Involutes to a Circle. oOl 



learn that there may be i cusps to the ith involute, or any less 



d? s 

 number differing from i by an even integer. Also, since -t- = G', 



the number of apses (in regard to the centre) may be any num- 

 ber inferior to and differing from i by an odd integer*. Also, 

 since G represents the perpendicular on the tangent, the number 

 of points where the tangent passes through the centre will follow 

 the same law (although, of course, the two numbers need not be 

 equal) as the number of the cusps ; at such points the curve, 

 after winding round the centre (it may be through one or more 

 complete revolutions and a part of a revolution, or through only 

 a part), will change its curve and wind round in the opposite 

 direction ; for it is clear that there can be no points of inflexion, 



ds 

 since -z-r can never become infinite. These points last named 



may be termed points of retrocession. The cusps, of course, can 

 only exist at points where the involute meets the parent curve. 



Between any two cusps of an involute evidently must be 

 comprised an odd number of the cusps of its parent curve; 

 but, of course, not vice versa-, thus, e. g., in the second invo- 

 lute, if there are no cusps, it will easily be seen that the curve 

 possesses a simple loop enclosing the cusp of the first involute (its 

 evolute), and consequently cutting the two branches of the latter, 

 and so in general the disappearance of consecutive cusps in any 

 involute will give rise to loops enclosing those cusps of the 

 parent curve on the branches adjoining to which (on each side) 

 cusps of the derived curve are wanting ; (by a branch, I mean, of 

 course, the portion of curve included between any two cusps, or 

 between either of the two terminal cusps and infinity ;) whether 

 the absence of cusps of the involute on 2i consecutive branches 

 of the parent curve implies the necessary existence of i distinct 

 loops, one round every alternate one of the 2i — 1 cusps in which 

 those branches meet, requires further consideration, It is clear 

 that in an analytical sense the length of the arc of the parent 

 curve included between any two cusps of the second curve must be 

 taken as zero; the correct view (at least, for the purposes of this 

 theory) being that the angle of convergence continually increases 

 or decreases up to positive or negative infinity as we pass in one 



* Thus we see that the apsidal distances from the centre are the arithme- 

 tical magnitudes of the roots of the equation formed by equating to zero 

 the discriminant of G-fr=0, which is of course of the degree (i— 1) in r. 



If we consider the apses and cusps of any involute to form a combined 

 group, an odd number of the points of this group will always be included 

 between any two intersections of the curve with a circle concentric with the 

 parent circle; for the limiting equation to G 2 +G' 2 — r 2 ==0 is G'(G+G") = 0. 

 Every point in this combined group is a point of maximum or minimum 

 elongation from the centre. 



