656 DR J. HALM 
F(n+2a, -n,at+4; =) 
(C4")'F(-n, $—n-a, 1-2n-2a; z ) 
2 liw 
= 4—a n+a—h 
(5°) | F(}-n-a, l-n-2a, 1-2n-2a; a) (16a) 
eso) * F(Q+n+a, t-n-a, 2-a; ==") 
3") (GS) F(1 -n—-2a, l+n, 3-a; = 
a ee F(1-n-2a, L_—n-a, 3B_q; 7) 
DB} 9 = w+i 
Note added on June 30.—The sixth integral in (16a) agrees with Professor 
CurystaL’s Lake function, which is obtained by substituting a= —3, n(n—1)=c and 
! = =z, so that Lic, z)=z F(n, 1—,2; z). The other corresponding integral is re- 
presented by No. 1 of (16a), viz., F(n—1, 7,0; 2), but it belongs to the exceptional 
class y=0 of the hypergeometric series, and has a logarithmic form (see Professor 
Curystat “On the Hydrodynamical Theory of Seiches,” § 41). The corresponding 
solution of the LecENDRE-function (a=4) is F(n+ 17, 1k 7 a well-known ex- 
pression for the LecrnprRe-function of the first kind. (See Waurrraker, Modern 
Analysis, § 118.) In this case Nos. 1 and 6 of (16a) are identical. 
Reverting to equation (11), we notice that it may be transformed into (3) by still 
another substitution, viz., by v=w’, a =5+ Gh | -5 , and y=4. Hence we obtain 
24 other particular integrals of the following type :— 
Group I, 
n n 
1 F (+a, io 4; w?) 
a ln ln 
2. (1-w?) xB(5-5-4, ata, 4; w?) 
2 Na w 
4, (l-w’) x F Ba oe 2 ce 77) 
