ico) 
-XXVI.—On a Group of Linear Differential Equations of the 2nd Order, including 
Professor Chrystal’s Seiche - equations. By J. Halm, Ph.D., Lecturer on 
Astronomy in the University of HKdinburgh. 
(MS. received May 20, 1905. Read June 19, 1905. Issued separately July 31, 1905.) 
It is readily seen that the two differential equations 
a 
(1 -w2) 54 +n(n — ly =0 (1) 
1 2 a Uy I 7 =0 92 
(1 +07)? Tetn(n + 2)y= (2) 
which play an important réle in Professor Curysrau’s mathematical theory of the 
Seiches, are special cases of the more general type 
d?y dy 
el — w?) a ~ (2a+ lw a +n(n + 2a)y=0. (3) 
With regard to the first, the Seiche-equation, this becomes at once apparent by writing 
a@=-—%. Equation (2), on the other hand, which we may briefly call the Sroxzs- 
equation [see Professor CurysraL’s paper on “Some further Results in the Mathe- 
matical Theory of Seiches,” Proc. Roy. Soc. Edin., vol. xxv.| will be recognised as a 
special case (a = + 1) of the equation 
d*y 
2)2 
(1 +2") da? 
d 
— (2a -2)0(1 +22) +n(n + 2a)y =0 ; (4) 
0 
which is transformed into (3) by the substitution x= aaa 
It appears, therefore, that the Seiche- as well as the Stoxns-equation belong to the 
same family of differential equations whose general form is given by (3). We may 
write the latter also 
te ay 
er 2a tan 27, + n(n + 2a)y=0 (5) 
ifwe substitute w=sinz or «=tanz. Corresponding to this equation we have further : 
GEO) d 
art 2a tanh 2 + n(n + 2a)y=0 (6) 
which for w=sinhz and a=-—4 leads to the hyperbolic Seiche-equation : 
q2 
(1 +102) TS + n(n —1)y=0 (7) 
and for «=tanhz and a=+1 to the “hyperbolic” Sroxus-equation : 
2 d?y 
(1 - x?) Gat Mn+ 2)y=0. (8) 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART III. (NO. 26.) 96 
