2,51 
compound lenses and object-glasses. 
Let this assumed value of the refractive index be put for 
[x in our equation 
/= ([x— 1) (r — r 2 ) + ({x 7 — i)(r 3 — r 4 ) -f- &c. — D 
and similar values for jx', [x", &c., and we get 
/=( (J-— n + (/— l) (r— r + ) + &c. — D 
+ ' {/’(M- — i) (r—rj +/(|x'— 1 ) (r— -r) + &c.} 
+ — ■ 0 ( r ,— r 2 ) + ?'((»''— ») ( r ~> r 4 ) + & c- } 
+ &c. 
or simply (L, L', See. designating the powers of the several 
lenses for the most luminous rays.)* 
/=L + L'+L"+&c.--D; (q) 
+ c [hp + L 'p'+ L "p"+ See . } 
c 2 + L'g'-f- L" &c. | 
+ &c. 
In order then that this may remain the. same for rays of all 
colours, it must be verified independent of any particular 
value assigned to c , a condition which gives 
o = Lp -f L '/>'+ L "p"+ Sec.^ 
o = ~Lq -{• L' q'-\- L" q"-\- See. V ; ( r) 
&c. 3 
These equations, being infinite in number, while the number 
of the quantities L, L', &c. is limited, it is of course impossible 
to satisfy them all, by any adaptation of the latter quantities ; 
but as q, r, &c. decrease rapidly, we may confine ourselves to 
satisfy one or two of the first, as the others can produce but 
* It were to be wished that in physical optics the most luminous rays were always 
employed as the term of comparison. The mean or middle ray of the spectrum varies 
in every different medium, and has no distinguishing property which renders it sus- 
ceptible of exact determination, while the others, by their presence or absence, uni- 
formly mark the maxima and minima of optical phaenomena. 
K k 
MDCCCXXI. 
