On Systems of Rays. 83 
ov 1, ow i ov OV ov IS 
eee Oe 0 zn 
Sal AG. nhOak TC aRy ACD aie DyA ced ow3B® \ ? eo) 
the given final ray becomes one of those which we have called principal rays in for- 
mer memoirs, and the point of convergence or divergence 0, 0, Z, is what we have 
called a principal focus. 
Second Application of the Elements. Arrangement of the Near Final Ray-lines 
from an Oblique Plane. Generalisation of the Theory of the Guiding Parabo- 
loid and Constant of Deviation. General Theory of Deflexures of Surfaces. 
Circles and Axes of Deflecure. Rectangular Planes and Axes of Extreme De- 
flexure. Deflected Lines passing through these Axes, and having the Centres of 
Deflexure for their respective Foci by Projection. Conjugate Planes of Deflexure, 
and Indicating Cylinder of Deflexion. 
17. The foregoing theorems respecting the mutual relations of the final ray-lines, 
suppose that the near final point B, is on the given plane which is perpendicular to 
the given luminous path (4, B), at its given final point B: but analogous theorems 
can be found for the more general case where the near final point B’ is not in this 
given perpendicular plane, by combining the solutions of the second and third of the 
four problems lately discussed ; that is, by considering jointly the second and third 
sets of coefficients (O°), and therefore by employing the following equations for a 
final ray-line, 
da. oa oa 
ox Peery ears az) : 
y=yte2 (@ ov +2 oy +2 ez ) - 
we =de +2( 
(A”) 
If, in these equations, we establish no relation between dv, dy, dz, then the system of 
these final ray-lines (4) is what has been called (in my Theory of Systems of Rays) 
a System of the Third Class, because the equations of a ray-line in this system in- 
volve three arbitrary elements of position, namely, the co-ordinates ea, dy, éz, of the 
near point JB’; but to study more conveniently the properties of this total system of 
the third class, we may decompose it into partial systems of the second class, that is, 
systems with only two arbitrary elements of position, by assuming some relation, with 
an arbitrary parameter, between the three co-ordinates dr, 4y, 8z, or, in other words, 
by assuming some arbitrary and variable surface, as a locus for the near point B. 
For example we may assume, as this locus, an oblique plane passing through the given 
point B, and having for equation 
oz = pea + goy, (B”) 
