therefore, by subtraction, 
¡3 (6 a 2 + a a + b y 2 ) = 0 ; 
and rejecting the value (3 = 0 which would make the moveable 
line always pass through the origin and so generate an oblique 
cone, we have 
a 2 -f y 2 + — a = 0...(2), 
by virtue of which, either of the equations (l) gives 
a a (b 2 - (3 2 ) = b (r 2 - a 2 )... (3) ; 
and it remains to eliminate a, ¡3, y between (2) and (3) and 
the equations to the generating line. Substituting the values 
of a and y given by the latter in (2) we find 
h co 2 + z 2 ax 
p = y + , and then a = — 
a x 
therefore, substituting these values of a and ¡3 in (3) and re 
ducing, we find for the equation to the surface 
\axy + h (x s + # 2 ) | 2 = b 2 r 2 x 2 + b 2 z 2 (r 2 — a 2 ), 
2. To find the equation to the developable surface gene 
rated by a straight line which constantly touches a Helix. 
The equations to the Helix being 
# . z 
x = a cos —, y = a sin — 
na na 
at a point for which z = a the equations to its tangent are 
a 1 . a 
x - a cos — = sin — (z - a), 
n na 
. a 1 a 
y - a sin — = — cos — (z — a) ; 
na n na 
« .a , a a v x* + y* — a* — xy 
x cos f- y sin — = a, and tan — = 
na na na y* ~ a