Plummer — On the Symmetrical Optical Instrument, 69 



providing for mirror's of the usual form. Tlic development of X will not be 

 carried to a higher order. 



Let the ray considered meet the refracting surface in P (X, Y, Z), and 

 let iV,-.,, N;. be the feet of the] perpendiculars from the fixed points 

 (Cj_,, 0, 0), (c,-, 0, 0). Then 



w i = IM _, . m.iP + ju,- . pi?i 



= -[fi\l{X-c) + mY+nZ\i 



= - [,i (mY + nZyf - [i, (1 - m* - w 2 )i («•; - c + . . . )]_ 



= - fn («,- - Cj) + m-i (en - a. ,) + JF ; "+ 0™ + . . . 



where Wi" represents the terms of the second order in m, n, Y, Z in the 

 development of W t , Wi" those of the fourth order. Thus 



wt = - 



ft \(mY+ hZ :i - i (at - c) (*»« + «») + 1 (P + #)] 



*' i J i- 1 



- g(o*~c)(m' + » ! ) s 



i ji-i 



It is evident that 7PV 7 is a sum of similar functions of (m, Y) and of («, Z). 

 Only the first need be expressly considered. Then 



1 

 If?' = - T (,j;mi - m-iPK-i) - ^ (fi - fH-i) I - + £L> («» - c)w( 2 ]. 



*j? i l ~V 



But this expression must be a minimum for small displacements of Y. 

 Hence 



Y = _ nljWf - Hi-iin-j^ Z _ /mil; - flL^ i-! ,n\ 



1'i fit - ih-x l'i Hi - M-i 



This expresses the law of refraction to the first order only. But the 

 correction which is required is of the third order, and, owing to the minimum 

 property, will affect W t in the sixth order only. Hence the law of refraction 

 is sufficiently represented when W t is developed to the fourth order only. 

 The first part of Wi' now becomes 



Wf = ir^" I '^'"' £ + \ m {a, - in)m? - |^(«, - c,J >n%. 



m - m-i 



Conjugate planes may be defined as planes perpendicular to the axis 

 through the points (c Uv 0, 0;, (c,-, 0, 0) when c,.j, <; arc such as to make the 



R.I. A. PROC.,VOL. XXXIV, SECT. A. [12] 



