234 Dr. J. McCowan on Ridge-Lines 



The conditions for a multiple point on any of the curves 

 y= constant are 



£=<>> g=°> ^ 



which are also the conditions for a maximum or minimum 

 value of 7. Hence the points of the surface for which 7, or, 

 say for clearness, the curvature of the projections of the 

 lines of slope is a maximum or minimum, are multiple points 

 of the particular curves on which they lie. In the case of the 

 ridge-line, 7 = 0, and for a multiple point this must be a 

 maximum or minimum value of 7. Thus, while in general 

 the curvature of the projection of a line of slope changes from 

 positive to negative in crossing a ridge-line, so that the point 

 of crossing is a point of inflexion, at a multiple point the 

 curvature will not in general change sign, and therefore the 

 point will not be a point of inflexion. Hence, finally, the 

 ridge-lines branch in general at points whose projections are 

 points of no curvature but not points of inflexion on the pro- 

 jections of the lines of slope which pass through them. 



§ 7. Definition of Outlines. Contact Theorems, 



The consideration of another class of lines connected with 

 the surface leads to a further interesting interpretation of 

 equations (15) and (16). 



Imagine a cylinder whose generating lines are horizontal 

 to envelop the surface. This cylinder touches the surface 

 along a line, the locus of the points of contact of the generators 

 and contours. This line may be called an outline, for it is 

 the line which would appear to bound the surface when seen 

 from an infinitely distant point on one of the generators. 

 There is of course a whole family of such lines, as the 

 generators of the cylindric envelope may have any direction 

 parallel to the base. Outlines corresponding to points at a 

 finite distance and in any direction might be considered, 

 but the more restricted definition chosen is sufficient for the 

 purpose at present in view. 



If 6, the inclination to the axis of x, of the generators of 

 enveloping cylinder, be taken as the parameter for the out- 

 lines, then (16) is the equation to their projections, for it may 

 be written 



cos 0^+ sin 0^=0, (21) 



which shows that the generators touch the contours wherever 

 it is satisfied. Thus it is to be noted that, as in the case of 



