112 Professor Hamitton’s Third Supplement 
which is intersected in its extreme points by another infinitely near path of the same 
colour. This general problem, of which the solution includes the general theory of 
the caustic surfaces touched by the straight or curved rays of any proposed optical 
system, may easily be resolved by the methods of the present Supplement. 
In applying these methods to the present question, we are to differentiate the 
general equations which connect the extreme directions with the extreme positions 
and colour, by the partial differential coefficients of the first order of the character- 
istic and related functions, and then to suppress the variations of xy za’ yz y. 
And of the partial differential coefficients of the second order, introduced by such 
differentiation, it is easy to see by (4°) that those of the characteristic function V, 
or at least some of them, are infinite in the present research: it is therefore 
advantageous here to employ one of the auxiliary functions 7, T, combined if 
necessary with the functions v, v’, or Q, ©’, which express by their form the properties 
of the extreme media. 
Thus, when the final medium is uniform, and therefore the final rays straight, we 
may conveniently employ the following equations, which involve the coefficients of 
the functions V7, ©, and were established in the sixth number, 
3W Pye} sw FY] sw rye) ” 
taytV yy arth yo tH atl ss: (W*) 
Differentiating these equations with respect to «7 v as the only variables, and sup- 
pressing the var iation of the first order of V7, as well as those of yz a’ y' z' x, we 
obtain 
OW cQ ow 7 80 
o=(5, + rae Sa ao +Ge+ Ve) Su, 7 
ow yee E Pay 
0= (55,7 V 3,5.) 8+ _ “) 2 (K") 
ow OW aw W 
iss =(ae + Vo) ie + Se Sho. Vo 20. or + 
and hence, by a symmetric Sanat 6 te and ies the forms of /V, Q, 
rr + en) Ge = ee ee Lae 
a aM may 2) 
V su 
Ww yko 1 
¢ Geter 2a.) Co. zay_ -Ge+r oe ee 
which is a form of the condition required, for the final and initial intersections of two 
near luminous paths, of any common colour, the final medium being uniform. The 
condition (L") is quadratic with respect to /’, and determines, for any final system of 
+ 
