compound lenses and object-glasses. 231 
intervals between the first and second, the second and third 
surfaces, and so on ; or 
t = A A ; t = A A e &c. 
1 1 2 2 2 3 
and let f,f, &c. be the reciprocal distances of the central 
focus after the 1st, 2d, &c. refraction, from the respective 
surfaces, or the values of ~~ , ^-1— , &c. We here suppose 
the positive values of r to correspond to surfaces whose con- 
vexity is turned towards the original radiant Q (provided its 
distance be positive) while the positive values of / indicate a 
situation of the focus q on the opposite side of the surface. 
With regard to t, its values are necessarily positive in cases 
of refraction, but when m = — 1 , which corresponds to those 
of reflexion, (which are thus equally included in the present 
analysis,) t has a negative value. 
This premised, if we make D = o, or the distance of the 
radiant point infinite, the focus for parallel and central rays 
will be assigned by the equation 
/= (1 — m) . r 
Let <p denote this value of/, and <?>,=( 1 — mj) r , &c. : then 
will <p t , <p 2 , &c. denote the reciprocals of the principal focal 
lengths of the several surfaces, in situ , i. e. supposing the 
adjacent media in each case continued to infinity. We have 
then in general the equation 
f— c p — mD. 
Suppose now/ and/', m and ml, a? and <p' to represent any 
two consecutive values of/, m, and q> in the series 
&c. &c. Then, since the focus after any refraction becomes 
the radiant point corresponding to the next, w r e have 
