302 G. H. KNIBBS. 



except by motion across the surface, or a solid from a surface 

 except by motion in it, are essentially impossible, but the generated 

 figure may be understood as representing the continuum as a row 

 of points may be regarded as representing a line. 



29. Involutional, evolutional, pedal, modular and umbilical 

 generation. — The terminal K of a string, unwound from the curve 

 KhL, Fig. 25, would trace out the dotted line Kkk'M, the involute 

 of the curve; in a similar way KM', the involute of KL' could be 

 defined. In the latter case if the length of the string were Kf, 

 or KO, instead of being originally zero, the involute would be ff' 

 or O'S. 1 Suppose the line hk, which is obviously equal to the hK, 

 to vary in any definite way, i.e. as some continuous function of 

 the angle through which it turns, the point K (or f, 0', or O) may 

 be said to generate a curve involutionally : such a scheme of 

 generation would obviously be continuous. 



On the other hand imagine a point, to which a string OKY is 

 attached, to move along the curve O'S, and to be kept perpendicular 

 to the tangent to the curve of the moving point; the successive 

 positions of the string would envelope the curve KL', called its 

 evolute? If the distance along the straight line which envelopes 

 the evolute, is any continuous function of its total length, measured 

 from curve to evolute, or if the length be any continuous function 

 of the angle through which the string turns, the curve traced by 

 the terminal may be said to be evolutionally generated. This also 

 is a continuous scheme of generation. 



Again let a line KG, Fig. 25, make some definite angle with 

 the radius of curvature 3 — it need not necessarily touch the curve 

 Kk'M — and let the point K be moved with the radius: or what 

 is the same thing let a string unwind from the curve KhL and the 

 line KG make a definite angle with the string kh : the curve 



1 The lines are equidistant measured along the string and are therefore 

 generally described as parallel. 



2 Observe the lines kh, k'L, enveloping the curve KkL. The distance 

 from the point to the evolute perpendicularly to the tangent is the radius 

 of curvature of the osculating circle. 



3 See footnote 1, above. 



