152. On Cog or Toothed Wheels. 
would be between the two circles placed in 
the same plane, and touching at the Lav n 
the same common tangent. 
For draw the lines DE, fd, dg, gf, ge and 
gD; and adverting to the known equation of 
‘the cirele, let dn = x, gn=y and Dg =a, 
the absciss, ordinate and radius of the circle; 
we have 2ax—x* = y?. From this equation 
we obtain a= a the denominator of this 
fraction (27) being the width, de,, of the 
touching surfaces, fdg, and feg of the two 
circles. But the numerator (y*+a*) is equal 
to the square of the chord g d of the angle 
EDg, which chord I shall.call)z; then we 
have a= a 3 from which RTI we derive 
this proportion, a :2:: 2: 2a¢= “es But in Pdi 
small angles, the sines are taken for the ares 
without sensible error ; and with greater rea- 
son may the chords; if then we suppose the 
arc dg, or the chord z, indefinitely small, we 
shall find the line de = 2 =~, indefinitely 
‘smaller ; that is, of an order of infinitessimals 
one degree lower; for it is well known that 
the square of evanescent quantities, are inde- 
finitely smaller than the quantities themselves. 
And to apply this, if the chord z represent 
