On Systems of Rays. 87 
this angle, therefore, is equal to the semidifference of the extreme deflexures multi- 
plied by the infinitesimal perpendicular distance from the axis of deflexion, and by the 
sine of twice the inclination @ of this perpendicular (or of the plane of deflexure 
containing it) to one of the two rectangular planes of extreme deflexure. In this 
general case, the deflected line ((*) does not intersect the given axis of deflexion, 
which we have made the axis of z; but the deflected line (Q") always intersects its 
own axis of deflexure (VV), in a point of which the co-ordinates may be thus ex- 
pressed 
CS “2. SL a cosig, z= i (Vv") 
the symbols f, ¢, and ey, retaining their recent meanings. It is easy also to see that 
if a near deflected line be projected on the corresponding plane of deflexure, the pro- 
jection will cross the axis of deflexion in the centre of the circle of deflexure ; and 
therefore that this centre of deflexure may be considered as a focus by projection, and 
that the planes of extreme deflexure are planes of extreme projection. 
The foregoing results respecting the deflexures and deflected lines of a eurved 
surface, near any given point upon that surface, and for any given axis of deflexion, 
may easily be expressed by general formule extending to an arbitrary origin and arbi- 
trary axes of co-ordinates. If, for simplicity, we still suppose the co-ordinates rectan- 
gular, and still take the given point upon the surface for origin, and the given axis of 
deflexion for axis of z, but leave the rectangular co-ordinate planes of xz and yz 
arbitrary, so that the coefficient s in the equation of the surface shall not in general 
vanish, then the equations of a deflected line become 
va=du—z(rdx+sdy), y=sy—z (sdx + toy) ; (W”) 
since the equation of the paraboloid (S”), to which they are nearly normals, and of 
which the ordinates measure the deflexions (G"*) of the given surface, becomes 
z=trrtsary + hty’. (X”) 
The deflexure for any plane ¢ is expressed by the general formula (H™) ; and in like 
manner the general formule (J/") (NV) determine still the circle and axis of deflex- 
ure. ‘The two principal planes of deflexure, ¢,, ¢2, are still determined by the for- 
mula (Z"), while the corresponding extreme deflexures, fi, f:, are the roots of the 
following quadratic 
SJ’ -f(r+O+rt-s=0: (Y”) 
and the angular deviation éy of a deflected line from the corresponding plane of 
deflexure, is thus expressed, 
=k (fi—fa). sin. (24 — 4). = (> -sin, 2p — s. cos, 2g) &i. (Z") 
