Mr. Ivory on the astronomical refractions. 467 
The supplemental part is less than 
2 x _ 
z i m l + e z ' 
it is therefore very small when m is a considerable number, 
in which case the value of the integral will be found with 
sufficient exactness by means of the first series alone. But, 
with regard to the foregoing expression, we must not omit to 
remark what is a very curious instance of the artifices that 
must sometimes be resorted to in order to bring an analytical 
expression within the boundary of arithmetical computation. 
If the supplemental part be developed in a series of the powers 
of e, it will consist of precisely the same terms, but with op- 
posite signs, as that part of the first series which is multiplied 
by c m . In reality, therefore, the exact value of the integral 
is what remains of the first series, when the part multiplied 
by c m is thrown out ; which is also very manifest from the 
mode of investigation. But the series so obtained is imper- 
fectly computable. It belongs to that class called semi-con- 
vergent ; which converge indeed to a certain degree in their 
first terms, but afterwards become divergent. By adding 
and subtracting the same quantity in two different shapes, an 
expression is produced consisting of two parts, that can be 
calculated separately to any degree of exactness. 
For the sake of brevity, let the supplemental part be re- 
presented by c~ ”*• R : then, if we separate from the 
first series the part of A (o) multiplied by c~ m , we shall have 
f t T~= vki x l e + A<1) * 3 + A ' [2) + &c - } 
