8 Professor Airy on the Splierical Aberration 



(^since the part independent of a must be the value of - when a = 0, that is, 



must = jA ' ^"tJ taking the reciprocal, and subtracting C, 



c-C=-^Ab-B) 



2 In ^ 4{n~i) ) 



Substituting this value of c — C in the expression for the proportion 

 of the tangents, it becomes 



^1, , B+C,, p.^aV/ 



4(n-iy 



£f n'Cf'^ \) 



+ -^(n + 2.tJ' + 4n + 4.ez; +3n + 2e-) + — - — ^—^A\ 

 n ^ 4 In — 1)V) 



This, for abbreviation, we shall call 7-, {1 + Q}- The first term of Q may 

 sometimes be more conveniently calculated by observing that 



1 1 1 B + C , ,, . B+C C C „ 

 F^B+C^-BiT' '"'^ ''^"■'f*''" -W = Bl = Bf- 



Prop. II. To find the proportion of the tangents of the 

 angles after refi'action through several lenses. 



The proportion of the tangent of the first angle to the second being 



B B' 



^ {1 + Q], that of the second to the third ^ {1 + Q'}, &c., the proportion 



/? /J' Jirn 



of the first to the last will be ^^, ' ' . {1 4-Q + Q' + &c.} : and it is only 



necessary to show how Q, Q', &c. are to be calculated. First, it will easily 



B' B" 



be seen that a'=-^a, d'=-^,a', &c. Then, since c + 6' = interval be- 

 tween first and second lenses = C + B', we have 



6' - 5' = - (c - C), 



