70 Professor Hamitton’s Third Supplement 
BB - BB, + BB,, (M) 
BBz being the projection on the element, and BB, the projection on the perpendi- 
cular plane, and we may consider separately the two near points Ba, B,, upon this 
element and plane, and the two corresponding paths, 
yr BRE Cee Beet (N*) 
instead of considering the more general near point B’, and the near path (4, B),. 
In this manner we are led to consider separately, as one subordinate class or set, sug- 
gested by the path (4, Bz ),, the system of the two coefficients . ; - ; distinguish- 
ing these from the eight other coefficients of the second groupe (L*), which corres- 
pond to the other near path (4, B,), ; and these eight may again be conveniently 
divided into two distinct classes, according as we consider the changes of final or 
of initial direction. 
We are then led to arrange the twelve retained coefficients of the expressions (D’), 
in four new sets or classes, suggesting four separate problems : 
First set & 5 eB Second, =, 2, 
x” dx oz” «OZ (0°) 
i da Sa 8B sp. aa ba’ ba’ op’ op’ 
Third, Sz? By? Sn? By > Fourth, EU? yp eae easy 
In each of these four problems, the initial point is considered as given, and may be 
supposed to be a luminous origin, common to all the infinitely near paths of which we 
compare the extreme directions. In the first problem, the final point also is given, 
but the colour y is variable ; and we study the final chromatic dispersion of the dif- 
ferent near paths of heterogeneous light, connecting the given final point with the 
given luminous origin : whereas, in the three remaining problems, the light is consi- 
dered as homogeneous, but the luminous path varies by the variation of its final point. 
In the second problem, the new final point By, is on the original path, or on that path 
prolonged ; and we examine whether and in what manner the final direction varies, 
on account of the final curvature of that original path. In the third problem, the 
new final point B, is on an infinitely small line 
v= BB, 2) 
which is drawn from the given final point of the original path, perpendicular to the 
given final element of that path, namely to the element 
ds= BB: ; (Q’) 
and we inquire into the mutual arrangement and relations of the final system of right 
lines which coincide with and mark the final directions of the near luminous paths, 
