of the binomial theorem. 247 
? (i+.t)=A'(x-^+ — )+A"L* (1+- c ) (2) 
nothing can be easier than to find the value of the remaining 
series ; for it is quite obvious, from the equation expressing 
the property of the function, that L( is a particular value 
of <p( !-}-*£)• The same also appears from equation (1 ), in 
which, if we putA^-C ”— : z )= — 1 — , A" = — -, A' — n — T, 
the left hand member vanishes. 
Make then, in equation (2), A'=i, A f/ = ~ , and it becomes 
L(iW = ^~+rf 4 -te-p’(«+i), whence 
x — ^ &c - = L(i+^) + -~L*(i + x), and finally 
$(i+tf) = A'{L(i +x)+ j-L-(i + .r)} +A"L'(i +.r) 
Let us now endeavour to develope the function next in order 
of the same class, viz. <p ( 1 -J-jt), having the following pro- 
perty, 
<p(i+w}+<p(i+a?) +<p( i+y>+^(i+«)==' <p(i+x. 1 +y) 
i+z)-f 
<p(l +y • 1 + *) + <P(i +w. 1 + x) + <p(l + w . 1 +y) + 
<p('r-fw.l+2r)— ......(3) 
. i+y . 1+%) — » <p( 1 + w . 1 +y . t+%) <p(i+w . 
1 -J -<37 . 1 +%) — 
0(14- w . i-fia? . i+y).+ < p( 1 “i“ w • 1 +‘* 7 • i+jy • 1 +^)- 
Assume <p(i+.r)=A-j- A^-J-A'V-J- +A #»+ 
and make the requisite substitutions ; we shall find A =0, A\ 
A" and A"' arbitrary. Then to have the law of the coeffi- 
cients, find, according to the rule, the coefficient of x in the 
coefficient of % in the coefficient of y, and, comparing in this 
the coefficients, of the powers of w, we find 
K k 
MDCCCXVII. 
