Mr. Knight on the construction 
232 
L ■ s "— 
(Borda's Series.) 
Prop. VI. 
12. To expand L . «"•••*» and L . $"••••* into series of monomials of 
A 
the form — . 
J x r 
L(. + „)=L., + M{i-^+i_ 
m +I T z' I \ K+It . x r f M+I M 1 , M+I 
— , L( x + W — 1 ) = — f-L.x+M{ — i-. — +•+- 
(« + »)«! 
±f”L(^ + «_ 3 )=^L.x + M {l^i.2= 
M+I M— I , M+I (M — l) a 
2X 1 
_ »+. («-o; + l 
‘ 1 rx* - — ) 
(m + i)m m— 2 (m+i)m ( n — 2) 3 
. _j_ (m + i)m (h— 2) 
• * ° * — 1.2 
&c. &c. 
These added together will give L . a"- M . It is easy to see 
that L . a: will disappear, because its coefficient =(1 — 1 ) n+1 ; 
we have then, putting % to represent the sum of the terms 
formed by the different values of r, 
{ r n + 1. 
„ — 
-} 
where for r we are to take every whole number from one 
upwards ; thus 
L .«= M{~+ -1 4.-L4. + r -F+} 
L.*'=— M{^+ + 7^ + &c - } 
&c. &c. 
Hence L . a! equals double the sum of the second, fourth, &c. 
