Involutes to a Circle. 305 



This is the polar equation to a known curve (of the kind used 

 by Captain Moncricff in his barbette gun-carriage). It is of the 

 class of curves generated by a fixed point on a wheel rolling on 

 a plane. Such a curve may be termed the convolute of a circle of 

 a pitch denoted by the ratio of the distance of the fixed point 

 below the centre to the radius of the revolving circle ; thus a 

 convolute of zero-pitch is the spiral of Archimedes, a convolute 

 of unit pitch the first involute to the circle : the general 

 equation to a convolute, when the distance below the centre is d 

 and the radius a, is given by the Rev. James White in the last 

 September Number of the Educational Series, and is easily 

 shown to be 



^sm-i/i + ^ZE?. 

 P a 



Similarly, we may define the pitch of the second involute to be 

 the ratio of the distance of its apse from the centre to the radius; 

 and then we are conducted to the observation that whilst the 

 convolute of full pitch is the first involute, the convolute of half 

 pitch, on applying to it one of the simplest forms of M. Chasles's 

 or Mr. Roberts's method of transformation (given in Dr. Salmon's 

 ' Higher Plane Curves/ p. 236), viz. doubling the vectorial angle 

 and squaring the radius vector, becomes converted into the second 

 involute of half pitch. Since for this curve 



& / » o t \ ds 

 r= g (*» + l)=# 



we see that it may be completely defined, without reference to 

 any theory of involutes, as the curve whose radius of curvature 

 at any point is equal to its radius vector reckoned from a given 

 origin. It is the curve which completely satisfies the equation 



dr 

 rd cos -1 -j- =s, the two arbitrary parameters which the complete 



uS 



integral of this equation should contain being furnished by the 

 linear magnitude and angle of swing of the curve round the given 

 origin*. 



* This evolute possesses the property, which serves to characterize it 

 completely, of cutting the originating circle (its second evolute) orthogo- 

 nally. For when r 2 = a 2 , G 2 =0, i. e. the tangent to the curve passes through 

 the centre. Moreover, since G = gives 0=1, it follows that the curve 

 cuts out of the circle an arc equal in length to the diameter. Summarizing 

 such of its principal properties as have fallen in our way, we see that it 

 bisects the line joining the centre of the originating circle and the cusp of 

 the first involute ; that it cuts the said circle orthogonally ; that its radius 



Phil Mag. S.4. Vol. 36. No. 243. Oct. 1868. X 



