9 REPORT—1843,. 
On a Theorem in the Calculus of Differences. By Sir Witt1AM Rowan 
HamILTon. 
It is a curious and may be considered ay an important problem in the Calculus of 
Differences, to assign an expression for the sum of the series 
X=u,, (w+n)"—u,_) “ - (@+n—1)®+4+u,_,. es (a-+n—2)"— &e.; (J.) 
which differs from the series for A” #” only by its introducing the coefficients u, deter- 
mined by the conditions that 
u;= +1, 0, or —1, according asx+i>0,=0,or<0. ..... (2.) 
These conditions may be expressed by the formula 
ao 
y= af Lee (CEA EB scalars aolainee ets (3.) 
awJo ¢t 
and if we observe yy 
T 
i. sin (at + 6) =a sin (at+o4+— 
n 
(+) sin (4¢+ 6) = a’ sin (att b +" 
we shall see that the series (1.) may be put under the form 
2 £7 at (ad ne n z) 
=— —(—) A tt——);...0 ee (4 
Kao Caz) aren (21-4 Y 
the characteristic A of difference being referred to x. But 
A sin (24a +4 @) = 2sine sin (2er+6+a4+2), 
A” sin (2ea + 8) = (2sin «)” sin (2ae+4 B+nat =): 
therefore, changing ¢, in (4.), to 2, we find 
if we make, for abridgement, 
eh g® ott Care Ye way, FMR ERIE REN (6.) 
© 
Again, the process of integration by parts gives 
da ae 7° de d®*A 
fs ees Sasso =if’ wrt da®-* 7 na? 
provided that the function 
ni mee 
a da 
vanishes both when # = 0 and when a = ©, and does not become infinite for any — 
intermediate value of «, conditions which are satisfied here ; we have, therefore, finally, 
Ka1.2.B.an (" da ° Hed nn* a: dna CE 
wy -and.c= 2a +n, tence eee es (8B) 
Hence, if we make 


