100 THE ROYAL: SOCIETY-OF “CANADA 
The identity (6) can also be proved by mathematical induction. 
It holds, as we can verify without difficulty, for s=1, 2, 3. Assume 
that it holds for s—1; then 
ee pe lip-2...p5 a1 Se (a) Pras perse Be 
Tin ee | DTS DETTE CARRE 
a Pas al 
ae ne Fat ee 
1 
SEE (or) =, Ps = rene .v+2s—2  (i) 
Replace p by p+1 and » by +2; hence 
De es 
ae pen NICE 
Sat aE Pastore Paes 
(a ae pa RES ee, mere aL 
—— DAS RATS. 
Ses Je 4 ,pv+5...r+s+3 PRE te 
1 (ii) 
ho er ES PES ETES 
Mn bre Ds : À p+s Soe 
Multiply (i) by ras and (i) by ER thus obtaining 
ORNS pou) Dame Poe Dors +2 
yes © “A >4O p+3...r-st+l: vts 
ie aa pres i A0 DELA RE 
en D De Ed y oo Es 


ee re Gi) 
and 
pts prl.pt2...ets ne ar ER a pe. prs Me 
PES vtl.v+2...v+s ae ) BC en D. TE 
— DRE EE À 
0: Je Dome ee à de 
Subtract (iii) from (iv) oy we get the identity (6), thus completing 
the proof by mathematical induction. Incidentally we have to use 
the entirely elementary but interesting identities 
(s—1) (v+s+1)+1.(v+1)=s(v+s) 
(s—2) (v+s+2)+2(v+2) =s(v+s) 
(s—3) (v+s+3)+3(v+3) =s(v+5), etc. 
Attention is called to the presence of the factors y, v+2, v+4, 
y+ 2s in (6); such factors are of common occurrence in the theory ‘of 
series proceeding according to Bessel Functions, but the algebraic 
identities such as (6) containing these factors in an analogous way do 
not seem to have been specially studied. 


