245 
determining the dispersive ratio of glass, &c. 
Observing that + ^7 > / being the focal length ; 
or writing A, B, C, D, for the several coefficients, making 
also p' = 2 p a f, and calling y = 1, it is 
P' = 
A q* + B q + C 
( 8 ). 
D (q + I) 1 
9. This equation in the common form of object-glasses 
belongs only to the plate or crown lens, which receives the 
direct or parallel rays ; therefore the value of a, which enters 
into it, may always be considered to fall within the limits 
a = -50, and a =*53- 
When a = *50, this in numbers reduces to 
4‘5 + g _ aJ 
(?+0* 
and the solution gives 
p'— • 50 ± V \ (p'—-s°)* + (p—i-16) (4*s — p')j 
?= L ^ 5=7 ( 9 )- 
When a = -51, 
p' — S3 ± V \(p'— *53 ) 2 + (/>'— 1 * 146 ) (+’47— p') i 
a — j 3 - - (io). 
* 4’4 7—V x J 
When a = ’52, 
P ' — - 56 ± V Up' — ' 56)*+ (p' — 1*127) (4-44 — p')l 
</ = — ! 1 fS=7 ' * t 11 )- 
When a = ‘53, 
P'—'S 8 ± (P— -S8) 2 + (p'— i-ii) (4-42— p')} 
q = ; J - - ( 12 ). 
H 4‘44 1 — P v ' 
10. Having thus (equat. 7.) found a general expression 
for the aberration of a lens when the rays emanate from a 
given point, and in (equat. 8.) the expression for the aberra- 
tion of a lens receiving parallel rays, the indirect method by 
which an equal and contrary aberration in the two lenses is 
produced may be thus described. 
