322 MrlMoseley on a new Application of the Method of [May, 

 Therefore, by the former, 



dx dy Ci.x ^ ' ^ 



Now let the tangents move up to one another until they 

 become consecutive, and ultimately intersect in the point (X,Y,) 

 of the required curve.* 



Then since A x and A y are evanescent, and X', Y', become 



X and Y, we have, 



rfj; £F £j/ _ Q 



d X d y dx 



.'.-J- dx + -r dy — 



d X d y ^ 



The above is the Equation (2). 



Also by the Equation F X' Y' a- j/ = 0, we have 



FXYa;y=0 

 which is the Equation (1); and (3) is given by hypothesis. — 

 Q. E. D. 



Example 2. — To investigate general forraulse for the determi- 

 nation of the involute to a curve. 



Let (X, Y,) ix, y,) be corresponding points in the curve and 

 its involute; and (s) the length of curve measured from the 

 common origin to the point {x, y,) ; then evidently 



s^ = (X -{-xf + {Y -yf (!) 



differentiating with respect to x, y, and s (a function of these 

 variables). 



n+ (ii)y=_i_ — . — iz- (2) 



Eliminating x and y between these equations and the given 

 Equation^ — f x of thee volute, we have the required equatiott 

 (in X and Y) to the involute. 



Thus in the cycloid, since 



s^ = 8 a X 



8 « a; = (X + x)* + (Y - yf 



4 a = (X + T) + (i/ - Y) Jl 



Also ^ = ir-^^i 



dx \ X / 



* The cun'emay evidently be considered as the locus of the ultimate intersections of 

 its consecutive tangents. 



