100 Professor Hamitton’s Third Supplement 
oy. S ox + gy ) —ey 
Se ee eg! 
= Ou Bats, oy oe 7 <S oy 
so that the angle between the line BP which connects this with the origin of 
co-ordinates, and the given final ray-line or axis of 2, is 
Bee ey ee ee ws 
a= (~L=) gyde. (be + ay) (3 an + =y) (X") 
and this being equal to the infinitesimal angle of rotation, that is, to the small line 8z 
or BB’ multiplied by = or by the final curvature of the given ray taken with 
its proper sign, we have the following equation for the locus of the near point B’, 
when the condition of intersection is to be satisfied, 
ot 5. a5 se (2 ar +P y) — (Se += Say): (¥") 
oz ox 
which is, accordingly, the equation of the former conical locus (/?"), only simplified 
by the condition (#"), arising from a choice of co-ordinates. Without making that 
choice, we might easily have deduced in a similar manner the equation (#2), under 
the form 
3 *8 8 
a aw (2 aw +22 ay) — ay(S ax +5 ay ) a 
aH op ; 
in which each member is an expression for the infinitesimal angle of rotation divided 
by the curvature of the ray. 
Another way of applying the foregoing geometrical pritieiplan to investigate the 
condition of intersection of two near final ray-lines, is to consider the infinitesimal 
angle by which the ray-line from B, deviates from the plane containing the given final 
ray-line and the near point B,. This angular deviation is expressed by the numera- 
tor of the fraction (Z"*), divided by 8, that is, divided by the small line BB,; and 
the denominator of the same fraction (Z"), divided also by 8/, is equal to the final 
curvature of the ray multiplied by the sine of the inclination of the line & to the 
radius of this final curvature: and hence it is easy to see, by geometrical considera- 
tions, that the fraction in the second number of (Z") is equal to the infinitesimal angle 
of rotation required for destroying the last mentioned deviation, divided by the curva- 
ture of the ray, and therefore equal to the ordinate dz of the sought locus of the near 
point B’, as expressed by the first member. We might therefore easily have obtained, 
by calculations founded on this other geometrical view, the same condition of inter- 
section as before, and the same conical locus. 
