NEW MODEL FOR OHTIIOCON' A I, I'Hi ).I EC I'lON. 373 



The model is, ot" course, only :i skeleton .structure of S stages. If 

 it could be completely developed, the number of semicircles would 

 become infinite, and they would form a smooth continuous surface 

 in three dimensions. Along the midplane Z O Y, all of these circles 

 would have the same level, raised 1 decimeter above the horizontal 

 drawing board plane of reference X O Y. The circles would increase 

 in radius without limit, and would cover the entire X O Y plane to 

 infinity, the hyperbola extending likewise to infinity towards its 

 asymptote O S, in the X O Z plane. The actual model is thus the 

 skeleton of the upper central sheet of the entire theoretical surface, 

 near the origin. 



The semicircles are also marked off in uniform steps of circular 

 angle. Each step is taken, for convenience, as 9°, or one tenth of a 

 quadrant. Corresponding angular steps on all of the eight semicircles 

 are connected by thin wires, as shown in the Plates. 



A front elevation of the model, taken from a point on the O Y axis 

 — 15 units from 0, is given in PI. II. It will be seen that any tie wire, 

 connecting corresponding circular angular points on the semicircles, 

 is level, and lies at a constant height sin 6^ decimeters above the 

 drawing board. That is, the tie wire that connects all points of 

 circular angle 6-2, measured from O X positively towards O Y, lies 

 at the uniform height sin d-i decimeters above the drawing board. 



A plan view of the model, taken from a point on the O Z axis, +15 

 units above 0, is given in PI. III. It will be seen that each and every 

 semicircle forms an ellipse, when projected on the base plane X O Z. 

 The semi-major axis of this ellipse has length cosh di, where Oi is the 

 hyperbolic angle corresponding to that semicircle. The semi-minor 

 axis is 



cosh di sin /3 = cosh di • tanh ^i = sinh d\ 



from the well known relation that exists between a hyperbolic angle 

 and its gudermannian circular angle; namely 



sin j8 = tanh ^i 



All of these ellipses have the same center of reference O. Any such 

 system, having semi-major axes cosh di, and semi-minor axes sinh di, 

 are well known to be confocal, and the foci must lie at the points +1 

 and — 1 in the X O Y plane, or the points in which the innermost 

 circle cuts that plane. 



Moreover, as is indicated in PI. Ill, the vertical shadow projections 

 of the tiewires on the reference plane X O Y, form a system of confocal 



