compound lenses mid object-glasses. 239 
but when this is not the case, perhaps it will be found more 
convenient to use one of the following equations 
A/'-^A/= 
(m) 
or - A/'-A^sA/= 
= t ( 1 — m ') (“') 
which are derived from it by eliminating t, either wholly or 
partially from the second member, by the help of equation 
(c). These equations are universally integrable, and suffice 
to assign the aberration in any proposed combination of sphe- 
rical surfaces, however placed. 
Theory of the aberrations of infinitely thin lenses placed in contact . 
8. Confining ourselves at present to this branch of our 
subject, it will easily appear on integrating equation (/) that 
A /= ;rj>i Qi+ fs Q 2 + • • : •• PnQn} 
where we have 
0— — m (1 — m ) (r — f )* ( m r — (1 4-m ) f l 
in which it will be recollected that the value of f o is — D. 
Let us now examine the composition of this function more 
particularly ; and first, supposing n — 2, the case of a single 
lens placed in vacuo, we have ^ =1, m 2 = — =p l and 
A/==p, Q,+ Q 2 
If then v/e write p and m for p, and m , and make all reduc- 
tions, we get 
Q i = / [™r\ + ( 1 -f 3 m)r\ D + (2 -f D 2 + ( 1 -{- m) D 3 ] 
