xlviii 
lar differential ; therefore the length of the projection of the 
same element on a line perpendicular to the focal vector, and 
drawn in the plane through the axis, is denoted by this other 
radical, / (— da” — dr’) ; but the length of this last projec- 
tion is evidently to the length of the element itself, as the 
length p of the perpendicular let fall from the focus on the 
tangent is to the length 7 of the focal vector of the point of 
contact ; such, therefore, is, by (59), the ratio of n—2 to 1, if 
the scalar n, in the equation of the normal (57), receive the 
value which corresponds to the centre of curvature of the me- 
ridian ; therefore we have 
R v—a sy aes 2 Oe of 
ae ai a adele: 6 slams opis Ti (61) 
n denoting the length of the portion of the normal which is 
comprised between the meridian and the axis, and r denoting 
the length of the radius of curvature of the meridian. The 
projection of this radius on the focal vector is evidently the 
focal half chord of curvature, of which half chord the length 
may be here denoted by c; we see then that if we again 
project this half chord on the normal, the result is the normal 
itself, that is the portion N, because this double process of 
projection multiplies x twice successively by n—* ; and if, once 
more, the normal be projected on the focal vector, the third 
projection so obtained is equal in length to the semiparameter 
p, because, by (15) and (16), 
(re — a) + (re — a) 0 = 2p; (62) 
hence 
rr Ir’ 
VGN SceypS eae ° 
(63) 
that is, for any conic section, the geometrical mean between 
the radius of curvature and the normal is equal to the harmo- 
nic mean between the two focal distances; of which distances 
the second, namely 7’, is to be regarded as negative for the 
