S62 THIRD REPORT— 1833. 



combination. A few results less general than these, and yet 

 themselves extensive, may not improperly, perhaps, be men- 

 tioned here. 



When we wish to study the properties of any object-glass, 

 or eye-glass, or other instrument in vacuo, symmetric in all re- 

 spects, about one axis of revolution, we may take this for the 

 axis of ^, and we shall still have the equations (1.), the charac- 

 teristic fmiction V being now a function of the five quantities, 

 x^ + y^) X x' + y y', x''^ + y'^, s:, z', involving also, in general, 

 the colour, and having its form determined by the properties of 

 the instrument of revolution. Reciprocally, these properties 

 of the instrument are included in the form of the characteristic 

 function V, or in the form of this other connected function, 



T = ao;' + /S?/ + y« - «'^' -|3'y - y's' — V, (3.) 



which may be considered as depending on only three inde- 

 pendent variables besides the colour ; namely, on the inclina- 

 tions of the final and initial portions of a luminous path to each 

 other and to the axis of the instrument. Algebraically, T is in 

 general a function of the colour and of the three quantities, 

 a? + ^^, a a' + ^ |3', u'^ + /3'^; and it may usually (though not 

 in every case) be developed according to ascending powers, po- 

 sitive and integer, of these three latter quantities, which in 

 most applications are small, of the order of the squares of the 

 inclinations. We may therefore in most cases confine our- 

 selves to an approximate expression of the form 



T = T^o) + T('> + T^"), (4.) 



in which T'"' is independent of the inclinations : T^^' is small of 

 the second order, if those inclinations be small, and is of the 

 form 



T(2) = P («' + /3^) -h P, (a a' + /3 /3') + P' (a'2 + ^'«) ; (5.) 

 and T^*^ is small of the fourth order, and is of the form 

 T^") = Q{u' + ^r + Q; (a^ + /3^) (« «' + ^ /3') + Q'(a^ + /3^) (a'2 + ^'2) 1 



+ Q„(«a'+/3^')'+Q'/(««'+/3/3')(«''+/3") + Q"(«'' + /3T; r '' 



the nine coefficients, P P, F Q Q, Q' Q,, Q'/ Q", being either 

 constant, or at least only functions of the colour. The optical 

 properties of the instrument, to a great degree of approxima- 

 tion, depend usually on these nine coefficients and on their 

 chromatic variations, because the function T may in most cases 

 be very approximately expressed by them, and because the 

 fundamental equations (1.) may rigorously be thus transformed ; 



