FORM OF TANGENTIAL EQUATION. 
423 
Therefore BQ is half the harmonic mean between BK' and BN ; and this gives a 
geometric construction for the centre of curvature at the point P'. 
116. From equation (235) we get 
BQ= 
BK'. BN 
K'N * 
Now if we make BQ'=BN, the point Q' would be the centre of curvature of an extern 
cycloid, see art. 86. Hence by subtraction 
QQ' = 
BN 2 
K'N* 
(236) 
That is, the distance between the centres of curvature of an extern cycloid and epicycloid 
is a third proportional to the lines K'N and BN. This vanishes, as it ought, when K' 
is at infinity. 
117. From equation (234) we get the value of D cos \{/, that is, of BQ ; thus 
and 
-RO — a (g ~ 2 «) cosMf cos4> + A(4>)) 
'** ac cos \J/— (g — a) AfiJ/) 
BP'=J(g cos\f/+ A\f/). 
Hence, remembering that a = hc , we get P'Q ; that is, the radius of curvature at P' 
b (g — a)\c cos \f/ + A^J/} 2 
accos^+fg — a) A\J/ 
(237) 
118. The following geometrical expression for the radius is remarkable for its sim- 
plicity and symmetry. 
From equation (235), 
BK^BN 
BU— K / N , 
and from art. 86, 
1 l_ l 
BP' BK~ BN ’ 
BK.BN 
~ KN ’ 
.-. P'Q=BN 
BK 
KN 
BK' ) 
K'N) 1 
(238) 
Cor. 1. The anharmonic ratio of the four points, 
N, K, B, K'=BP':BQ (239) 
Cor. 2. 
BP + BQ=BK+BK ( 24 °) 
119. If P' be a point of inflection, the radius of curvature at P' will be infinite, and 
mdccclxxvii. 3 o 
