96 Blake on the Teeth of Cog-Wheels. 
Prop. 7.—Any curve being given on the plane of a cir- 
cle it may be ascertained whether it is isosagistic and its 
fellow isosagistic curve may be found. 
For since all isosagistic curves can be generated (prop. 6.) 
and all curves which can be generated are isosagistic (prop. 
4. Cor. 1.) the given curve is isosagistic if it can be genera- 
ted and not otherwise. But it may be ascertained whether 
it can be generated, and if it can be its generating curve 
may be found (prop. 5. Cor. 2.) And this being applied in 
the prescribed manner to any other circle will generate a 
w isosagistic curve. 
Cor.—lIt may be determined with regard to any given 
tooth-wheel whether its teeth are isosagistic. For thev are 
so then, and then only, when sectionsof their acting faces can 
be ~~ on the given pitch circle, if it be given 5 or on 
any circle concentric with the wheel assumed as the pitch 
circle, when it is not given. 
Pror. 8.—In simultaneously describing any corresponding 
portions of fellow isosagistic curves, the describing point 
main in the line of centres. 
‘If the describing point remains in the line of centres it is 
either at rest or in motion in that line. First, let d, Fig. 5, 
be the describing point, and let it be at rest while the circle 
a turns on its centre through the arc eh. Then the genera- 
ting curve is necessarily a circle, and the describing point, its 
centre, andan arc di of acircle concentric with the circle 
a will be described on the plane of the circle a. This, as 
‘res and move in that line while it traces isosagistic curves. 
