72 
ON A SEXTIC TORSE. 
[652 
showing the cuspidal edge £=0, U = 0, viz. z— x 2 — y 2 , x 2 -f- y 2 = 1. Moreover, along the 
cuspidal edge the surface is touched by £ — U {x 2 — y 2 ) = 0, that is, by z — (x* — y 4 ) = 0 ; 
and at the points where this tangent surface again meets the surface we have 
(¿r 2 — y 2 ) 2 (x 2 -f y 1 + 3) — 4 = 0; viz. the surface contains upon itself the curve represented 
by this last equation, and z — (od 4 — y 4 ~) — 0. 
As a verification, in the form 
[z (x 2 — y 2 ) — 3# 2 — 3y 2 + 2} 2 + 4x 2 y 2 (z 2 — 4x 2 — 4y 2 + 3) = 0 
of the equation of the surface, write z = sc 4 — y 4 . If for a moment x 2 + y 2 = X, x 2 — y 2 = ¡x, 
then the value of z is z = X/x, and the equation becomes 
that is, 
('X/x 2 - 3\ + 2) 2 + (X 2 - y 2 ) (X 2 fx 2 - 4\ + 3) = 0, 
y 2 (X 4 - 6A 2 + 8X- 3) - 4X 3 + 12X 2 - 12A + 4 = 0 ; 
or, what is the same thing, 
(\-l) 3 {[L 2 (A, + 3) — 4) = 0, 
so that we have (\ —1) 3 = 0, or else /x 2 (X + 3) — 4 = 0 ; viz. (x 2 +y 2 — l) 3 = 0, or else 
(or — y 2 ) 2 (x 2 + y 2 + 3) — 4 = 0, agreeing with the former result. 
In polar coordinates, the surface is touched along the cuspidal edge by the surface 
z = r 4 cos 26, and where this again meets the surface we have r 4 (r 2 + 3) cos 2 20 — 4 — 0. 
For the model, taking the unit to be 1 inch, I suppose that for the edge of 
regression we have 
x = 2 cos (f>, y = 2 sin cf>, z = 5 + (’45) cos 2cf); 
viz. the curve is situate on a cylinder radius 2 inches. And I construct in zinc-plate 
the cylindric sections, or say the templets, for one sheet of the surface, for the several 
radii 2, 3,.., 8 inches; taking the radius as k inches, the circumference of the cylinder, 
or entire base of the flattened templet, is = 2&7t ; and the altitude, writing 20 in place 
of 2# — /3 as above, is given by the formula z = 5 + (/45) \f(k 2 — 3) cos 20, so that the 
half altitude of the wave is =( - 45)\/(& 2 — 3); having this value, the curve is at once 
constructed geometrically, 
then are 
We 
have, moreover, cos /3 = 
3 k 2 - 8 
ifc 2 - 3) ’ 
the numerical values 
k 
2kir 
(-45) V(¿ 2 - 3) 
CO 
1 
00 
№ 
k 2 V(k 2 - 3) 
2 
12-57 
0-45 
100 
0° 
3 
18-85 
110 
•86 
15 
4 
25-13 
1-62 
•69 
23 
5 
31-42 
211 
•57 
6 
37-70 
2-59 
•48 
m 
7 
43-98 
3-05 
•42 
32¿ 
8 
50-27 
3-51 
•36 
34 
the altitudes in the successive templets being thus included between the limits 5 + 0’45, 
5± 110,.., 5 ±351.