244 
Mr. Barlow on the rules and principles for 
a ( c + q) * v C + (ff + 2) q 
a(c'+ l) : 
(be' - f i)* 
c (a c ' + a + i ) * 2r 
(c ' + 2 — b) i 
2 r 
>xy 2 =p; 
and ultimately to 
(c + ?) 2 c + (a + 2) g 1 
‘ a+ i)* [ 
(ac — q ) 1 
c(ac' + fl+i)* l 
(c'+2 — 6) q j 2 r' ” (?)• 
(6 c' + i ) 2 ~ 
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 
d y 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 
(ac'+n + i) 
0 
l ( c< + 0 2 (c *4 2 ~b)q 
T (6c'+ l ) 1 * c 1 
and since also in this case 
d — l a + l ) r and c' — 
l 
a v . 
= p ; 
2 r ' £ 
(a + l)q 
this equation after farther reduction, that is, after substituting 
c in terms of a t becomes 
4 a 3 
6 a* 
4 a 
+ i 
2 a 
<7* +5 :i 
q -j- 2 a 3 V 
+ I J 
a 
2 a 
a 
(9 + 0 
y - 
2 af 
— P 
