662 DR J. HALM 
are, however, unsuitable on account of the presence of complex factors. The hyperbolic — 
Sine- and Cosine-functions are again represented by the first series of Groups I. and V. 
These series contain only real terms, as was shown already by Professor Curystat. We 
~ 
find for the hyperbolic Cosine-function : 
Oh MG Ch 108, 5 
Bet Gy a > 29 a) 
2 
mn? + a? aig, Ce Eee alae ; (n? + c?) [n? +024 4(a+1)| [72 +0? + 8(a + 2)] 
— =< = _ Wr — 
ee a. 1.2.34 1.2.3.4.5.6 Ze 
The Seiche Cosine is obtained by substituting a= —4} and 17+4=c, hence % 
(c+1.2 (c+ 1.2) (e+ 3.4 
hyp. Seiche Cosine @(¢, w)=1 — 3 ab ae - wt — a 5 2 S ee ois (27) 
while for the hyperbolic Seiche Sine we have 
S(e ; Ww) = wE( : ae m1 We . : = v2) 
DS a) 
De Oe Gree Oa) 
= 12a ra ORIEL DSEASIGMI ee ne 
Similar expressions are obtained for the corresponding hyperbolic LeGENDRE- and 
SroxEs-functions. The series, by which the functions are represented, converge very 
slowly. For this reason the theory of convex lakes with parabolic floor has so far 
remained incomplete, chiefly owing to the difficulty in determining the roots of the 
equations G(¢ , 1)=0 and G(¢ ,1)=0. To avoid this dithculty Professor Curysrat, in 
his second communication, proposed a different assumption as to the contour of the floor 
of the lakes, by which he was enabled to express the problem by the solutions of the 
SroKEs-equation 
(ary 4 4 cy =0, 
which, as we have seen, are represented by elementary transcendents. In this case, 
curiously, the convex lake offers the least difficulty, doubtless owing to the fact that 
its equation belongs to the class which also contains the more tractable parabolic 
concave lakes. A disadvantage of the Sroxrs-equations, however, is that the quartic 
lake profile seems to be a less close approximation to the actual conditions than the 
parabolic. Even apart from this, it is certainly of importance to discuss the problem 
on at least two different assumptions in order to estimate the influence of the form of 
the lake on the periods and nodes. Now the preceding investigation enables us to find 
any of the infinite number of roots of the equations G(c , 1)=0 and G(c , 1)=0 with 
at least a sufficient degree of approximation, and thus places us in a position to find 
also without any difficulty the periods and nodes of the seiches in lakes with parabolie 
convex floor. 
