. AND ON DEFINITE INTEGRATION. 485 
z—| n+] 
Z . (mtav) "afectea aorta "ts 
a a (n+1) 
|ovtan) cote) 489 Sf Noten) bap SGaeey) ‘ 
:: 2 [2] 
—a'.n(n—1)....(n— oj(meas) pes vt 
Any IA bed 
n—3 
_ a@.n(n—1) (n—2) , J(mbae) —(m+ay) 
30 [4] 
+a.n.(n—1)...- (n—4), }(m-ta) —(m-+ay) 
42 . [6] 
=, n—9 
5a’.n(n—1)..... (n— o\(m--ar) —(m-+ay) 
66 [10] 
BMC OE nan are oo to tai aac Sde sodok ts een ss (18) 
This series, which will terminate when 7 is 
any whole number, was, I believe, first given 
between the limits v=1 and a=, by Simpson, 
who obtained it by means of a very elaborate 
investigation (see his Essays, page 101.) 
* By changing this sign from minus to plus, we have 
z n a-1 n 
z. (m-+az) , instead of 2 .(m-+ar) ; which is a very 
y y 
singular property. 
38 
