Equations of the Second Order. 33 



Let x 3 , y' be the coordinates of that point of the evolute which 

 is the centre of the curvature at the point of the involute whose 

 coordinates are x, y. Then we have the known equations 



2/-y = -^-, a:-x/=z -p(y-y<). 



Since the equation A=0 gives 1 +p*=qy, it follows that ?/=2^, 



da/ 

 and x—x 3 =py. Consequently, because^}= ^— n we obtain 



_ t ?/ da;' 

 X - X ~2 t dy r 

 Therefore 



dy dy*_ dx 3 



di/ 

 Hence it will be readily found, putting// for -j- ti that 



dy 1 dp 1 _ 



y> + ^(l+i/ 2 }~^ 



which equation gives, by integrating, 



^'=±dy'(f-lf ! 

 k being the arbitrary constant. Consequently, by integrating 



'+>-±scs-y-imf±<s-tf> 



w 



which is the equation of the evolute. From this equation the 

 differential equation of the involutes may be found as follows : — 



-U 2 ) dy'" dx~ 1% 

 Hence 



Consequently, since x 3 = x+p(y— y 1 ) —x-\-p(y— k\/l +P" 2 ), we 

 get by substituting in the equation of the evolute, 



kp 



.r-fc^-^+^v/l+^+glog^i+^+i?). . (/3) 



This is the required equation, which, as might be inferred from 

 the reasoning, embraces all the involutes of the curve whose 

 equation is (a). It will, however, be proper to substantiate this 

 inference by independent reasoning ; which I now propose to do. 

 Phil. Mag. S. 4. Vol. 4.2. No. 277. July 1871 . D 



