504 Mr. T. Chaundy: Method of Line- Coordinates for 

 Again, since u = ly — mx, it follows that 

 y=.(u + mx)\l, 

 = {u + m(u 2 + v 2 )j2r} { 1 - (m 2 + n 2 ) } "*, 



= u + m(u 2 + v 2 )/2r + w(m 2 + n 2 )/2, 



to the third order on expansion. 

 Accordingly 



{cos 0' — (cos #)///,}?/ 



= [ (fi — 1)///, + (//, — l)///, 2 {m 2 + n 2 + 2(wm -f nu) /r 



+ (w 2 + ^ 2 )/V 2 }] x [> + m(u 2 + v 2 )/2r + w(m 2 -f n 2 )/2] 



= u( fJ L-l)/f J L+(f J L~-l)/2fjL{u(m 2 + n 2 )(l + ll f M) 



-f 2(mw -f nv)/fir + (m/r + u//j,r 2 )(u 2 -f v 2 )} 



to the third order. 



Hence by substitution in the first two equations of (11). 



m' = mjfju - (1 - l/fi)u/r -Qi — 1)/2/a{(1 + l/fi)u/r(m 2 + n 2 ) 



+ 2w(mw + ^v)/yLfcr 2 + (m/r 2 + w//^r 2 )(w 2 + ?; 2 )} . (14) 



4- 2(u/jnr)(mu + nv) + (m/r + u/fir 2 )(u 2 + v 2 ) }. (15) 



These are the equations of the second approximation, 

 correct to the third order. 



(6) The General Optical Instrument. 



The first order formulae given for a single refraction by 

 equations (12), (13) contain, of course, no more information 

 than is accorded by the Gaussian first order geometry. For 

 a general optical instrument they will clearly take the form 



m' = Pm + Qw, 



u' = Rm + Si£, 



where P, Q, R, S are optical constants of the system. In 

 fact, applying these equations to a ray passing in the plane 

 yOx and recognizing that, for such a ray, to the first order, 

 m is the inclination of the ray to the optic axis and u the 

 ordinate of its point of crossing the plane of reference, we 

 see that we have merely reproduced the fundamental 

 equations of Cotes's method, as demonstrated, for example, 

 in Herman : ' Optics' (Chapter 6), with m, u written for a, y. 

 I shall therefore suppose the theory of the calculation of the 

 optical constants P, Q, R, S sufficiently known and pass on 



