On Systems of Rays. 69 
to compare the extreme directions of a given path of ordinary or extraordinary light 
of the colour x, from a given initial point 4 to a given final point B, which path we 
shall denote as follows, 
(A, B), bd (H*) 
with the extreme directions of an infinitely near path of infinitely near colour xX + 0x 
from an infinitely near initial pomt 4’ to an infinitely near final point B’, which near 
path we shall in like manner denote thus 
(A, B+ ox (1) 
we may do so by comparing separately the extreme directions of the given path 
(A, B), with those of the three following other infinitely near paths ; 
Astin CAMB), sho; 32h CE BY, S. 8d. C4 3B), : (K*) 
which are obtained by changing, successively and separately, the colour y, the final 
point B, and the initial point 4. We are therefore led, by this consideration, to 
examine separately and successively the meanings and uses of the three following 
groupes, out of the twenty-four coefficients of (D*) : 
sa SB &a Op’ 
1st groupe i a ioe A 
2 da da da 8B OB OB sai oo =P BB’ 9 
2d groupe we? ay’ 82? Sx? oy? See Ria e Sy a? dy > (L’) 
da da OB OB a!’ bo’ OB’ OB’ OB 
82" t) by ] 8a" ’ by ’ 8a’ ’ by > 82’ > 82’ >’ 8y/ > 82’ 
3d groupe 
But we may simplify and improve the plan of our investigation, by means of the fol- 
lowing considerations. 
Of the three comparisons, of the given path (/Z°) with the three near paths (K°), 
the third is evidently of the same kind with the second, and need not be treated as 
distinct ; because, of the two extreme points of a luminous path, it is indifferent 
which we consider as initial and which as final. We may therefore omit the third 
comparison (J<’), and confine ourselves to the first and second, that is, we may omit 
the consideration of the third groupe (L”), in forming a theory of the general rela- 
tions of infinitely near rays. For a similar reason we may omit the consideration of 
the two last coefficients of the first groupe (L’), and so may reduce the study of the 
whole twenty-four to the study of half that number. 
On the other hand, the second comparison (A°) may conveniently be decomposed 
into two: for instead of the arbitrary infinitesimal line BB’, connecting the given 
final point B with the near point B’, we may conveniently consider the two projec- 
tions of this line, on the final element or tangent of the given luminous path, and on 
the plane perpendicular to this element : that is, we may put 
VOL. XVII. a 
