^ Blake on the Teeth of Cog-Wheels. 



whether any given curve possesses this property, and of 

 finding its Atripsic fellow. 



In pursuing this plan, I shall endeavour to advance noth- 

 ing which is not susceptible of rigid mathematical demon- 

 stration, and to take no point for granted without proof 

 which will not be sufficiently obvious upon slight considera- 

 tion, to those who are versed in mathematical and mechan- 

 ical science. 



IsosAGisTic Curves. 



Prop. I. — If two circles which are in contact have equal 

 power at the point of contact to revolve about their centres 

 in opposite directions, and if a curve be attached to one of 

 them, acting on a curve attached to the other, the circles are 

 in equilibrio when the curves are perpendicular to a line 

 drawn from their point of contact to the point of contact of 

 the circles. 



Let the two circles (Fig. 1 ,P1. IV.) have an equal tendency- 

 ate, their point of contact, to move in opposite directions, the 

 one in the direction sA, the other in the direction e/and let 

 the curve gh attached to the circle 6, acton the curve gi at- 

 tached to the circle a, and at the point of action or their 

 point of contact g let them both be perpendicular to the line 

 gt. These circles will be in equilibrio. Draw the lines 6c, 

 ad perpendicular to the line gt produced, and let the pow- 

 er at e in the direction th be P, and at c in the direction 

 fg-, p ; and let the resistance at e in the direction ef be R^ 

 and at ' in the direction dg^ r. 



By the principles of mechanics P I p'. '.be I be and R ', 

 r'.'.ad : ae. But by similar triangles bclbe'.'.ad t ae. 

 Therefore P I p'.'.R : r. But by the hypothesis P=R. 

 Therefore /9=r. That is the power at c in the direction eg 

 is equal to the resistance at d in the direction dg. But 

 these forces are respectively the same at ^ as at c and d. 

 Therefore the forces are equal and opposite at g and the 

 circles are in equilibrio. 



Definition. — Curves attached to circles, acting upoa 

 each other, we shall call acting curves, and the point at 

 which they are in contact, the point of action. 



Prop. 2. — If the acting curves are such, that while one 

 drives the other, they both keep constantly perpendicular 



