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, V ( — 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 r of the focal vector of the point of 

 contact ; such, therefore, is, by (59), the ratio of n~i to 1, if 

 the scalar w, in the equation of the normal {57), receive the 

 value which corresponds to the centre of curvature of the me- 

 ridian ; therefore we have 



n _ V— a _ _^_ ^''' _rr^ ,p . 



N"~"r£ — a" ~"p^~ pp' ~ pa' ^ ^ 



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 r 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), 



i(^E _ a) 4- (re - a) £ = 2j9 ; (62) 



hence 



x/ (rn) = c-=znp = — =: — ■ — „ (o.i) 



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 ;', is to be regarded as negative for the 



