664 DR J. HALM 
©-FUNCTIONS. 
y= Vl) g | Ca- Veg $1.8) Ca Le £1 3Ne4+85) 5 | ee an 
Riera 2.3.4.5 9.3.4.5.6.7 spear (32) 
FVD gy Colt LIM e + 3:3) 95 Cot LIV + 3.30(e + Bry 
H aa 9.3.4.5 2.3.4.5.6.7 eae 
op GFF), Gt 3G + 5.3), (+ 81NG + 5.3)y + Try 
2.3 23.45 9.3.4.5 6.7 
_ a+ 51) wey + SING, + 7.3) 5 (Cy + BANG) + T3)ey + 95) rg 
73 2.3.4.5 2.3.4.5.6.7 
Let us now find the roots of the equations ©,(c,,1)=0 and S,(¢,,1)=0 for the eig t 
functions given in (29) and (31). For the 6-functions we have the conditions : 
tan [n log (1+ /2)] = 2/2; ¢,=n?+1. 
cos[mlog(1+ /2)] = 0 3 =n. 
cos[zlog(1+ /2)]}= 0 ;¢ =n?+l1. a 
tan[nlog(1+ /2)]=J/2; ¢ = n?+4. ; (33) 
and for the G-functions : 
cotan [mlog (1+ ,/2)] = —n./2;c,= +1. 
sim mloa(i 2) COMME 
sinfnlog(1+ /2)|}= 0 3;¢e =n?+I1. 
eotan [zlog (1+ ./2)] = —n/2; ¢ =m+4. 
From these conditions the roots c, are easily obtained. We find 
(S-PUNCTIONS. ©S-FUNCTIONS. ~ 
e_,=2°35012 ; 27:9681; 78: UOC tae c_, = 12°06756 ; 50°2082; . 
Cy) =8°17627 ; 28°5865 ; 79-4068; ... . €) =12°70508; 50°8204;.... 
4, = 417627 ; 29°5865; 80°4068;. . €4,=13°70508; 51°8204; . ..., (34) 
Ci, = 0 DIOLZ OOLIOS Mero TOG. on. C45 = 15°06756 ; 53°2082 ; . 
But obviously the roots in each vertical row may be considered as special values of 
a certain unknown function of @, so that generally 
Ca =f(a) ’ 
from which equation, if f(a) were known, we would obtain the values c_,, G@, ete. by 
substituting a= —1,0, etc. Now, as long as f(a) may be considered as finite and con- 
tinuous, the well-known formule of numerical interpolation allow us to find intermediate 
values of c from the given data, without knowing the analytical character of f (a), 
if a sufficient number of equidistant values of c are at our disposal. We are thus in a 
position to determine, at least approximately, the roots c_, of the hyperbolic Seiche- 
functions € and © by interpolating between the numbers of the above table (34). 
Performing the necessary calculations, we obtain the following roots : 
CG a0 p eS 274 5 28251, ote OO 
Gi(c,1) SOc =1280 h. .. o0Ow at (35) 
