ett 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 mechaa- 
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 acurve 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,P1.1V.) have an equal tendency 
ate, their point of contact, to move in opposite directions, the 
one in the direction eh, the other in oe direction ef and let 
point of contact g let them both be peupeotists to the line 
ge. These circles will be in equilibrio. Draw the lines be, 
ad perpendicular to the line ge produced, and let the pow- 
er at ein the direction eh be P, and atc in the direction 
eg, p3 and let the resistance at e in the direction ef be R, 
and at - in the direction dg, +. 
By the principles of mechanics P : tp: ibe 3 be and R: 
:iad 3 ae. t by similar triangles be} be::ad ? ae. 
Therefore Pip::Rir. But by the hy pothesis P=R.. 
Therefore p=r. That is the power at c in the direction eg 
is equal to the resistance at din the direction dg. 
these forces are respectively the same at g as ate and d. 
Therefore the forces are equal and opposite at g and the 
circles are in equilibrio. 
Derivition.—Curves attached to circles, acting upon 
each at we shall call acting curves, and the point at 
they are in contact, the point of action. 
Prop. 2.—If the acting curves are such, that while one 
drives Kos other, they both keep constantly perpendicular 
