ear 
vV°p=g Se ee, oe (IB 
: 1 1 i 
In the particular case that A = — oe and B and the relation 
q 7 
between the two equations (11) and (12) is evident. The function 
Ae~-vt Bet eu” — 17 
ren ic! becomes: EP If we substitute q /—1 
r qr 
for 7, the latter expression becomes: 
eyV—1 — er! _ singr 
Z2qrY¥—l qr 
This function is a special case of the more general 
A, sin (qr + @) 
7 
1 
g(r) = > Av=— and: e= 0. 
1 
By substituting in equation (12) gy/—l for g, we get equation 
(11). If g=0 the two equations yield the well known equation: 
9 
7? == 0 5 
The funetions 
A enor + Ber 
At ; 
g(r) = : and g(r)= sist 2 2e) 
7 
are solutions of two different partial differential equations of the 
2nd order, but we have seen that they are also common solutions 
of the same problem. 
We might also have deduced the partial differential equation (12) 
in the following way: 
Aer AMA og or gr 4 
aie ee a 
A 4) A g°r° 
SS ee —_—- —— 
cae 2 3 
and 
Bev 7° ye q° rè 
=—(14¢r—- 5 — + j= 
a” Tt 4 73 
B Bar Bor 
nr Oe eae Ge 
r wz 3 
