256 Mr. J. F. W. Herschel on the aberrations of 
o= L | ar 1 — b%r ] + L '|aV a — f bf -j-Lr') r'J 
+ L" { a"r"‘-(b" ( "+(L+ V)e") r" } 
-J- &c. 
+ L c£+ Ucf+ L"c , y i + &c. 
+ L' L/' e '+ L" (L + L')/" f"+ &c. 
+ L'LY+ L"(L+L')*g"+ &c. 
and finally, substituting in this equation for £>', &c. their 
values , &c., and for b, c, e,f, g , &c. their values 
deduced from the equations of Art. 8, we obtain ; [w) 
= L{(« + i)r-^Lr} 
+ L' {(am'+i) / '•— ( (4 m'+ 4) L + i£±i L') r' } 
+ L" { ( 2 m"+ 1 ) r"’_ ( ( 4 m"+ 4) (L + L') + ^£± L L") /' } 
+ &c. 
+ (pc-if + (**'— 0* + 0"-i) A + 
4- - 3 £i! L L' ! + (L + L') L"*+ &c. 
+ («»!'+ s) L* L'+ (2 »/'+ 3) (L + L'V L" 4 - &c. 
In this equation it will be observed, the quantities r, r, &c. 
relative to the several lenses are not combined with each other 
by multiplication, nor do they rise above the second degree. 
If then we assume, or determine from other conditions the 
powers L, L , &c. the equation takes a form of great sim- 
plicity. Now, as we have already seen, the destruction of 
the chromatic aberration depends on relations between the 
focal lengths only, without any regard to the curvatures of 
the surfaces, and therefore furnishes equations tending to this 
very point. It i > a singular circumstance, and it cannot but 
be regarded as a very fortunate one, that the introduction of 
