654 DR J. HALM 
This is ScHLAFLI'’s contour-integral of the LEGENDRE-functions, which thus is seen to 
satisfy the differential equation : 
dy d 
(1 = 2) 4 98 n(n On (15) 
Our general equation (3) includes therefore also the LecENDRE-functions. If we denote 
the solution of (3) by the symbol C,*(w), the Seiche-functions are represented by 
C,-(w), the Lecenpre-functions by C,}(w), and the Sroxxs-functions by C,'(w). | 
Since (3) is a special case of the hypergeometric differential equation, its integrals 
can be at once expressed by hypergeometric series of the type F(a, 8, y¥; w). From 
Jacosi’s schematic table of particular solutions given in his “ Untersuchungen tiber die 
Differentialgleichung der hypergeometrischen Reihe,” Crelle’s Journal, vol. lvi., we 
obtain thus the following 24 possible integrals of equation (3) :— 
1+w\i-¢ l-—w 
2. aes xF(g-n-a, f+nta,at4; 3") 
+ w\-n-2a w—1 
3. ee xF(n42a, busta, at; ) 
2 w+ 
4. (") xF (=n, k-n-a,ath; ary) 
2 w+il 
Grove II. 
1 — w\-?-24 w+_ti 
il. ( 5 ) xP (n+2a, ktnt+a, at+4; e+) 
1-w\" w+l 
2) = em = 
;. ( 2 ) x F( en PO aa w 7) 
x ey eam 
3 F(u+2a, —N, a+5; 9 
4-0 ; 
4, i x P(S+n+a, $-n-a, a+}; a 
Group ILI. 
i (45%) xF(n+2a, d+nt+a, 2n+2a+1; a 
2 l-w 
=p=Gi= —a 2 
2; io (iy x F(1+n, d+atn, 2n+2a+1; ) 
2 2 l-w 
: l+w —n—2u e 1 9 p 2 ) 16 P 
3, ( ) x F (n +20, 4+n+a, a comer (16) 
a 4-a —n—a-} 9) 
ae —— xF(Jtnta, l4n, n+2a41; >) 
4 
