280 New Algebraic Series 
ee 
efficient, or 12p* 
Now for the first of these co-efficients we have 
a(a+-k)(a+2k)(a+3k) - - - (a+(p—2)k) 
mes +b. a(at+k)(a+2k) - - ° (a+(p—3)k) 
Pah £ b(b+k). a(a+hk) - (a+(p—4)k) 
Rain = dees | 
+ s = a(a+-k). (b+) - (b+(p—4)k) 
74 — a. (bk) b+9k) - = (b+(p—3)k) 
+H(b+k)(b+2k) - - - = - (b4+(p-2)k) 
And for the second 
a(atk\(at%) - - - -- (a+(p— 1)h) 
+. Sige gail - = = (a+(p—2)k) 
oo P— b(b+&). a(a+k) - (a+(p - 3)k) 
Xc. 
2. Pe a(a+h). Bb+k) - - (6+(p—3)k) 
‘+ a bb+k)(b+2k) - - - (b+(p—2)k), 
FOOTE b+) = + = 4 (b+ (p—Mk) 
But as ee 
P: a fe. p-—2 pot 
1 Po ae 
Pp p-i vei pao par et Ti fae i 
i 
2 Bde er B 
&e. ‘ 
ok is plain that the above co-efficient can be decomposed » 
parts, the first of whichis 
