157 
177* To find the equation to the surface generated by 
a straight line subject to constantly pass through three given 
fixed curves. 
It is easily seen that the motion of the generating line 
will be completely determined. For conceive any point in 
one of the fixed curves to be the common vertex of two 
conical surfaces, having the other two fixed curves for their 
directrices; then these surfaces will intersect one another in 
a finite number of straight lines, each of which passes through 
the three fixed curves and is therefore a position of the gene 
rating line; and as we take fresh points for the vertex, the 
successive generating lines belonging to the same sheet will 
pass through points contiguous to one another on the three 
directrices. Also the surface will generally be twisted, because 
the tangents to the three directrices at the points where a 
generating line meets them, will not, except in very particular 
cases, be in the same plane. 
Let the equations to the generating line be 
x = a% + y, y = (3% + <5, 
containing four variable parameters, a, (3, y, S; and, as 
explained in Art. 173, let the three relations among the 
parameters necessary for the line passing through the three 
given directrices, be obtained and reduced to the form 
(3 = (p (a), y = (a), S = 7T (a) ; 
therefore, substituting for ¡3, -y, S in the equations to the 
generating line, we have 
x = a% + \f/ (a), y = %<p (a) + tt (a) ; 
and it remains to eliminate « between these equations, in 
order to get the equation to the surface; but this cannot 
be done without particularizing the functions, or the direc 
trices on which they depend. Therefore we must retain the 
above system of equations to represent this family of surfaces; 
regarding a as an indeterminate quantity to be eliminated, 
after the forms of the functions in each individual case have 
been determined.