98 Professor Hamitton’s Third Supplement 
To understand more fully the occasion of such composition, let us remember that 
our theory of the Elements of Arrangement enables us to pass from the extreme 
directions of a given luminous path (4, B),, to the four following sets of near 
extreme directions, by the solution of the four problems considered in the fifteenth 
number. 
First. The extreme directions of the near path (4, B),,;,, which has the same 
extreme points 4, B, but differs by chromatic dispersion. 
Second. The final direction of (4, B,),, that is, of the original path prolonged at 
the end, and the initial direction of (4,, B),, that is, of the same path prolonged at 
the beginning ; these near extreme directions being in general affected by curvature. 
Third. The final direction of the path (4, B,),, and the initial direction of 
(A,, B),; the small lines 24,, BB,, being perpendicular to the given path at its 
extremities. 
Fourth. The initial direction of (4, B,),, and the final direction of (4,, B),. 
We saw also that the initial direction of (4, B,), and the final direction of (4,, B), 
do not differ from the corresponding extreme directions of the original luminous 
path. 
If then we would apply this theory to determine the final direction of an arbitrary 
near path (4’, B’),+3,, we have to consider and compound, algebraically or geome- 
trically, the following partial deviations from the given final direction of the given 
path (4, B),: first, the chromatic deviation of the final direction of the near path 
(A, B),+;, from that given final direction; second, the deviation of curvature of the 
final direction of (4, B.),; third, the final deviation of the path (4, B,),, to be 
determined by the theory of the final guiding paraboloid ; and fourth, the deviation 
of the final direction of (4,, B),, to be found by the theory of the guiding planes 
and conjugate guiding axes. A similar composition of four partial deviations is 
required for the determination of the initial direction of the same arbitrary near path 
(4, B tax: 
Now to compound in a geometrical manner the four preceding partial deviations 
of the final ray-line, we may proceed as follows. We may construct each partial 
deviation, by drawing the deviated final ray-line corresponding, or a line parallel 
thereto, through the given final point B; the line thus drawn will differ little in 
direction from the given final ray-line or axis of z, and if we take its length equal to 
unity, then its small projection on the given final plane of xy, to which it is nearly 
perpendicular, will measure the magnitude and will indicate the direction of the — 
deviation : and if we compound all these projections according to the usual geometri- 
cal rule of composition of forces, the result will be the projection of the equal line 
which represents in direction the resultant or total deviation. And similarly we 
may compound the four partial deviations of a near initial ray-line. 
