212 Mr. Tredgold 07i the Nature of the Curves 



fixed points in another plane, and let the latter plane be moved, 

 so that while one of the points moves along one of the straight 

 luaes, the other point may move along the other straight line : 

 it is required to determine the nature of the line that v/ould 

 be described by a tracer fixed in any part of the moveable 

 plane ? 



Let AD and DB 

 be the two straight 

 lines on the fixed 

 plane; and A and B 

 the two points on 

 the moveable plane; 

 and, in the first 

 place, suppose the 

 tracing point C to 

 be situate at any 

 point in a straight 

 line passing through 

 the points AB. Put 

 AB = a; AC = na; 

 A G= s; and denote 

 the angle ADG by fl ; the abscissa and ordinate of the line 

 described being, m any position of the point C, denoted by x 

 and 3/. 



Now, CE = wfl4-GD— ^'; and 



{na-^GI)—xy = n^a^-{s+i/y. 



The sign + applying to the case where the tracing point C 

 is out beyond B. 



x = )ia-\-GT)- 



Hence, 

 But, 



y 



a; 7ia :: s: s jf-j/; or 5 = 



Also, GD= — ^r; consequently GD = --— ~ — -. 



' - tan. ^ 1 J (l'^H)tan.^ 



Therefore, a' = na+-J^^-— - /^Vi^-yf-l- l^lV. 



If we suppose the ordinates to be taken parallel to DA, then 

 GD=0; and. 



Hence it appears that the conic ellipse will be described by 

 the tracer, if it be placed any where in the straight line passing 

 through the points A B : excepting when placed so as to de- 

 scribe either a circle or a straight line. 



A circle 



