-*' + 9W'*|4-i ...... (io.) 



a 



Aberrations of a thin Lens of Uniaxal Crystal. • 291 



leave that quantity T expressed as a function of a! and <r only ; 

 £ being here treated as a known function of £, y' of a', and 

 v of <r. And since, by (1.) (5.) (6.), 



ST = ^8(7 + zSu-^'Sa'-z'Sy', .... (9.) 



we have, by (3.) and (4.), the following equations, for the in- 

 cident and refracted rays respectively : 



V 4 <T 



4. The approximate equation (2.) of the section of the re- 

 fracting surface, which gives £ as an explicit function of £, is 

 now to be combined with the following analogous expressions 

 for the functions y' and o, deduced from the relations (3.) 

 and (4.) : 



y = 1-^-; „ = ,,-!£;...... (12.) 



and thus,- the expression (7.) for T becomes, if we neglect 

 terms which are small of the sixth dimension with respect to 



T m T< 2 > + T (4) ; (13.) 



T W = £(a-_ a ') +$rr , G*-l); (140 



T W = i 5 r(^-l)-irr(^-«' 2 ).. • (15.) 



And to eliminate £, it is sufficient to employ the equation (8.) 

 under the approximate form 



ft TY2) 



o= ^~ ^-«' + rJH; (16.) 



for although the complete expression for the abscissa £ of in- 

 cidence contains terms of the third and higher dimensions 

 with respect to a! and <r, yet the introduction of these terms 

 of £ would only introduce terms of the sixth and higher di- 

 mensions, in the expression for the function T. 



5. Retaining therefore £ as an auxiliary symbol, of which 

 the meaning is determined by the formula (16.), and making 

 the abscissas x' and x to vanish in the equations of the two 

 rays, (10.) and (11.), in order to discover the relation between 

 the ordinates z' and z of the intersections of those two rays 

 with the axis, we find 



U2 



