^1^—. 
— - . 
NOTES ON LAGRANGE S THEOREM. 
[101 
(s being less than r). Or in a somewhat more convenient form, writing p, q and k 
for p — s, r—p and r — s, 
-(^)(^)=0, 
(ö) 
\(p +*) [p] p M* 
where s is constant and p and q vary subject to p + q = k, k being a given constant 
different from zero (in the case where k = 0, the series reduces itself to the single 
term -). The direct proof of this theorem will be given presently. 
s 
II. 
The following symbolical form of Lagrange’s theorem was given by me in the 
Mathematical Journal, vol. ill. [1843], pp. 283—286, [8]. 
If 
then 
x = a + hfx, 
Fx={Ì-Y dh 1 F’a e h f*. 
\da 
(7) 
Suppose fx = (f) (b -f ky\rx), or x — a + 1uf) (h + kyfrx), then 
Fx = h * ÌL ~ l F'a é^ (6+ ** a) . 
But 
/ J \ fa „ 
ghiji (b+kÿa) — I j dk ß h<t>b+k^a 
\dbj 
(In fact the two general terms 
[</>(& + kfa)} m and [^Y dk e k * a 
d\*ì k 
of which the former reduces itself to e Ha di (<f>b) m , are equal on account of the equiva 
lence of the symbols 
Hence 
e V db and (-^-i dk e k * a ). 
x = a + hcf) (b + k-tyx), 
Fæ = (J^ k dh~ 1 (J^j'^F'aehÿb+k+a. 
(8) 
and the coefficient, of h m k n is 
1 / /V \ m —i / d \ n 
1 (jt) F'a(^a) n . ^\) (#)” 
[m] m [n] u \daj vr 7 \dbj vr