652 DR J. HALM 
From (5) and (6) the solutions of the SroxEs-equation may at once be derived. a 
we write, in accordance with Professor Curysrat’s notations, n(z+ 2) = 40 a: and 
consider that (5) is identical with 
d 
ETE ae +1)? [y cos z]= OF, 
we find the general solution 
Y= = |. cos (202) + D sin (202) if 
COS @ 
or since z= tan~’ x, 
y = (1+)? (C. cos (20 tan?) + D sin (20 tan-1 2’) ]. 
In the same way we find from (6) 
d*ly cosh ly, 
de 
n?—2n—1)[ycoshz]=0, 
and, writing n’?+2n—1 = 40’, 
Vea ae cos (20z) + D’ sin (2%) | 
which, since z= tanh™* «= 1 log 1 +4 i= , becomes 
y= (1-28 ¥, cos( 8 fog 1 = + D’sin (2 log a as alk 3 (10) 
(9) and (10) are identical with (26) and (i2) of Professor CuRysTaL’s paper quoted 
above if - is substituted for z. 
It is also at once evident that if we express the Stoxss-functions by means of the 
variable w instead of «, we find the general solutions : | 
d d dY 
W=AT | cos (20 sin w) | ea [sin (20 sin7! w) ] = al , say, 
and 
dV, 
d 
=A 55 rp | 008 (2 sinh"? w) [+B Fg | sin (26 sinh™ tw) |= ae 
dw 
This result is made evident if we consider that Y, and Y, satisfy the well-known differ- 
ential equations 
PY,  a¥, 
(1 ~w) =F — w= 4 4eY, =0 
aFY, AY 
ene ease) ONE == 
+ w ae +4Y,=0, 
(1 + w?) ho 
which, by differentiation with regard to w, assume the form : 
oh pe ee 
ayy 
yes 
(1 ee 
(1+ wry TY yop 30 aoe (407 + 1)y,=0, 
and become identical with the ane if x is introduced instead of w. 
