372 KENNELLY. 



visualized; so that once it has been reahzed by the student, the three- 

 dimensional artifice is rendered superfluous, and he can roughly trace 

 out a complex sine or cosine on an imaginary drawing board, with his 

 eyes closed. The model, however, will be shown to enjoy certain 

 interesting geometrical properties as a three-dimensional structure. 



A photograph of the model is shown in PI. I. On an ordinary 

 horizontal drawing board 53.5 cm. X 31.8 cm., and 2.2 cm. thick, is 

 a horizontal half rod A B of brass, which merely serves to support 

 the various brass-wire semicircles, and a semihyperbola, in their 

 proper positions. It should be understood that the axis of A B in 

 the X Y plane, on the upper surface of the board, is a line of symmetry 

 for the structure, which, if completed, would be formed l)y full circles 

 and a complete hyperbola. For convenience, however, only half of 

 the structure above the X Y plane is presented, the omission of the 

 lower half being readily compensated for in the imagination of the 

 beholder. 



The eight wire semicircles are formed with the following respective 

 radii, in decimeters: 1.0, 1.020..., 1.081..., 1.185..., 1.337..., 

 1.543. . ., 1.810. . ., and 2.150. . ., which are the respective cosines of 

 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4 hyperbolic radians, according to 

 ordinary tables of real hyperbolic functions. These successive semi- 

 circles therefore have radii equal to the cosines of successively increas- 

 ing real hyperbolic angles ^i, by steps of 0.2, from to 1.4 hyperbolic 

 racHans, inclusive. All of these semicircles have their common center 

 at the origin 0, in the plane X O Y, of the drawing board. The planes 

 of the semicircles are, however, displaced. The smallest circle of 

 unit radius (1 decimeter), occupies the vertical plane X O Z, in which 

 also lies the rectangular semi-hyperbola X O H. Angular distances 

 corresponding to 0.2, 0.4,. ... 1.4 hyperbolic radians, are marked off 

 along this hyperbola at successive corresponding intervals of 0.2. 

 The cosines of these angles, as obtainable i)rojectively on the O X axis 

 are marked off between C and B along the brass supjwrting halfliar, 

 and at each mark, a semicircle rises from the X Y plane, at a certain 

 angle (3 with the vertical X O Z plane. This displacement angle is 

 determined by the relation 



cosjS = — ; — = sech 6i nmneric (1) 



cosh di 



where di is the parliciilar liyixrbolic angle selected. This m(\iiis, as 

 is well known, that the (hsj)lacein('iit angle /3 between tlie i)lane of 

 any semicircle and the vertic a! j)laiu> Z () X is eciual to the guderman- 

 nian of the hyperbolic angle 6i. 



