and Lines connected icith them. 235 



the ridge-lines, the outlines may he derived directly from the 

 equation to the surface by differentiation only. 



This interpretation of 0, as an outline parameter, leads to a 

 corresponding interpretation of (15), namely — aline of slope 

 touches an outline wherever a ridge-line crosses it. This 

 points to a simple method of tracing the ridge-lines on a 

 model of any surface if the lines of slope (or lines of curvature) 

 are already traced on it. It is only necessary to view the 

 surface from a point at a considerable distance in the base 

 plane, and then, turning the model slowly round, to mark the 

 points where the slope-lines (or lines of curvature) touch the 

 continuously changing outline. The ridge-lines on a moun- 

 tain could not be traced in a corresponding fashion, by a 

 person moving round it at a sufficient distance, for there 

 would be no means of identifying either slope-lines or lines 

 of curvature on its surface. 



Two general theorems with respect to the contact of the 

 various lines considered have now been proved : — 



I. At every point on a ridge-line, the outline, line of slope, 

 and one of the lines of curvature have a common tangent. 



II. At every point on a ridge-line, the isoclinic or line of 

 contant slope, the contour-line, and the other line of curvature 

 have a common tangent. 



It is an immediate inference from the first of these theorems 

 that if any line has at once the characteristics of any two of 

 the lines there named it must also have the characteristics of 

 the other two, and it easily follows that such can only be the 

 case when the line is that of a section of the surface by a 

 vertical plane. For example, suppose an outline to be a line 

 of slope : it is therefore by I. a ridge-line, and therefore, 

 again by I., a line of curvature. Further, since it is an out- 

 line p/q is constant, therefore, since it is also a line of slope, 

 dyjdx is constant for the projection of the latter, which is thus 

 a straight line, and the line itself is the section of the surface 

 by a vertical plane through this. 



Again, it follows in similar fashion from the second of 

 these theorems that if a line has the characteristics of any two 

 of the lines there named it must have also those of the other 

 two, and in this case, being a contour-line, it is a section of 

 the surface by a horizontal plane. 



An obvious illustration of these results is afforded by sur- 

 faces having a plane of symmetry : if the plane is vertical the 

 section by it is evidently a ridge-line, an outline, a line of 

 slope, and a line of curvature ; but if the plane is horizontal 

 the section is a ridge-b'ne, a line of constant slope, a contour- 

 line, and line of curvature. 



