Successive Involutes to Circles. 461 



reasoning from general mental impressions in the absence of all 

 suggestive visible representation of geometrical forms *. 



Let us compare the cusped cyciodes of the second order in 

 figs. 2, 4, where the tail-f is negative, with the natural one in 

 fig. 1, for which this tail is zero. It is interesting and instruc- 

 tive to trace the passage from the one to the other by following 

 the fortunes of the double tangents, which in the figures 2 and 

 4 will be seen to connect the sort of Moorish arch that consti- 

 tutes the middle finite branch with each of the adjoining infinite 

 branches. 



How is either of such double tangents to be determined ? At 

 the two points where it meets the curve the angle of contingence 

 is not the same, but has increased by 180° in passing from the one 

 to the other. Accordingly p, the perpendicular, if represented 

 by ¥<f) when the double tangent is regarded as a tangent at 

 one point, will be represented by — F(0 + 7r) when that line is 

 regarded as a tangent at the other point of contact, and the equa- 

 tion for finding <£ becomes F<£ + F(0 + 7r) = O. Thus we see 

 incidentally that p must become zero, i. e. that a point of radia- 

 tion must necessarily exist somewhere between the two points of 

 contact. And here I may remark incidentally that this throws 

 light on the notable equation, applicable to any curves whatever, 



ds _ d q p 



d$~ P + d^ ; 



for -r^2> by virtue of the remark made in a footnote to the former 



paper on this subject, is the perpendicular to a tangent to the 

 second evolute at a point corresponding to that, in the curve 

 itself, for which p is the perpendicular to the tangent; but at 

 corresponding points in a curve and its second evolute, the tan- 

 gents, although parallel in direction, are opposite inflow. Hence 



d T) If) 



p+ ~2 ( anc ^ notjo— vfg) is the distance between these two 



tangents ; and it is obvious that such distance is identical with 

 the radius of curvature corresponding to the perpendicular p ; so 

 that, viewed in this light, the differential equation above written 

 is reduced to a truism. Returning to our cyclode of the second 



* I believe I am correct in saying that in like manner a mistake made by 

 Steiner in his description of a surface viewed only by himself " in the 

 depths of his inner consciousness," was first discovered by Professor Rum- 

 mer after the construction of an actual model. So impossible is it to prove 

 demonstration, and to make oneself absolutely safe against the fallacy of ig- 

 noring entities on the one hand, or unduly assuming their existence on the 

 other. 



t In general the tail is the distance of the cusp of the first involute 

 from the corresponding points of the involutes successively engendered 

 therefrom. 



