258 Mr. J. F. W. Herschel on the aberrations of 
of the same order with the powers and curvatures of the 
glasses, it may be a matter of some moment. 
21. The equation (m) being developed becomes 
0— ( 4 * w “j" 4 ) ^ L ,/ r // -J-&c.-| 
— j 3 ft±I L * + 3^+_i 3 ^+i L //l + &C. } 
l (A.— I 1 /x/ 1 1 [A, I 1 i 
— {(4».'+6) LL'+(4,m"+6) (L+L') L"+&c. } 
which being of the first degree, adds nothing to the algebraic 
difficulty of the problem. 
22, Let us apply these results to the case of a double ob- 
ject-glass, and putting, as in Art. 16, for the ratio of the 
dispersive powers, and writing for L and L' their values 
and 
Zjl 
the equation ( w ) becomes, 
o=(2ra-fi)r’. 2 t*+ l 
/A. I I ZJ 
lr 
(*) 
+ { fcrr,)*— ( 2 m '+ 3 )™ + -^rr v '~ 
+ { ( 4 «*'+ 4 ) — • rzz 1 ’ 1 — «'(«'+>)* 
while (jy J reduces itself to 
o=:(4m + 4)r-(4ff^ , -^. 4)^rJ , 
— (4 m + 6 > + ^zrw‘}rz5j’ ^ 
To reduce these equations into numbers, we may ob- 
serve that the value of sr is that which varies within the most 
considerable limits. If we combine the least dispersive flint 
with the most dispersive crown or plate glass which have yet 
been observed by Dollond, Boscovich, Robison, Brewster, 
&c. and vice versa , we shall find 0-51 and 0782 for the 
minimum and maximum of this quantity ; but it is rare to 
meet with the extremes. Mr. Tulley was so good as to 
communicate to me the highest and lowest dispersions of 
the two sorts of glass, which had occurred to him in the 
