=34
if n = a, b, =0 if 2n =& B, etc, Written out '
=14 0x+rx® 40+t HL1+0+ qua® 40 +rox® 404 000 ] vooe
The only element in the sequence {q,} whose weight is one, is q4.
The only elements of weight two are ¢, and ry. In general the only
elements of weight w are g, and r, if ¥ is an integer. The only
terws which go into C, are products of g¢'s and r's such that their
weight is w, Then

., ee p { 24 = :
hrhrh ..er where Zi + 22; v, and

Co = 205 950000 4
TT =1+ qux + (92 + 72)2% 4 (95 + 9292)%° 4 (P2 + gzry + Gags + gg)a® 4 *ooee
In a similar manner we find, in general,
TIL + @™ 4 rax®™ 4 £5a®t 4 e0ee ] =
149+ (qn +ra)x®+ (90 + 9202 4 22)2° 4 (G4 + G105 + ro + qgrs + gaty + ug)ateres

in which C(y= Zqia"'qih'j,""jktl,

‘ tl,"."
Bhopes 23 ¥ B2 4 T3V isee e in Tl

The product of a finite number of power series may be
written out as an spplicaticn of Theorew II, looking, for convep—
ience, at equations (12) fer the expressions of the €'s in terms
of the #'s. As an example, multiply
(1 + ayx + b,x' 4 **ce0)1 4 ax + b,x’ + °*** (] +-agx + bax’ 4+ *ovee ),
If the sum of the subscripts of Pires.r 1is greater than 3, the
valye of ¢ is gero. Expressed in a's, b's, c's, =v-- , We have
Cy=a; +ag +ag, '
Ca = @yap + ajag + agag + by + by + by ,
Cg = agagag + a,b; + agby + a3bg + aghy + 0zbs + agbg + ¢3 + ¢ + ¢35,

RN | R T ST N ITE R SR Wil 1 M A W Thi i gl e R
.

These are the results which would be cbtained by the use of Cauchy's
rule for multiplying infinite series.
In the case of a fipite number of terminating factors, the
formulas give the results which wouid be obtained by multiplying by
ordinary algebra. Both the finite and infinite cases are inecluded 4

in the one treatagent.