FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS. 33 
From these equations we obtain 
B, = l(a) 
B, = (a+ (1)) — Il(a) 
B= 2s mia + (2)) -™?2m(a4+(1))+ 22 ma) . ; . (20) 
Pi t+ P2P2 PiP2 PyP2t Py 
By=p,! | 2oa 2 ee ) M(a+(1)) (a) 
To establish the form of the ties 7 we assume that the law of formation holds 
for B, B, Bz... . . B,_1, namely, that 
= W(at+t(n—1)) Wat(n-—2)) ean II(a) j 
BasPel| Geil @ ea aloe) CP 
a T(a+(n—2)) W(a+(n-3)) no Ua) | 
Ba=Pot! | G2 ON a (aea ye | a ve (0)! ie 2}! f 
From (19) 
T(a+(n))=B, +B,” + Oey Mev + Be peda: + Bn 
Pr Pr! Pn} 
Replacing B, B, . . . . B,., by the expressions (21) we obtain 
B,=AM(a+(n))+A,M(at+(m-1))+...... +A, (a+(m—s))+.... +A,II(a) - (22). 
where 
— Pri 
“Gy 
x =, Pn! (7)n—1_ 
: (n)!(m — 1)! {O}! 
x =p | (%) a1 z a (1)n—» \ 
2 (ny!) (w= 2)1{ 1}! (w= 2)1{0}! 
—il si Pn! Roa J coe 2 oo s (ye. G 
a , Gan {s—1}!  (n=s)! cai i elie ie osaes PDS) 
{s—1}! in the first term of 2, is formed out of the set of elements p, p, . ~~. Pn-y 
and is. 
(Pn—1 + Pn—2 + Cece + Pn—s41)(Pn—2 + Ce FOP a + Dn—s41) Cele Conky Pn—s+1 
{s—2}! in the second term of A, is formed out of the set of elements p, py... - Pn-o 
and is 
(Pr-2+ epee ee + Dn—st+1)(Pn-s+ ms 1 +f Dregenl) ie Ais a at Pn—s+i+ 
We see that A, which is 
f ' ; Dn! (1) n—1 i 
2 (nx)! (n— 1)! {0}! 
may be written 
TOW Walon eect 29) aoe wins ene Up aigtOs) 
(n-1)! (pi+ ame » +Pn)(po+ sie ae 0006 Gia ae:, 
= Pr! = 
(2-1)! Py Ty iy 
since p, is {1}! for the n elements p, py... - Dn- 
