Mr. Herschel on circulating functions, &c. 167 
° = S x{ A -" B ,-,+ “ A x-z} + S t -1 { B x~ 6 A x_, + ^ B x- 2 } 
whence we find 
A -4- a. A = a B 
X * X—2 X—I 
B +/3B — b A 
X 1 X—2 X—l 
The first of these gives B^ — 7 { A a , + 1 "+ * A x —i } > ( E ) 
which substituted in the second multiplied by a produces 
A, +I + (a + /3 — ab) A t _ i 4* « P A x—s ~ ° 
0r ’ A x +i — ( ab — A x + z + “P . A x =z O 
This equation corresponds to the equation ( D) of the fore- 
going articles, and if we take p and q such that 
(% a — p 9 ) (V — q*) = z*—(ab — a,— ft) z* - j- «/3, 
we shall have for the complete value of A 
A X= S x{ c. (/+ f) + G*. (/- 2 +V 2 ) } 
+ S I _,{ C . ( / + /) +,'.(/- 2 + /- 2 ) } 
This expression substituted in (E) will give the value of 
B , but as it is sufficiently evident that the expression so ob- 
tained, as well as the value of u thence derived, will all have 
the same form, the constants only differing, we may at once 
suppose u x equal to the second member of (F) and determine 
the relations between C, C , c, c' by substituting it in the 
proposed equation. This gives the following four equations 
of condition among the constants. 
( c + J.H 1 + — °fi c = ° 1 
(«+?)(!+ 7 ) -f(c+|:)=o 1 
(C + P)(i+p)-j(e+£) = o, 
( e + ?)( 1 +?)-;( c +F>=°i 
