New Algebraic Series. 281 
has: (a+2k) = - = (a+(P—1)h) 
+ = Hak) (a+2k) « . (a+(p —2)k) 
—ilp—2 
+P FE 6(b-+h)(a+h)(a+(p—3)k) 
a 
es ee 
—1 
+P (ath)(b+K) = - +(p-3)h) 
4b(b+K)(b+2k) = = (b+(P—2)h) 
And the Second. . 
a(a+k) (a+2k) - - - (a+(p—2)k) 
=Y 
+= (b+h).a(a+k) - - (a+(@—3)4) 
oo - - ie Pes 
: Be ) 
EP aa h(O+h) -- O+(0—3)®) 
| Pa a(b+h (b+2h) = - 0+(P- 2h) 
| +(G+4)(b4+2k) - - - (6-+(p—1)k) ) 
hat the multiplier of a, 
_ In these expressions it is evident t 
in the first of these two parts, is that which would become 
—I 
ficientof—— -, when aischanged into a+4; 
and that the multiplier of 6 in the second, would become 
the same co-efficient, when 4 is changed into b+k. The 
co-efficient of 5-7. being therefore, reducible to 
e2et 
the form sch 
(a+b) (a-+b+k) (a+b+2k) - - (a-+b+(p—2)h)5 
the multipliers ofa and 4 will each become 
(a-b-k) (a+b+2k) (atb+3k) - - (atb+(P— 1)k) 
Both these multipliers being united, the co-efficient of 
oe will therefore be (a+) (a+5+4) (a+b+2k) - - 
fa+-b+(p ~1)k). | 
Vou. ViL—No. Pz 36 
