Mr, Hersohel on circulating functions , &c. 155 
be the case, when the progression n A x , 1 A X , . . . l A x is 
exhausted, instead of going on to write “~ I , ~~ z A x , we 
must begin over again with n A x , &c. and so continue in 
regular rotation as often as may be necessary. Now to de- 
termine these functions, let each term be separately made to 
vanish, and a system of equations will be obtained as follows: 
•A,+V ■*,_.+ V —A_,+ " ax .""+*A + 
'A.+'i,- 'A x _,+ 2 &, • ”A m + ™b, . 
>(C) 
“A I +'^-"- , A I _,+^."- 2 A J _ z +. 
mi 7« — m 
K- m = m + I k x \ (») 
These equations are of that peculiar class which Laplace has 
distinguished by the name of “ re-entering equations " 
(equations rentrantes ). The elimination of 1 A X> 2 A X ,. . . . n ~“ m A x 
from them may be performed by the ordinary process for 
elimination between algebraic equations of the first degree, if 
we only prepare them by substituting x + 1 for oc in (a), 
x~i-2 for x in (3), and so on throughout ; because only one 
value of each of these functions is contained in each equation, 
and by this substitution that value becomes the same through- 
out the system. This done, the system is reduced to another, 
consisting of m equations, between the remaining m unknown 
functions, among which the process of elimination must then 
be pursued by the rules peculiar to equations of differences. 
Without encumbering our pages with the detail of this ope- 
ration (which a possible difference in the numerical forms of 
m and n renders necessarily very complex, when taken up in 
