96 Blake on the Teeth of Cog- Wheels. 



Prop. 7. — Any curve being given on the plane of a cii*- 

 cle it may be ascertained whether it is isosagistic and its 

 fellow isosagistic curve may be found. 



For since all isosagistic curves can be genenited (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.) A.nd this being applied in 

 the prescribed manner to any other circle will generate a 

 fellow isosagistic curve. 



Cor. — It may be determined whh 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 generated on the given pitch circle, if it be given ; or on 

 any circle concentric with the wheel assumed as the pitch 

 circle, when it is not given. 



Prop. 8. — In simultaneously describing any corresponding 

 portions of fellow isosagistic curves, the describing point 

 does not constantly remain 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, and an arc di of a circle concentric with the circle 

 a will be described on the plane of the circle a. This, as 

 has already been shown, is not an isosagistic curve. 



Secondly, during the same motion of the circle a let the 

 describing point d move through a segment dg of the line 

 ab. Then the curve described will be ig, which when the 

 arc eh is taken infinitely small, may be considered as a 

 straight line. Now since i d is the arc of a circle whose 

 centre is a, it is perpendicular to the Hne a d. Therefore 

 the angle i c? ^ is a right angle. Consequently the angle 

 i g d is less than a right angle. But g being the describing 

 point and e^ the describing radius, i^ is perpendicular to 

 eg. Therefore the angle igd is a right angle. But it can- 

 not be both equal to, and less than, a right angle. Where- 

 fore the describing point cannot remain in the line of cen- 

 tres and move in that line while it traces isosagistic curves. 



