

Vol. IV, No. 10.] A General Theory of Osculating Conics. 509 
(N.S. ] 
Again let p, p,’ p,” p’” be the radius of curvature and its 
three successive dnote, on the supposition that the are is 
the independent variable 
Then by (11), "(95) and (96) 

3 8 5 
sl P? p’ Pp? P*p’ ‘98 
pt... R=dQ=- (TP te Se Es 1) 
oe p wee p* p* p* 
2p’? p” 
Also S+R’=dR=P re aioe 1 
psp 
6 43 6 Pe we (99) 
T+2S’=@R=P* (- eee) | 
p* ep - 
By the above substitutions (98), (99) any expression in 
z set &, S, &e., can be readily converted into another in P, p, p’, p” 
ste 9Q* + (38QQ, -PR)?= A(9+5 Pp =) (100) 
3Q9 —5R? +12QR’ Maat -e) (101) 
p* z r ; 
40 B8—45 QRS+9 Q2T-90 QRR’ +45 Q28’ 
pt 
ee ee ey p’ +9 p® p'” +36 P p’ , (102) 
P 
Therefore the differential equation of a conic in p and ¢ is 
4 p’®—9 pp’ p'’ +9 p? p’”” +36 P p’=0 
or, 
2p 
(4 Py — 9p Te 998 8 F +36 2—0 (103) 
