386 
PKOFESSOE A. R. FORSYTH ON THE DIFFERENTIAL INVARIANTS 
after substitution and reduction, we have 
Also, we have 
\/ 
L — 2K ~ ^ ^ 4- ^ 4- 
B ds 1 ' 
<"/ J J ^r3) 1 _ J (wj, iv^) j K?f;2J «’h) 
c/^V I V4 ( ' A}'4 2 ^tT 
I r 
A 
= 9 ^ ■! ' -'"''='’1 + — J (“'=> ®"=) J "’ 3 ) + ' 
thence a value of J {iv^, iv^) is obtained in the form 
^ A \ _j_ 1 K _ lb® 1 fZ /I \ 2 
BW^ dsdn\n')~^ - d” L'/ 4 
p '/ ~ T p ds 
dB [ d /I 
dt \p‘ 
2 /I r/B\2 
B ds 
+ 
B r/.s \dn\p', 
+ f^-o'yn" 
p/p 
Comparing this value of J {w.,, iv^) with the value that was obtained in § 41, we find 
d:^ /1\ r/2 /I 
ds dn \p'/ dn ds \ p' 
= 1T('T\ + 
4 4,, ' ^rn ' 
1 r/B d l\ 
_ _ _, K _ 4 /I fZBV 
r' cZ?i I p''] B ds dn \p'J p'' dt \p'l ^ r' r Ib ds j 
1 tZ^/1 
_ o 
Lastly, we have 
V /ilT i'lfe’JLs)! _ 1 
~ (Zn 1 J V^' 
IV. 
yr 2 y4 
TC 
1 C D f 
A_- ' V . V'" - 
[ 2 ;r 2 ?r' 2 '^^~ — I {iv.., ivd) tty} 
_ J (w.2, ^v"d , , 3 T -r . . 
-- ^6 ‘b + oye J («h> I (^b> 2 ) 
3 
+ bh, V 4 w^)]- 
jL> tVs\ V 
When we substitute the values of the invariants in this expression and reduce the 
result, we find 
(Z 2 /I \ (Zd-I 2 \ 1 (ZH d~ A \ 
did \p 
2 / 
ds- 
K II - 
+ 
1 
B 
P 
2 rZ-B 
C dn dt~ 
P 
_ ('■yBcZH (ZB (Z /I 
y dnds ds ds ds dt 
+ j(H-V,®}- 
-b as [\ p j ds T d)i J 
70 T> 
It is to be noted, from the results obtained in this section, that ^and , „i- 
ds- dn-\p 
are expressed in terms of the other magnitudes retained ; or, if we choose, we can 
r/~B 
regard the last relation as determining j —, in terms of the other magnitudes 
® dnds ^ 
retained. 
rZ^ /I 
