On Systems of Rays. 85 
ZSpU+Yy 5 a) 
and the deflexion from this tangent plane, measured in the direction of the arbitrary 
axis of z, which we shall call the axis of deflexion, or in any direction infinitely near 
to this, is, for any point B’ infinitely near to the point of contact B, 
Deflexion=3.8z=h réa? + sdudy + $ td’. (G”) 
This deflexion depends therefore on the perpendicular distance é/ of the near point 
B from the axis of deflexion, and on the direction of the plane containing this point 
and axis; in such a manner that if we put, as in (/?"), 
dr=dl. cos. ¢, Sy=el. sin. ¢, 
and give the name of deflexure (after the analogy of the known name curvature) to 
ha ’ 
pendicular distance from the axis of deflexion, we shall have the following law of 
dependence of this deflexwre, which we shall denote by f; on the angle ¢, 
the quotient that is, to the double deflexion divided by the square of the per- 
Deflexure =f= < =r cos. ¢?+2s cos. ¢ sin. p+ésin. ¢”. (aks) 
There are, therefore, wo rectangular planes of extreme deflexwre, corresponding to 
angles $,, ¢., determined by the following formula, 
2s 
on — ——- 12 
tan. 2p = 3 (es) 
and if we take these for the co-ordinate planes of az, yz, and denote the two extreme 
deflexures corresponding by fi, f2, we have 
P= 80, o—fas (K”) 
and the general formula for the deflexure becomes 
SHfi cos. ¢ +fz sin. ¢? : (L”) 
which is analogous to, and includes, the known formula for the curvature of a normal 
section. And as it is usual to consider a system of circles of curvature, for any given 
point of a curved surface, namely, the osculating circles of the normal sections of 
that surface, so we may now more generally consider a system of circles of deflexure : 
namely, in each plane of deflexure ¢, a circle passing through the given point of the 
surface, and having its centre on the given axis of deflexion, and its curvature equal 
to the deflexure f; so that the radius of this circle, or the ordinate of its centre, 
, ; Sua) Fi : 
which we may call the radius of deflexure, is 7? and so that the equations of the circle 
of deflexure are, 
y=autang, @t+y+2?=—. (M”) 
VOL. XVII. Z 
LP 
