> ■ (16) 



Investigating Aberrations of Symmetrical Optical System. 505 



to consider the second approximation, involving terms of the 

 third order. 



I shall show that for this approximation the formulae for 

 m' } u' are always of the form 



m [ = T>m + Qu + i{ (Am + Bu) (m 2 + ri l ) 



— 2(Cm + Bu) (mu + nv) + (Em + Fu) (u 2 + v 2 ) } 

 u' = Rm + Sw + \ { (Gra + Ru) (m 2 + rc 2 ) 



— 2 (Ira + Jw) (mu + nv) + ( Km + Lw) (u 2 + v 2 )}, j 



where A, B, 0, D, E, F, G, H, I, J, K, L are optical 

 constants of the system. 



Now these forms are invariant for any number of coaxial 

 spherical refractions so long as we are content to retain 

 small quantities up to the third order only. At any rate it is 

 evident that the equations (14), (15) are of this form, though 

 certain of the constants, namely A, C, G, I, are absent, i. e. are 

 zero in this case. 



For a series of refractions suppose m', n', u\ v' expressed 

 in terms of intermediate coordinates M, N, U, V by equations 

 of the form (16), at the same time that M, N, U, V are 

 themselves expressed in terms of the final coordinates 

 ra, n, u, v by a second set of equations of similar type. 



Now, in the third order terms it is sufficient to substitute 

 the first order expressions in M, N, IT, V, i. e. expressions of 

 the form pm + qu, pn + qv, rm + su, rn + sv. Thus the third 

 order terms in m', n r , u', v' have the same form in m, n, u, v 

 as they had in M, N, U, V. Again, in the first order terms 

 we replace M, N, U, V by expressions in ra, n, u, v of the 

 form of those on the right of (16). Hence, after both these 

 substitutions we have expressions for m', n' , u', v' in terms 

 of m, n, u, v of the form stated in equations (16). 



Hence the form persists through a pair of refractive 

 systems, if it persists through each separately. Since we 

 have seen that it holds for a single refraction, it follows that 

 it holds for our general optical instrument. 



The third order functions of m, n, u, v, 

 namely 



±{(Am + Bu)(m 2 + n 2 ) 



- 2 (Cm 4- Du) (mu + nv) + (Em + Fu) (u 2 + v 2 ) J- 

 and 



i{(Gm + Hw)(m 2 4n 2 ) 



— 2(Im + Ju) (mu + nv) + (Km + Lu)(u 2 -f- v 2 ) \ , 

 which must be added to the first order expressions ~Pm + Qu 



