Involutes to a Circle. 299 



Thus, then, when the origin is at the centre, whilst for the ith 



involute to a circle the arc is any quantic \ d(f>F of degree 



(i+ 1) in </>, r 2 is a quantic in cf> of degree 2i of the particular 



form (G</>) 2 +(G'</>) 2 , where G((jf>) may be supposed, if we please, 



ds 

 to be any quantic of the order i or </> * ; and then -rr the radius of 



curvature at s, </>, is expressed by G<p -f G ! (f>f. 



It may be here noticed that substituting for <p, (f> + \, where 

 X is arbitrary, amounts only to a rotation of the curve through 

 the angle X, so that, as regards the intrinsic form of the curve, 

 no generality is sacrificed by imposing one condition upon the 

 coefficients in G<j>, or, if we please, in making any of the coeffi- 

 cients in it except the first to vanish. 



nitude to the two : — 



(!) g^jj. (2) r^ + (±)\ 



the last of them more familiarly known under the form 



The third and fourth equations show respectively that s and p are each 

 quantics in 4> ; the first gives the connexion between the constants which 

 enter into these quantics,, and the second, combined with the first, the relation 

 between s and r (in other words, the rectification of the curve), that between 

 r and 6 (where 6 is the vectorial angle) being contained in a fifth equation, 



<9 = d>+ sin-i£. 

 r 



* Accordingly we see that the spiral of Archimedes, as is well known, is 

 the locus of the feet of the perpendiculars upon the tangents to the first in- 

 volute from the centre of the circle ; and, much more generally, if we substi- 

 tute for each radius vector in this spiral any given quantic thereof, we obtain 

 the corresponding first pedal to an involute whose order of derivation is 

 the degree of the quantic. Ex. gr. by squaring or cubing the radius vector of 

 the spiral of Archimedes (of course leaving the vectorial angle unchanged), 

 we may form the pedal to particular species of the second and third invo- 

 lutes respectively. 



t Since the radius of curvature, radius vector, and perpendicular on the 

 tangent arc are all known rational integral functions of the same quantity, 

 it becomes a simple problem of elimination to determine the central force 

 competent to make a body describe an involute of any order to a circle. 

 Thus it will be found that the half-pitch second involute may be described 

 under the action of a central force varying as the inverse cube of the short- 

 est distance from the generating circle. So, again, the first involute may 

 be described under the action of a central force, the component of which in 

 the direction of the tangent to the generating circle (or say the centrifugal 

 force) varies as the inverse cube of this tangent, the centre of force in each 

 case being of course situated at the centre of the circle. 



