462 TRANSACTIONS OF SECTION A, 
the total coefficient AOD . piut+) being obtained by giving ¢,, t,,. . ., tm 
all possible positive integral or zero values consistent with tn +tnot+...+¢,=¢, 
and adding the corresponding values of P. 
Now P Sky) SF ky) Akin) x pe 
=i} AC = t,) *At) Ak a t,) fees *Htm)- Fig — tn) : 
where e = $4 (¢,+1)+4t,0@,+1)+ ... +3tn(tnt 1) + 4,4, + t,(k, + ky) 
te. thnk that... + hm) 
= $¢(t +1) + (u,—#,)(¢-4,) + (u,—2,)(2t-t,- t,) 
+ (us — ts)(8t — t, — 2t, — 3t,) + re 
Hence P=Nplu+»; where N is the number of subgroups with ¢,, invariants 
m, tm-1 invariants m—1,..., ¢, invariants 1 in an Abelian group of order 
Mog (m—Ditm—y+ +++ 4% with Um invariants m, w%m_, invariants m—1,..., uw, in- 
variants 1. The index of each subgroup is p'-'; for k—-t= muy, + (m—-1)upn-1 
Fie es $U,—Mty—(M—1)tpi—...—t. 
Now the equations vw, +t, i+... +Us—tm—tmi— ...—t4,;=h% may be 
put in the form Us —t541 = Ky — Kg 41(Um =him). Hence if we take every abstract 
Abelian group whose order is a power of p with no invariant greater than m, and 
find every subgroup with ¢ invariants such that (for all values of s lying 
between 1 and m inclusive) the number of invariants s+ 1 of the subgroup differs 
from the number of invariants s of the group by a given integer /,(=k,—k,,1), the 
number of subgroups so obtained is 70 6D » Where p*-' is the index of 
each subgroup. 
If we take every possible abstract Abelian group with no invariant greater 
than m, and find for each group all subgroups of index p'~' with ¢ invariants 
which possess no more invariants s+1 than the group possesses invariants 
s(s=1,2, .. .,m), the total number of subgroups so obtained is 
( ; 
FO). AE=0 “re 
SG=9) (=F) 2 A) 
For ai: the number of ways of selecting positive or zero integers such that 
L4+21,+...+ml =k. 
where @;, is the coefficient of y* in the expansion of 
7. Factorisation of the Pellian Terms (Tny Uny dc.) 
By Lieut.-Col. Attan Cunnincuan, R.2. 
Let r,2—Dv,2= +1, and—(when possible)—let one of 
/ 7 Qe 2 
"Eo Dybe a2, Di tee Besa cy f where Dy Dy=D 
Taking =1, 2, 3, .--. gives the successive solutions of above. 
(i) Then vs, =27,v, always 
V_n—1=2T,,U, or= X,Y, always 
where (T,,, Un) = (r'n, v'n) OF try %p) | formed by same rule as Tp, vp, 
(Xn, Yn) = (En, gn) or (2,,, Yn) 1.0.5 T+] = 27, « Ta—Tn—-1, When 2 >2, 
Also ra.) = (2u’n—1)(2v’, +1), when D =2, 
(ii) As to 2! forms: v’,, =a?+b2, To = 2r,,?>—1 always. 
and when D=2, 7’,=e—2f%, v’',=a?+0?, r,=c?+2d? and ron has the three 
2i¢ forms, 
' Proc, Lond. Math. Soc., 2, v. p. 3. 
