916 APPENDIX. 



69j. Patent Hatching Ruler (for shading by cross-lines in 

 drawing and engraving). 



E. 0. Richter and Co., Chemnitz, Saxony. 



A ruler is attached to a cylinder which, rolling on a plate, draws the 

 former after it. By means of an endless screw, working in a wheel, the 

 cylinder is moved forward. A cogwheel is attached to the endless screw, 

 in which a bar catches, and by the depression of which the mechanism is 

 put in motion. The depression can be regulated by a screw, by which means 

 the various distances of the lines are obtained. One finger is sufficient for 

 working the ruler by simply pressing on the screw, and holding it until 

 the line is drawn. 



69k. Six Setter Diamonds. 

 691. Five Machine Diamonds. 

 69m. Six Scratching Diamonds. 



E. 0. Richter and Co., Chemnitz, Saxony. 



123d. Model representing the Bight Lines on a 

 Surface of the Third Order, having a tangent-plane touching 

 along a conic (the singularity dualistically corresponding to a 

 double-point of the second order). 



Elling B. Hoist, Stipendiary of the University of 

 Christiania. , 



The model is composed of twenty-one wires, six of which, painted light 

 red, lie in the same plane and touch the conic in points painted dark red. 

 Through the fifteen points of intersection of these six lines the others, 

 painted white, pass, again intersecting three and three, and are the lines in 

 which the surface cuts itself. All points on these lines have therefore two 

 tangent-planes ; where the latter are imaginary the lines are black. The 

 black is in part laid on schematically, especially where the black part con- 

 tains the point infinitely distant. The parabolic curve consists of the conic 

 aforesaid and two species of cuspidal curves, viz. : 



1 . One curve passing the dark red points and having cusps in those six 

 limiting-points between black and white which are nearest to the conic, the 

 curve therefore having a zig-zag course. 



2. Four closed branches having cusps in the other twelve limiting-points. 

 All these parabolic curves together separate ten distinct elipsoidically 



curved parts from the surface everywhere else hyperboloidically curved. 



166a. Stereograms of the Lines of Curvature of 

 Surfaces. Drawn by the exhibitor. Prof. J. Clerk Maxwell. 



Lines of curvature of the cyclide of Dupin (4). 

 Curves on a sphere (4) : 



a Two systems of orthogonal circles. 



j8 Concyclic spherical ellipses. (This represents a spherical har- 

 monic of the second degree.) 

 7 Confocal spherical ellipses. 

 5 The projections of a spherical ellipse on the three principal 



planes. 

 Quadric surfaces (4) : 



Elliptic paraboloid, hyperbolic paraboloid, ellipsoid, and the surface 

 of centres of ellipsoid. 



