CONTINUOUS CALCULATING MACHINES. 
377 
therefore 
or 
to 
o y 
= t an a 
ft) — ft» 1 tan a 
These may be called the “sphere and roller” arrangements or mechanisms. The 
relation between the two forms of disk and roller is now clear. The first is derived 
from the more general sine or cosine (or secant, or cosecant) form of sphere and roller 
mechanism, the second may be in the limiting positions compared with the tangent 
or cotangent form. 
For in the first case, when 
a=0 then ft) — 0 
a=90° ,, (y = ft) 1 
or as before by changing the rollers 
a=90° then <y =w 1 
a—0 ,, ft) 1 =oo 
So that the range in the sine form is either from 0 to ft), or from Wj to oo . 
In the second case, when 
a=0 co=0 
a=45 &) = ft) 1 
a=90 ft)=co 
So that the range in the tangent form is (theoretically) from 0 to oo. 
These results were brought by the author before the Physical section of the Bristol 
Naturalist Society in November last, and illustrated by a model with a wooden sphere 
6" in diameter. It was, however, in endeavouring to apply them to practical purposes 
that the author was led to the investigation which has resulted in the present 
communication. 
In fig. 6 take any point P in the axis of rotation of the sphere. Draw P F in a 
direction perpendicular to the plane of rotation of the roller B. Then by a suitable 
mechanical device consisting of a cross F, sliding upon a rod O T through which 
another rod turning about the centre P freely passes, it is evident that for any value 
of the angle a 
PF 
OP 
= sin a 
but by previous reasoning it may be arranged that 
&) dz 
— = — = Sin a 
ft) L dx 
