On Systems of Rays. 99 
The geometrical synthesis of the partial deviations may also be performed in other 
ways. or example, we may consider each partial deviation as arising from a partial 
or component rotation, and we may compound these several rotations by the geo- 
metrical methods proper for such composition. 
In particular, we may compound the final deviation of curvature with any of the 
other partial deviations, by making the deviated ray-line, obtained without considering 
the final curvature of the ray, revolve through an infinitely small angle round the 
axis of final curvature, that is, round the axis of the final osculating circle of the 
given final ray. By this rotation, the projection B, of a near final point B’ on the 
final perpendicular plane, will be brought into the position B’; and, by the same 
rotation, the near final ray-line, which had been obtained by abstracting from the final 
curvature, and by considering B, as the final point, will be brought, at the same time, 
into the position of the sought ray-line, which corresponds to a final point at B’. 
Applying now these general principles to the particular question respecting the 
condition of intersection of two near final ray-lines, from two near final points B, PB, 
(the colour x and the initial point 4 being considered as common and given,) we see 
that if the projection B, of B’ be given, the small projecting perpendicular B’B, or 
éz and therefore also the near point B’ itself may in general be determined s0 as to 
satisfy the condition of intersection: for the final ray-line from B, may in general 
be brought to intersect the given final ray-line, by revolving through an infinitesimal 
angle round the axis of curvature of the given final ray. We see also that the angular 
quantity of rotation and therefore the length z= B,B’ depends on the position of 
the projection B,, that is, on the co-ordinates dr, 8y ; and therefore that there must 
be some determined surface as the locus of the near final point B’, when the final 
ray-line from that point is supposed to intersect the given final ray-line. 
To investigate the form of this locus, by the help of the foregoing geometrical 
conceptions, we may observe that the only point, on the near ray-line from B;, which 
is brought by the supposed rotation to meet the given final ray-line, is the point con- 
tained in the final plane of curvature of the given final ray ; and that if we call this 
point where the ray-line from B, intersects the given plane of curvature the point P, 
the angle of rotation required is the angle between the line BP and the given final 
ray-line ; because the same infinitesimal rotation which brings the near ray-line from 
B,, that is, the line B,P, into a new position in which it intersects the given final 
ray-line, brings also the line BP into the position of the given final ray-line itself. 
Translating now these geometrical results into algebraical language, and taking the 
given final plane of curvature for the plane of xz, so as to satisfy the condition (R"), 
we find the following co-ordinates of the point P of intersection of this plane of cur- 
_vature with the ray-line (V”°) from B,, 
