1826.] Parameters to the Determination oj certain Cttroes. 325 



The above are the equations to the involute of a caustic by 

 reflexion. 



Example 3. — If a plane curve roll on another similar and equal 

 curve to determine the path of a given point in the rolling curve. 



Let there be taken for origin a point in the fixed similarly 

 situated to the given point in the rolling curve. Whence we 

 have 



(X - xY- + (Y - yf =-. -r^ + ^ 



Or, X^' + Y''-2{Kx + Yi/) = (1) 



And since the principle we have demonstrated is manifestly 

 applicable to this case, we get differentiating with respect to x 

 and w. 



Xdx+Y dy = (2) 



Thus if the curves in question be rectangular hyperbolas and 

 the generating point, the centre of the rolling hyperbola 



x* _ 3^2 _ ^1 



.•. d i/ = -dx 



and by (2) Xi/ + Y x = 



/.X Cr^-a*H+ Yi = 0. 

 Hence we have ^ 



Xa 



*' ~ (X-^ - Y«)* 

 Y a 



y — (Y^ - Y»)^ 



,*. by Equation (1), observing that y is greater than Y, 



\2 4. Y- ss 2 (X- a - Y^ a) 

 (X- - Y-)^ 



.-. X- + Y' = 2 a (X» - Y'-=)^ 

 the equation to a leminiscata having double the axis of the 

 hyperbola. 



The Equations (1) and (2) are precisely those found for the 

 involute of the caustic by reflexion, we have, therefore, the 

 following very singular property. 



If a curve similar and equal to the reflecting curve be made to 

 roll in its plane and upon itsperiphery (similar points having first 

 been in contact), then a point similarly situated to the radiating 

 point will trace out the involute to its caustic. The above pro- 

 perty may be demonstrated very simply on principles purely 

 geometrical. The following are examples of its application. 



If a parabola be made to roll on another equal parabola, the 

 vertex of the former curve will generate the cissoid of Diodes, 

 the caustic formed by rays diverging from the vertex of the latter 

 curve, and reflected at its periphery, is, therefore, the involute to 

 the cissoid, and similarly the caustic formed by rays diverging 

 from the centre of an hyperbola is the involute to a leminiscata 

 of double the axis. . 



