On Systems of Rays. 75 
of the paraboloid, of which the line 8/ is anelement ; and the normal may be brought 
to coincide with the ray-line by being made to revolve round the element él, through 
an angle & proportional to él, and equal to that element multiplied by the constant n : 
the direction of the rotation depending on the sign of the constant. On account of 
this simple law of deviation of the final ray-lines from the normals of the paraboloid, 
we shall call this paraboloid the guiding surface : and the constant n, we shall call 
the constant of deviation. And we may consider this theory, of the guiding parabo- 
loid and the constant. of deviation, as containing an adequate solution of our third ge- 
neral problem, in the discussion of the geometrical relations of infinitely near rays : 
since this theory shows adequately the general arrangement of the final system of ray- 
lines (7°), and the geometrical meanings of the third set of coefficients ( 0"), namely, 
du da 38 8B 
bz” dy’ 82’ Sy 
The geometrical construction suggested by this theory may be still farther simpli- 
fied by observing that the infinitely near normals to the guiding surface, all pass 
through two rectangular lines, namely, the axes of the two principal circles of curva- 
ture of the surface ; it is therefore sufficient to draw through any proposed point B, 
two planes containing respectively these two given axes of curvature, and then to 
make the line of intersection of these two planes revolve round the proposed small 
line 3/ or BB, through the same small angle nel as before, in order to obtain the 
sought final ray-line for the proposed final point. 
Finally, to compare, as required in the fourth problem, the initial system of ray- 
lines (/7”*) with the corresponding final points B, on the given final plane, we may 
denote these initial ray-lines by the equations 
x =2'80'. cos. 4, y' =2'd0'. sin. ¢', (P”) 
if we put 
? Sa’ = 60’. cos. ¢, 33’ =80'.sin. ¢' : (Q”) 
and if in like manner we put 
sv =8l.cos.¢, oy=dl.sin. >, (R”) 
we shall have the following relations, between 4, ¢, ol, 80’, and the fourth set of par- 
tial differential coefficients ( O°), 
80. cos. ¢' = (= cos. @ + = sin. > )a, 
; : (S") 
30’. sin. ¢’ = (2 cos. p + 2 sil. al. 
These relations give 
