1003 
4. In some cases it is possible to derive another relation from 
(7) and (8), in which only first derivatives occur, a socalled 
first integral. For this purpose we multiply the equations (7) 
successively by p’, q’, and s’, and we then add them. The result 
may be written in the form 
OL OL OL ÒL aL SH q ae 
Al trae S aot alter e a (Zp+2te+"s). 
From (6) we find that 
OL OL OL cay 
cage Sea ds’ ==} FD 
OL p? q” s!? 
so that we get in connection with (8) 
d 12 13 '2 / 2 \ 
GIN 
dr p De ne p 
p ‘4 q? 3? = 5 ; dP 
Ee a eet rame Q) 
Pp de 
For the equations (7), written in full, we find, after having 
multiplied them successively by p, $q, and s, 
d p! ( 
— (ern!) —4L—4Fp (1 — 1) a5 xn PE 
dr p p/p 
d ! 
(ero) ae apt — 1) 2 wg Dat ey, EN 
r q P/P 
d ! 
= (vr) +340 = x's. 
dr 8 
We now add twice the second equation to the first, and get in 
this way 
and 
(9) 
S| 
ala (2 42 al NL ar EE OE = 5 GAD 
dr p q 
When we subtract twice (11) from 7 times (9) we find 
a 12 s/2 2 
(5 ae +). —!) | |- 
Pp q s p 
/ p' el q'? s? a \2 
= Hae + 22) 4 bep iG TE. ‚) a s(1- L) | = 
dr p q Pp gnd p 
