Blake on the Teeth of Coz-Wheels: 101 
’Pror, 12.—If the velocity of the circles be uniform at 
the point of contact, and the velocity of the describing point 
be also uniform, the curves described aré Logistic Spirals. 
' For since the velocity of the circle a, Fig. 9, is uniform at 
point p, and of the point p. Divide the arc of a circle gf 
into indefinitely small equal parts, and through the points of 
division draw the lines ad, ap, &c. Now in the triangks 
aed, adp, apc, the angles at a are equal, because they sul- 
tend equal arcs, and the remaining angles have already been 
shown to be respectively equal. Therefore the triangles 
are similar each to each. Hence ac? ap:iap : ad::ad: a, 
&e. That is, the radii ac, ap, ad, ae, &c. are continued 
Proportionals ; which is the fundamental characteristic of the 
‘ogistic Spiral. The same might be shown of the curve 
hi on the plane of the circle b. iat! 
Scuo..—The Logistic Spiral is perhaps the most simple 
form for atripsic teeth of wheels. 
Prop, 13.—Any curve on the plane of a circle may be 
driven by some other curve without friction, when evely 
Successive part of it, commencing with that extreme whica 
48 nearest the centre, recedes continually both from the cer- 
tre and from the radius in which it commences. : 
Let a, Fig. 11, be the centre of the circle, to the plane cf 
Which is attached the curve pe answering the condition cf 
Proposition. Through p, the nearest extremity of the 
Curve pe, draw the line ab, and through the other extremity 
€, draw the arc ec of a circle whose centre is a. Now it 
the circle a revolve on its centre till the point e comes into 
the line of centres at ¢, the point of intersection between the 
furve pe and the line ab will advance from p toc. F 
