82 Proceedings of the Royal Irish Academy. 



for all values of 0) ', £ '). But when 



•fn = £ = »J = Zj = °. .'/o = z o = IU = */ = °> 

 and 



m 



a 



.'/o 



Hence if 



it follows that 



k;l; HjHj ■ 



^0 



= 3£ 

 'Jo' 



?9 



IV 



Kj" 



■ k; 







A'," 



H", 









"'o "o M» 



KJ' 



AV 











(13) 



I.' = 'l/'. ?«' = £,-', 

 and the conditions (12) are clearly .satislied. Now 



&i 'i , = Zo 



A',' //, ,Jj ' .,„ //y 2,- 



t>> the first order, and this ratio is the reciprocal of the magnification. Hence 

 ( L3) represent- the Bine condition, and when it is satislied the instrument is 

 free from coina or the effect (of any order) which depends on the first power 

 of the coordinates of the object in the plane perpendicular to the axis. 



11. The equations 11 are general and reduce the problem of calculating 

 the errors for any order to the development of the function S. For a single 

 refraction omitting the similar terms in 



... A7A ( /. •, A,.'/,,- -'/, Iff,, , J,f . ) 



" . , ,/. A-,i A" KT A" '■ - Liy ^\ 



I,- I' \ I- '■ i i ' i \ 



~ " ■.//• 1 /( -- " 1i-i' • y v - ( ' '' l I'M -'l>-i>- 



But 



// , L , , , // /. 



//,., A",,, //,' A 

 Therefore in the sum, since W=lW 





$ + 2 Kin - 'ioX'// - 'M-ii + (&* - ?.)(?.•' - £'.-,.}• 



