
51. A General Theory of Osculating Conics 
(Second Paper),! 
By Pror, Syamapas Muxnopapnyaya, M.A, 
INTRODUCTION, 
Abel Transon in a classical memoir, published in Liouville’s 
Journal (vol. vi, 1841, Researches on the curvature of lineg 
and surfaces), gave the first impulse to the study of osculating 
conics and higher affections of curvature. 
im we owe the important discovery, that if O be the 
middle point of an infinitesimal chord PQ, and T the summit of 
the are PQ, then the line OT’, in its limiting em makes an 
angle 6 with the normal, such that tand =356. He calls the 
line OT, in its ultimate position, the azzs of oleae bai takes 
tand as the measure of the rate of deviation of the curve from 
circular form, or, of the second affection of — 
more exact interpretation of tan 6% seems, to the present 
writer, to be what he has called se paltalk gtr of variation of 
curvature, and the formula tan = P follows at once from this 
Sds 
interpretation 
ranson notices that the deviation axis is the locus of centres 
of osculating conics of four-pointic contact. e determines the 
centre of the conic of five-pointic contact, as the intersection of 
two consecutive deviation axes. The istance Ff of this centre, 
from the point of contact, he first expresses in terms of 
p 
dp dp ’ 
ca? aan? and then reduces to an expression in q, 7, s, taking p to 
be zero. His result is— 
Bet (FY +998 fF 
R 
(2) op 9p 
_3q (79 + 9q4)# 
3g3 — 5r? 
He gives elegant geometrical constructions for completely deter- 
mining the osculating parabola and the oseulating conic, after 
tan 6 and =e have been determined. 
is quasi-geometrical. His chief aim was to discover 
‘the iascnait and third affections of curvature.’ His discovery 



: Continued from Journal A.S.B., vol. iv, No. 4, New Seri 
2 Vide ‘ The Geometrical Theory of a Plane peer pen finite as 
well as infinitesimal’ (J.A.8.B., vol. iv, No, % New . 
