Mr. Herschel on circulating functions , &c. 
It will be unnecessary to enter into the detail of the pro- 
cess of elimination in this case ; it is always practicable, and 
leads to a final equation with constant coefficients, when those 
of the circulating functions which enter into the proposed 
equation are constant, just as in the case of one variable. 
( 14 ). lam unwilling to occupy the pages of the Philoso- 
phical Transactions with examples of the application of the 
processes here delivered to the various problems in the pure 
and mixed mathematics where they afford either a remark- 
able simplicity in the result, or great neatness in the investi- 
gation. Such instances occur frequently in the evaluation of 
continued fractions and other similar functions where the 
denominators (or other elements) recur in a certain order. A 
variety of complicated questions relative to the simultaneous 
employment of capital in different mercantile transactions, 
can scarcely be treated with perspicuity in any other way, 
and other instances will readily suggest themselves to the 
reader whose experience in enquiries of this nature has led 
him to feel the inconvenience which these pages are designed 
to obviate. I will therefore merely subjoin one example of 
the integration of a circulating equation of the second order, 
with constant coefficients, by way of illustration of the methods 
themselves. 
Suppose then we have 
u — (a S 4- bS )u 4- («S 4-/3S )u =0 
where S is ■§■ the sum of the x th powers of the roots of 
z* — l — o assume 
u =A . S + B . S 
X X X 1 X .T— -I 
and we get by substitution 
