4 THE REV. F. H,. JACKSON ON 
Differentiating the series (14) with regard to a” we see that the differential 
coefficient of the first term is zero and we obtain the series 
roa Ante" Dy eer a 
Sas 
y=0 Artery Pir ypin+2r—1) 
2 (2), (2) nee[? = 1}! [n + r|! 
Nrtert ly pln ter] 
= \zx —pin) Se eR a lg a 
po 2),(2) Vnevrl]! |n +1+ r|! 
= Agu 8 Fan (2,0) 
which establishes 
d 
qm { ersten) | = wafer) 18) 
By a change of the independent variable this may be written in the form 
n+l 1 yeti 
Jinn” 2 ney ‘ 
Ua |e PMNS (ae?) } : . 
from which we have by repeating the operations 
pri n-+1 
NYt1y ] ee) ree x) zs 
ly pie 7), eee | ee pet | 
See d(x?"") 4 as! dP") ) Se nan dr aoe bs Fe oP dx. i ees \ \ ; (18) 
a theorem analogous to 
4. 
Similarly we may easily verify that 
MeIg@t ye 2 of aJu(ePd) ) ; . (19) 
and in general that 
LMM (ar) = = \ oJ (a2) t (20) 
from which by repeating the operation 
n -—= d ics 
AHopIy ae" Dr) = oh a? — 7 { aa \.. 
x lie da") To AT eae A) ; . os (21) 
In the case p= 1, this becomes 
r+ { 
Ny pe) (aX) — (- as { "Tnss(ed) f 
