80 Professor Hamiiton’s Third Supplement 
lines will be the centre of curvature of the corresponding normal section. We may 
therefore call the curvatures of these two diametral sections the ¢2vo vergencies of the 
final ray-lines ; and the two corresponding planes P, P, we may call the two planes 
of vergency. 
The same conclusions may be deduced algebraically from the equations (A), which 
give the following conditions of intersection of a near ray-line with the given ray- 
line or axis of 2, 
O=(27'—r)dx—ndy; O=(z-'—#) dy + nda; (1") 
2 being the sought ordinate of intersection, and therefore z—' the vergency: for thus 
we find by elimination the following quadratic to determine the ratio of 8a, dy, that 
is the direction of 6/, 
(t—1) dady =n (dy? + 82"), (K") 
which may be put under the form 
sin. apa, (L") 
the angle ¢ being, as in (F"), the inclination of 8 to the axis of x, that is, to one of 
the tangents of the lines of curvature, while 7, ¢, are the two curvatures themselves, 
of the guiding paraboloid ; there are therefore two real directions of 8/, or one, or 
none, corresponding to the mtersection supposed, according as we have 
9 
t—r7\? 
( ) PP OMS jos < 7B (M") 
so that we are thus conducted anew to the same conditions of reality, and to the same 
symmetric directions of the two planes of vergency, which we obtained before by a 
reasoning of a more geometrical kind. The same conditions may also be obtained by 
considering the quadratic for the vergency itself, namely 
(z2-'—r) (e¢-'—f)+n7=0, (N") 
which results from the equations (J") and shows that the sum and product of the two 
vergencies may be thus expressed, by means of the curvatures 7, ¢, and the constant 
of deviation 7, 
ate arté; 2cetsrt+n. (O") 
The equations (J'') give also, by elimination of 7, 
z'=r cos. ¢' +# sin. 97; (P") 
we see, therefore, as before, that the two vergencies, when real, of the final ray-lines, 
are the curvatures of the two corresponding sections of the guiding paraboloid. In 
general the centre of curvature of any section of this surface, made by a normal plane 
drawn through the given final ray-line, is the common focus by projection of all the 
