361 
of Edinburgh, Session 1873-74. 
the curve is raised to such a height as to have always seven normals 
from a point in the axis, wheels of 14 teeth, developed by its help, 
have their outlines mechanically complete. 
Putting r for the length of OA, the half-arc of oscillation, the 
equation of the curve is 
y - A . sin (r sin u ), 
and the length of the HN subnormal is given by the formula 
x — i A 2 . cos u . sin (2 r sin u), . . (1), 
A 
which gives, at the same time, the form of the contact path. 
Hence we have EN, the length of the pitch-line, corresponding to 
the contact at P, 
EN = a - u -f -i rA 2 . cos u . sin ( 2r sin u), . (2). 
2 
Hence if we denote by U that value of u which corresponds to 
the extreme position of the point N, we must have 
= sin U . sin (2 r sin U) - 2r . cos U 2 . cos (2 r sin U), (3), 
and when the number of contacts is to be n, we must have the 
corresponding value of EN equal to ; wherefore 
tan (2rsinIJ) 
tanU , tan(2rsinU) - 2 r . cosU 
+ U = 
> + l 
( 4 ) 
by help of which equation we can determine the values of U and A, 
corresponding to any assumed value of r, and to any desired number 
of contacts. 
For seven contacts, and when r = 
3 
we obtain 
A = 3*469167 and the maximum value of y , 3 - 205089 ; by help of 
which dimensions fig. 8 has been drawn. 
If the point P be carried along the saddle-shaped curve EX of 
fig. 3, the subnormal HN lies first on the one and then on the 
other side of the ordinate PH, so that we may have two stationary 
positions of N as N x and N a , and these may be placed so that the 
part N 2 N x is traversed thrice, as actually happens in the figure. 
By lessening the ordinates the whole curve may be flattened, and 
