22 5 
compound lenses and object-glasses. 
pound lens is the sum of the powers of its separate com- 
ponent lenses/' than to express the same thing by saying 
that “ to obtain the focal length of a compound lens, we 
must divide the product of the focal lengths of its component 
lenses, by the sum of all the similar products which can be 
formed by combining them, omitting one in each combina- 
tion ;" or to announce that “ the power of a lens is equal to 
a certain coefficient multiplied by the difference of the curva- 
tures of its surfaces," than to assert that “ the focal length is 
equal to a certain coefficient multiplied by the product of the 
radii, and divided by their difference." This contraction in 
language is so convenient, that I hope to see it generally 
adopted. 
The formulae in the following pages extend no farther than 
the second term in the developement of the aberration, or 
that depending on the squares of the semi-apertures. It 
would have been easy to have carried them to the fourth, and 
even higher powers ; and should object-glasses of very great 
aperture, in comparison with their focal lengths, be ever con- 
structed, it may become necessary ; but the dimensions of 
our present telescopes are far indeed from calling for the 
immense complexity of algebraic symbols into which this 
attempt would plunge us ; not to speak of the tediousness of 
the numerical computations, where equations of the tenth 
and higher degrees are to be resolved. The general value 
of the aberration for any number of spherical surfaces placed 
at any finite distances from each other, is assigned by means 
of an equation of finite differences of the first order. The 
integration of this presents no difficulty ; but I have thought 
it unnecessary, in the present paper, to pursue it farther in 
