Caption: Mathematical Models Catalog of a Collection of Models of Ruled Surfaces
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22 where l,m,'?i, are fixed quantities known, if the direction of the generating line be known ; but a,fi,and y are variable quantities depending upon the director curve. W e may write the equations as x y a fi I m I m m n m n a fi y Of the three quantities - - ~, one is absolutely arbitrary, as j we only have to do with their differences. We may therefore consider aft to be a function of 7 m n I m the particular form of the function depending upon the character of the director. W e may therefore write the equation of cylindrical surfaces as / 3 y 0 m \m nj in which F i a functional symbol. s It might seem at f r t sight that the common equation of is a right cylinder x2, + y2 = o? is not of this form. But any change of variables such as will introduce the third variable z will at once reduce i to t this form. Conical Surfaces. A right line always passes through a fixed point. Calling the point afi7j i s equations are t X— a 1 whence y — fi __ z •— y m n X— a I y — fi __ m y •—fi~~ m ' z — y ~~ n ~ I 11 7 and — must be some function of —. This gives m n °
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