92 Blake on the Teeth of Cog- Wheels. 



ING POINT ; the curve to whose plane the describing 

 point is attached, the generating curve ; that upon which 

 the generating curve rolls, the ba.se curve; that part of 

 the base curve with which the generating curve comes in 

 contact, the base of generation. 



Cor — During the action of isosagistic curves, the veloci- 

 ties of .heir circles are equal at their point of contact. 



Prop. 3. — Any curve generated by one curve rolling up- 

 on another is at every point of it, perpendicular to a line 

 drawn from that point to the point in the base, which was 

 in contact with the generating curve when that point was 

 des< ribed. 



Let a b. Fig. 2. be any base curve ; c d any generating 

 curve rolling upon it ; f any point in the plane of erf, and 

 gh the curve described by/l Now during the time when 

 a small part/of the curve hg is described, the describing 

 point revolves about the point e as a centre. Therefore if 

 a circle be described from e as a centre with a radius ef, the 

 circumference of the circle will coincide aty with the curve 

 hg. But the circumference of a circle is every where per- 

 pendicular to the radii. Therefore the curve hg, which coin- 

 cides with the circumference of a circle whose radius isye, 

 is perpendicular toye. 



Corollary. — -Hence it is manifest that no curve on the 

 plasie of a circle is susceptible of being generated by anoth- 

 er curve, rolling on the circle, unless perpendiculars from 

 every successive point of it, commencing at one extreme, 

 fall on corresponding successive points of the circle. 



Prop. 4. — Any curve whatever being taken as agenerating 

 curve and rolled upon any two circles with one side toward the 

 centre of one, and the other toward the centre of the other, 

 will by any point in its plane, as a describing point, trace on 

 the planes of the base circles, fellow isosagistic curves. 



Let the two circles, whose centres are a and b, Fig. 3. 

 and auy generating curve cde touching each other at the 

 point rf, be rolled together in such a manner that their mu- 

 tual point of contact shall be kept constantly in the point d 

 of the line a b, until the generating curve come into any oth- 

 er position as fdg. Then if the point d in the curve cde 

 be taken as the describing point, it will have moved top 

 and described the curve pi on the plane of the circle « 



