( 752 ) 
Ar 2h 
a x h=w(—1)h (=) 
a’ b xv? BAT 
J’ (az) J” ba = sin?’ pd Lies 2 ee 
(aa) gg” be 2Ya I (v+3) dine ( -) pay F(v4h1), 
0 
i. 
or 
a’ bP a * J? (Qz) 
2 VaToth)) 2 
0 
J* (az) J* (bz) = sin?’ p dp, 
by which the indicated theorem of GEGENBAvER has been proved. 
2. With the aid of this theorem we can extend some well- 
known results concerning discontinuous integrals, in which functions 
of BrsseL appear; particularly do the two following theorems *) 
aa B 
i oH (uc) J’ (ua) du = Nae Bee 
a’ 0 for a >c, 
A 
ct? du 
J*H? (uc) J’ (ua) — = 
a’ v 
0 
lend themselves to this extension. 
The theorem of GEGENBAUER namely allows on the ground of these 
results to determine in certain supposition the value of the disconti- 
nuous integrals 
4 (c?—a’) fora<e, 
0 for a >. 
DR 
+l Ee du 
WZ | J’! (uc) J? (ua,) J’ (ua,) . . . J” (wan) ——— 
Meren dn” i un; 
0 
7 or : yo : 3 5 du 
W ss aar. La, | Use: (uc) Je (ua) B fe (wa,) ae je J (ua) na-D Ek] 
in which the number of ./-functions is arbitrary and v is > — 4. 
Let us think the positive numbers a,, a,,...a@, to be successive 
sides of a broken line OA,A,...A,, let us put 7 OArAri == pt 
and OA, = sk, then we find successively 
7 
1 Paya ) 1 J? (uss) os : 
-— J” (ua) J” (UA) = = sn” pr AP, 5 
Ga” 1 { 2 2» Va MG 44 8,” P, Py 
Be Re ec : TG) si d 
—— oh (US) ed (Ua 3 KIDS = 
sau” US, oe 2° Va T(v+3) se Sin Gig OP 
0 
1) Nietsen, p. 198, 
