112 THE REV. F. H. JACKSON ON 
The term of degree 27 is the following expression— 
{ nor sf 2), 2p? k N aa yp? 
{2r}1{2r}! {2r -2}NOr—2}NfQH apr + Part {2r}i{2r}! 
[4] ‘eine Ns rseave pee Wate sonar ; 
+h peaceen’ Groot opts: © 
eplBl{ as ars ee ie heres} 
PF} (arr — 4}I{4}! * (Br = 2dr — OVOP!” {4} {Or — 4} Ory! 
rir), 47] NA," 
ee [27] {2r}!{2r}! } 
We have shown in Art. (4) that in case \ =A, this expression is 
(A4ANMA+HAp2)(A+Ap*) 2 2. (A+AP")- (A+Ap?)(A+Ap4) (A + Ap”) 
{2r}1{2r}! 
It has been directly verified that for particular values of r (1, 2,3) the forms, in case - 
A be not equal to A,, are 
(33) _ 
_(A+A)(A +A, p?) 
{2} {2}! 
(A+A)MA+ AV? )MA +A p?)(A + Ay p*) 
{4} {4}! 
_A+A)At Ap) +A pt) A+AWM A+ APYA+ Ap®) 
{6}! 6}! 
respectively. This indirectly establishes the form of the coethcient of degree 2r in 
A and Ay, <A direct proof of the algebraic identity would, however, be preferable. 
Writing now 
x re 1)"p i al) ) =1+ Sn yn s aoe ss LE Aa z ae = -. = a 2! sas a a! a 1b”) 
(34) 
n=0 
If p=1, we obtain the addition theorem of J, 
Jj(UHAL) SSI AONTROw) SLInCOUR AMEE ae... . (85) 
The analogue of LommEt’s theorem 
2 
TQ) = (= 1} Sanaald) = MEE Teen F ee ee ; 
I have shown by two distinct methods * that 
(pi=*-— pra ea =) ieee ee ee Sas (py-B+«-1_ ]) 
K=o (pY-2=8 =") (pre-B . (py-o-Bte-1— J). (py = 1)(pytt - 1) a: (py+*-1— ]) 
Bee cc. ieee ae 
Seciieiee. TEAC GIIGESE y 
* Proc, Lond. Math, Soc., series 2, vol.i. pp. 71, 72, 1903, and Amer. Jour, Math., vol. xxvi., 1904, 
