[652 
71 
= r 2 ) the 
radius r. 
die values 
r — 1, we 
‘S to \nt, 
)tained by 
3) cos 26; 
^ 2 ] ON A SEXTIC TORSE, 
or, what is the same thing, it is 
rV - 2e (3tf- - 2) (a? -,/)+ (.V - 2)‘ - (4 (»•« - 1)* + (3r‘ - 2) a ) = 0, 
viz. the term in { } being ?- 4 (4?’ 2 -3), this is 
r 4 * 2 - 2 z (3 r 2 - 2) (x 2 - if) + (Sr 2 - 2) 2 - W (4r 2 - 3) = 0, 
or say 
0 2 O 2 + iff - 2 z (Sx 2 + 3 if - 2) (x 2 - if) + (Sx 2 + Sif - 2 ) 2 - 4 x>if (4# 2 + 4 f - 3) = 0. 
This may also be written 
{2 (x 2 - if) - 3 x 2 - Sif + 2j 2 + 4 ay (z 2 - 4a? - 4 y 2 + 3) = 0, 
a form which puts in evidence the nodal curves 
x = 0, xif = — 3f + 2, and y = 0, zoo 2 — Sx 2 — 2. 
It shows also that the quadric cone 2 2 - 4x 2 - 4y 2 + 3 = 0 touches the surface along 
the curve of intersection with the surface 2 (x 2 — y 2 ) - 3 (x 2 + y 2 ) + 2=0. This is, in 
fact, the curve of maxima and minima of the cylindrical sections, viz. reverting to the 
form f(4r 2 -S)cos(26 +/3), or, if for greater clearness, attending only to one sheet 
of the surface, we write it z = f(4r 2 — S) cos (26 —/3), we have a maximum, z = f(4r 2 — S), 
for 26 = ft (or 27r + /3), giving 
Sv 2 — 2 3r 2 — 2 
cos 20=cos /3, = —, = —-—: 
r 2 y(4?’- — 3) r 2 z 
and a minimum, z = — f(4r 2 — 3), for 2#=7t + /3 (or 37r -f ¡3), giving 
3r 2 — 2 3r 2 — 2 
cos 26 = — cos /3 = — 
r 2 V(4r 2 - 3) ’ 
r 2 z 
viz. the locus is z 2 = 4(r' 2 — S), z(x 2 — y 2 ) — 3^—2; and for # = \/(4r 2 — 3)cos(20 + /3) we 
find the same locus, viz. the equations of the locus are 
z 2 — 4# 2 — 4y 2 +3=0, z(x 2 — y 2 ) — Sx 2 — 3 y 2 + 2 = 0, 
as above. 
To put in evidence the cuspidal edge, write for a moment ^—z — a? + y 2 , the 
equation becomes 
{? (x 2 - y 2 ) + (r 2 - 1) (r 2 - 2) - 4 x 2 y 2 } 2 + 4 x 2 f f 2 + 2£(x 2 - if) + (r 2 - 1) (r 2 - 3) - 4 x 2 y 2 } = 0 ; 
viz. this is 
^r 4 + 2£(# 2 - y 2 ) (r 2 -1) (r 2 - 2) + (r 2 - l) 2 (r 2 - 2) 2 - 4x 2 y 2 (r 2 - l) 2 = 0, 
or writing the last term thereof in the form 
- {r 2 - (x 2 - y 2 f \ (‘r 2 - l) 2 , 
and then putting r 2 — 1 + U, the equation is 
£ 2 (1 + 2 U+U 2 ) + 2£U(U- 1) 0v 2 - if) + U 2 (U-If - U 2 {(U+ l) 2 - (a? - y 2 ) 2 } = 0; 
viz. this is y*)Y + 2U{? + SU(x 2 - f) - 2 U 2 } + ?U 2 = 0,