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. Fundamental Formula of Mathematical Optics. Design of the present Supplement 
. Fundamental problem of Mathematical Optics, and solution by the fundamental formula. Partial 
CONTENTS OF THE THIRD SUPPLEMENT. 
differential equations, respecting the Characteristic Function V, and common to all optical com- 
binations. Deductions of the medium-functions Q, v, from this characteristic function V. Re- 
marks on the new symbols a, 7, v.--+++.+-- 
. Connexion of the characteristic function V, with the: nies i letersi ms re general 
equations of a curved ray, ordinary or extraordinary ..........seseessseeeseeeees 
. Transformations of the fundamental formula. New view ae the ransshinn faye W: new auxi- 
liary function 7. Deductions of the characteristic and auxiliary functions, V, W, 7, each from 
each. General theorem of maxima and minima, which includes all the details of such deduc- 
tions. Remarks on the respective advantages of the characteristic and auxiliary functions ...... 
. General Transformations, by the auxiliary functions W, 7, of the partial differential equations in 
V. Other partial differential equations in VY, for extreme uniform media. dias of these 
equations, by the functions IV, 7... 
. General deductions and Stare ee of the differential cad, aed ee of a “eet or 
straight ray, ordinary or extraordinary, by the auxiliary functions W, 7. . Soprnccn 
- General remarks on the connexions between the partial differential Sais ob the Ee cee 
of the functions V, W, 7. General method of investigating those connexions. Deductions of 
the coefficients of V from those of 7, when the final medium is uniform.. 
. Deductions of the coefficients of W from those of V. Homogeneous ae eruations eenereee ea: 
Deductions of the coefficients of T from those of W, and reciprocally .. ta soasedenraone 
General remarks, and cautions with respect to the foregoing deductions. rs of. a S85 cache 
medium. Connexions between the coefficients of the functions v, Q, v, for any single medium. 
General formula for Reflection or Refraction, ordinary or extraordinary. Changes of V, W, 7. 
The difference A V is = 0; A W=A T= a homogeneous function of the first dimension 
of the differences A o, Az, A v, depending on the shape and position of the reflecting or re- 
fracting surface. Theorem of maxima and minima, for the elimination of the incident variables. 
Combinations of reflectors or refractors. Compound and component combinations........... 
Changes of the coefficients of the second order, of V, W, T, produced by reflection or Seen 
Changes produced by transformation of co-ordinates. Nearly all the foregoing results may be ex- 
tended to oblique co-ordinates. The fundamental formula may be presented so as to extend even 
to polar or to any other marks of position, and new auxiliary functions may then be found, ana- 
logous to, and including, the functions W, 7’: together with new and general differential and 
integral equations for curved and polygon rays, ordinary and extraordinary..........1006seeeeeereees 
General Geometrical relations of infinitely near rays. Classification of twenty-four independent 
coefficients, which enter into the algebraical expressions of these general relations. Division 
of the general discussion into four principal problems...........cssseeseeeeseseeceeceeceeteeeescesenseaeees 
VOL. XVII. A 
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