al 
a Te 
———— 
Pes cee . Bee EA. oeapecoer mae 
ee 
Vol. IV, No. 10.] A General Theory of Osculating Conics. 507 
(N.S.] 
all eran of three-pointic contact, pass through the same 
point (86 
ee as two consecutive parabolas, of four-pointic contact, 
may be conceived to possess three consecutive points common, 
their directrices meet at (86), and, therefore, the envelope of the 
directrix of the osculating parabola i is the locus of the point (86). 
If a and b be the semi-axis of any ellipse of the system 
of conics of four- . contact (56), then from (72) 
want ——— {9Q++3(QQ,—RP)?+ Pa} 
a 3A2Q2P 
Py» 
sec *y + —— 
— (87) 
3Q§ 
~ Pr 
e® 
But (¢+ 7) a= +7 aa 
a. Oo. 7a ; ic 
Therefore hr is a Minimum when e is a minimum, 
a 

the ellipse of minimum eccentricity of the system 
ce, 
(56) i is icikeanid by 


, 94+ (3QQ,- RP)? 
Pe 
ee ae 
ba cosy P 
Therefore, the centre of the osculating ellipse, of minimum 
eccentricity, i 1s a point, on a line of centres, towards the concave 
side, at the same distance, from the point of contact, as the 
centre of me Iihiersr8 siriclakiral hyperbola. Here, evidently 
OP=CD=p 
Again, “f , and A, correspond to equal values of the eccen- 
tricity, and, therefore, to equal values of 5 $4. , then from (87) 
J/. ia We ene (89) 
Therefore, if C, O,, C, be the centres of the ellipse of 
nimum eccentricity and of any two ellipses of equal eccentricity, 
then, ore OF. 0,P=O0P (90) 
where P is the point of contact. 
Analogous results hold for the system of hyperbolas of four- 
on c contact. 
ee be the centre of the osculating equilateral hyperbola, 
and a Q, the centres of any two osculating hyperbolas whose 
