360 Proceedings of the Royal Society 
the oscillation is augmented, the curve shows its tendency to flatten 
at the vertex X, and when the oscillation extends over the semi- 
cumference BAO, the curve, as shown in fig. 2, becomes quite flat 
at X, the radius of curvature there being infinite. 
Where OA extends beyond the quadrant, the ordinate rises to 
be equal to the radius 00, and then decreases to reach a minimum 
value at X, after which it again rises, and so produces the saddle 
form seen in fig. 3 ; and when the oscillation extends over a com- 
plete turn, the vertex X of the curve comes down to S, as shown 
in fig. 4. If the oscillation extend over more than the whole 
circumference, the vertex X passes to the other side of the axis, as 
seen in fig. 5 ; and when the extent is one turn and a half, the 
curve is again flattened on the opposite limit, preparatory as it 
were, to the return towards S, when the oscillation is still farther 
extended. Thus this genesis produces a great variety of phases, 
beginning with the curve of sines, passing to the curve of second 
sines, and continuing in an endless series of variations beyond. 
As soon as we pass beyond the flattened vertex, these curves lose 
all interest to the practical mechanician, who can hardly contemplate 
the use of wheel-teeth with hollowed tops; yet to the speculative 
engineer they offer the attraction of peculiar phases in the con- 
figuration of the relative contact-path, and in the convolutions of 
the tooth outlines; but their real interest is centred in this, that 
amongst them we find the best known form for the rack. 
When the arc OA is three-fourth parts of a quadrant, and when 
