118 



rf* rf^F . dW 



da da^ da.dc 

 rf/3 d^V d^V 



da da.db db.dc 



(c) being the other coordinate of the perpendicular surface, connected with a, b, by the re. 

 lation 



r= const. 



which gives 



that is, 



^.da + ^.db + —.dc=0, 



ctda 4- ^db -(- ydc =: : 



and if we represent by (5) the portion of the ray, intercepted between this perpendicular surface 

 and the plane of aberration, we shall have 



By means of these formulas, combined with the equations 



a.dX + ^.dY ■\- y.dZ=.0 

 *.d^X + ,8.cP y 4. y.d^Z = 



«.rf"X+ /s.^" y + y.d'^z = 0, 



we can calculate the partial differential coefficients of the five quantities X, Y, Z, a, /3, consi- 

 dered as functions of (a, b) ; and if we wish to deduce hence, their partial differential coefficients 

 relatively to one another, we can do so by means of the following formulae, 



dX dX , 



dX = -T — . da + —r— . rf/3, 

 d» ^ d/i 



d^x= ^. ^« +^. d^^ + ^. d.^+2^-^.d».d,^-.d,^ 



together with the corresponding formulae for Y and Z. 



rsS.] As a first application of the preceding theory, let us suppose the distance between the 

 two rays so small, that we may neglect the squares and products of (»„ ,3,); let us also suppose, 

 that the perpendicular surface of which (a, b, c) are coordinates, crosses the given ray at the 

 point where that ray meets the mirror, and let us take that point for origin, the given ray for the 



