304 Mr. J. J. Sylvester's Note on Successive 



we have the transition case, for which 



and the arco-radial equation becomes 



Another and more remarkable case occurs when G 2 + G' 2 becomes 

 a perfect square, for then the degrees in s and r will sink to 



half their usual values: this occurs when &=— -, which is the 



case of a looped curve bisecting at its apse the radius drawn 

 from the centre of the circle to the generating first involute. 

 We have in that case 



6=1(^-1), 

 r-J^+lJorf-^/EE* 



9as s =(2r— fl)(?-+a) s , 



a form even simpler than that of the first involute. 

 We may write this equation under the form 



V r V i« 



or, turning round the line from which is reckoned through a 

 quarter of a revolution, 



whence 

 and 



»-»-vW r -i? : 



Let now 



0=2$, %.r=p* 



(which is the same thing as if for x and y we substituted x*—\f 

 and 2ocy), then 



2 



d— sin- 1 .— - + 

 P 



a-V"-©" 



