1001 
be taken starting from the real (sought) values of the g’s and the y’s, 
and so that they are zero at the boundary of the region. 
Let us first calculate H. We must then differentiate the g’s and 
the y’s with respect to the coordinates; then we can take all the 
quantities as they are in a point of the z-axis at a distance r= x 
from the origin, and thus we find 
In = 5 Jaa Jas UV 9 Jaa FW 0 Yn =P sv Van = Yan SI > Yaa — 8 
09,4 =) 0959 Beas: a) 0944 ae 091, = Os; Hae 0915 _ 9951 es 
Oe — 9 de Ow ” Ox Open Oye de os ken 
OF; et Dis bd Os = ire per ee ae. Oy; fd Oi eae a BEG 
we | OR Ox Ow " Oy Oye da OE be nee 
FS WO. 
In this the accents denote differentiations with respect to 7; the 
values that have not been given are zero. 
Let us call V—g r for brevity. Then on account of (4) 
Bong Sia ar nL Sd oe A ek) 
We find for H 
Fo} 
HS Ar 
pup! + ed + ws) + Aedo: 
as in virtue of (4) 
jee 3 
gep — 4) — 9 É ener =H 
pr ej p 
this becomes 
H=—tFp 
p q 3! 4 q> 2 
TE GR BRED pat EE PA 
ette 
We now apply the thesis of the calculus of variations to the region 
Oy ETS ota ye ees 
bi ’ 
then the first variation of [Adr becomes 
tg 72 rz 
óf Hat = re Aard. H = — Aa (tt) d [ Lar, 
1 
ry ri 
Prater afl. © 
P q 8 P 
re | de = PES Tav Oy pv 
py 
if we put 
L=—H?r =} Fp 
For 
