TRANSACTIONS OF THE SECTIONS. 361 



this bent path to the positive semiaxes of coordinates, as func^ 

 tions oi X y z, af y' %', that is, of the six extreme coordinates 

 themselves, the colour being here indifferent. And Mr. Ha- 

 milton's general solution, for this and for all other questions 

 respecting combinations of ordinary reflectors, — a solution 

 which is itself a particular case of a more general result, extend- 

 ing to all optical combinations, — is expressed by the following 

 equations ; 



8V rv _ sjv 



(1.) 



y 



the characteristic function V representing, in all questions re- 

 specting combinations of reflectors, the length of the bent path 

 of the light, and being for the present mirror of the form 



'-\ 



V = a/ (^ - x'f + {y- y'f + (^ + ^f, (2.) 



but being different in other cases. Thus, for a reflecting sphere, 

 or for a Newtonian telescope, the length of a bent path of light 

 would depend differently on the extreme points of that path, 

 and we should have a different form for the characteristic Junc- 

 tion V ; but by substituting this new form in the equations (1.), 

 we should still deduce the connected forms of the six direction^ 

 functions or direction-cosines, a.^y, u' /3' y', and so might deduce 

 all the other properties of the telescope ; at least, all the pro- 

 perties connected with its effects upon systems of rays. 



It may be perceived from what has been said, that Mr. Hamil- 

 ton divides mathematical optics into two principal parts : one 

 part proposing to find in every particular case the form of the 

 characteristic function V, and the other part proposing to use 

 it : as in algebraical geometry, it is one class of problems to 

 determine the equations of curves or surfaces which satisfy as- 

 signed conditions ; and it is another class of problems to discuss 

 these equations when determined. The investigations which 

 the author has printed in the fifteenth, sixteenth, and seven- 

 teenth volumes of the Transactions of the Royal Irish Academy, 

 contain examples of both these inquiries, although they relate 

 chiefly to the second part, or second class of problems, namely, to 

 the using of his function, supposed found. He has endeavoured 

 to establish, for such using, a system of general formulae, and has 

 deduced many general consequences and properties of optical 

 systems, independent of the particular shapes and positions 

 and other peculiarities of the surfaces and media of any optical 



