Mr . Herschel orc circulating functions, &c. 
we may neglect in the expression of *A all those terms 
which have not for a multiplier; in that of Z A X , all but 
those multiplied by S _ ; and so on. This is a simplification 
of considerable moment, and will enable us in any assigned 
case to dispense with the troublesome and complicated pro- 
cess of determining l A x , &c. by elimination from (C) after 
n A x is obtained. In fact, if we suppose ( * ) 
’A, = S I { , C,.i I + *C I ' + + 
= { '*7. ' s x + 2 C 2 . S x _ n + • • • ’"C 2 • s x — mn+ n } 
&c. 
the relation between the constants which enter into these 
expressions may be assigned by merely substituting them 
in the equations (C), and after writing for s its value 
(V7r+(v?r+ &c} 
and for s , s , See. their values. 
x — n ’ x—-z n' 
+ + } ,& c . 
equating the coefficients of each separate term ( n ^/7Y, 
See. to zero. This will give m %n equations of relation 
between the constants, but it will be found that m of them are 
necessary consequences of the others, in virtue of the equa- 
tions (m in number) 
* In these equations the constants 1 C 1 > 2 C 1 , & c. 1 C 2 > &c. are not intended to 
represent the same with those in the expression above given for n A x which are 
denoted by the same letters. They are functions of them, whose form it is of no 
moment to enquire, it being sufficient for our purpose, that the one system of con- 
stants consists of an equal number with the other, viz. m x n. 
