146 



[68.], and are to be deduced from the characteristic function of the system in the manner 

 tliere described. To develope these equations (T""), we have, for the first members, 



d.r. cos. V „ . d.r. sin. v o / -• 



T— = 2^/r-. COS. V, r-/ = ^V'"- sm. v; 



d»/r d^r 



d'-.r. COS. u ^ ef'.r. sin. u . 



-^-^=2. cos.., ^-^_=2.sm.v; 



d^. r. cos. V d^.r. sin. u 



= 0' T~,Z3~ — ^ '• 



d^r' ~ ' d^r" 



and in general, wlien s > 2; 



d'. r. cos. '0 _ f. d'. r. sin v _ _ 



and for the second members, 



d'. x' ^ ^ ^ d^-'+^^J 

 'V'^ da^'.db^^ 



d\ 1/ ^ . ^ d^"+SA„' 



(U"") 



if we put for abridgment, 



,,.-( day, f cPa y. (.^Ys^fJLY^ (J^Y^ (J±_Ys 



X ([0]-^)-'- (corO"^.. (coi-'hx ([o]-)'''. (cor^... ([or> 



X [0]""'. [0]""'... [0]""^ X [0]"'*'. [0]~^^... COr^'; (V"") 



«,, «j, ... «s, /3^, /3s, ... /3„ being any positive integers which satisfy the following relation, 



«= «, + 2«2 + 3<»3 + ... i.«j 

 + /8. + 2/3, + 3^3, + ... s.^, ; (W"") 



and 2 being the symbol of a sum, so that 



2« = «, + «i + ... «„ 2/3 = ^, +/3s + ... /3,. 

 Developing in this manner the equations (T""), and observing that by our present choice of 

 the coordinate planes, we have, [68.], 



^=0- i^-0- ^-0-' il-o- 



db -"' da -"' db ~"" d^r-^- 



we arrive again at the same equations which in the preceding paragraph we deduced by substi- 

 tuting in ( S'"), for the components a, b, of aberration at the mirror, their assumed developments 



