244 
Mr. Barlow on the rules and principles for 
° + g) * „ c 4 (a + 2) g _i_ 
c (a c'+a + I) ^ 2 r 
, a(c'+ i)» (c' + z — b) _i_ 
+ c' ^ zr' 
and ultimately to 
(c 4 g)° c 4 (a 4 2) g "I 
(ac — q)^ c (ac' 4 o 4 I)* ! ^^ x 
, (c'4i)% (c'42 —6)g r 2r'—^ ■ - W;. 
' (6c'4 I)® c' J 
And this I believe is the simplest form to which the general 
expression for the aberration of a single lens can be reduced. 
In the above form it applies to the case of diverging rays 
and for a double convex lens ; but it may be rendered appli¬ 
cable to every other case by attending to the proper signs of 
dy r, and r '; d being negative for converging rays, and r, r' 
being positive or negative accordingly as they are either or 
both convex or concave. 
(8). When the distance is infinite or the rays parallel, 
then c being infinite, this expression becomes 
I _I_ 
a®^(ac'4fl4 1 )® 
, (c*4 1 ) 2 (c"4 2 —^)g 
(6c'4 ^ c' 
and since also in this case 
(/= and c'= , 
a a 
this equation after farther reduction, that is, after substituting 
c in terms of a, becomes 
