he drawn , the points of 
contact are in a right line. 
Demonstration . We shall simplify the proof by taking z = unity. Let x' y' he the 
point of contact, then the tangent is 
and if (a (3) be a point where this meets the curve again, we have the equations 
iL. A = 
I 3 ^ y 13 
■ 1, * . 
• (1) 
+ 
"^1 
II 
=1,. . 
• (2) 
x lil y' C l 
1. . . 
. .(3) 
Hence from (1), (3) . . 
ct-rf 
x 13 
, P-y 
1- y!3 
0, . . 
• (4) 
,, „ (2), (3) . . 
a 2 — P 2 
, P°~-v n 
0 
. (5) 
aV 2 
1 /3 V 2 " 
— u, . . 
„ „ (4) and (5) . 
cr 
L P 
= 0, 
(cc + x')x' 
1 (P+y'W 
or (f3x'—ccy')((3x'-\-ay , —ci{3)=0. Therefore the line j3x'-\-ay' — aj3~0 passes through 
the points of contact, and the proposition is proved. 
Cor. 1. The envelope of the line through the points of contact is a conic section ; for if 
XV . ..11 
we seek the envelope of — 4-^=1, subject to the condition ^ 2 +^ = 1, we get the conic 
section 
a?+f=l. 
The reader is not to imagine from its form that this equation represents a circle. 
Cor. 2. The anliarmonic ratio is constant of the four points in which the chord of 
contact meets the curve. This follows at once by considering the pencil of four tangents 
from a point infinitely near the former one. 
Cor. 3. If four tangents he drawn to the evolute of a conic at the points where any 
tangent of the evolute meets it , these four tangents are concurrent , and the locus of their 
points of concurrence is a conic passing through the six cusps of the evolute. 
169. In the sphero-quartic WU, if P, P' be inverse points with regard to one of its 
MDCCCLXXI. 4 U 
