2g6 Mr. J. F. W. Herschel on the aberrations of 
thin lenses in contact. Let us next examine how these re- 
sults will be modified by taking into consideration a small, 
but finite thickness in each of the lenses. To this end we 
may proceed as follows : 
If U=o be any equation of differences involving/,/ 7 , and 
t , where t and its values are so small as to permit their 
powers and products to be neglected ; suppose (/) to be the 
value of/ deduced from the equation on the supposition that 
t=o , then in general we may take 
/= (/) + « 
where u is a quantity of the same order with t. If this be 
put for/ in U=o, the equation, by developing, and rejecting 
the powers and products of u and t, will take the form 
o= V + W. u + X.u' + Y. t 
V, W, X and Y being functions of (/) and (/ 7 ), and it is 
evident that V vanishes by reason of the values assigned to 
these quantities. There remains then a linear equation of the 
first order between u and u which is easily integrated. In 
the case before us, we have 
which developed becomes 
f= <$> + m 'f + m'f 2 t 
In this, writing (/)+« for/, and retaining only the terms 
multiplied by w 7 , u, and £, we get 
u ' — m'u — m! (f) 2 1 
and integrating 
u = m 2 m } ....m n (/,)* t t + »„(/,)*«,+ •••• 
4 - m (f ) 2 / 
n w n — i ' n — i 
Hence it is easy to conclude, that if we have n lenses placed 
