398 
PROFESSOR A. R. FORSYTH ON THE DIFFERENTIAL INVARIANTS 
B 
ds^ 
= 2K - 
j^n\ _ cm 
drdXp'j f/.s-3 
-L i_ IA d (i\\ 
B 1 r dnd.s ds ds ^ ds dt \ p')\ 
_^c^r/„_ 2\f/B I dB\ 
B^ ds p'j7js ~ ?A^r 
and the values of ] and 
dt. [p' / 
72 p 73 
unnecessary to retain - „ , -~ 
ds' drd 
d~ 
dd 
o{ ,) have been given in §§ 36, 41 ; hence it is 
I ' 
P ' 
We therefore retain the quantities 
1 A, 
1 
mill 
p' ' (Is ' 
\p'J 
’ (In ' 
\p'J’ cm 
V p / ’ dsdn^ 
^ cm dR dm d-H cm 
’ ds ’ dn ’ d.s-" ’ dsdn’ dir ’ 
^ cm dB dm dm 
ds " dn ’ ds dn ’ dn~ ' 
j cl i±\ A_/i\ 
A’ cls \p'V drAp"'^ 
1 
being 20 in all; their associatioii with tlie 20 algebraically independent differential 
invariants set out in § 53 has already been made. 
56. These lesults would seem to have an important bearing when we proceed to 
the next higher order of derivatives. As is rejected from the aggregate 
of quantities, the quantities ^ and -5^ ^ o ’'vill also be reiected ; also, as -AA 
ds^ dnds~ dnds dsdn 
IS expressible by quantities of lower order, the quantities and - will be 
ds-'dn chidsdn 
rejected ; thus, in this order, the only derivatives of B to be retained are 
c/"B d^B jim 
dsd7ids' dn^ds^ ds chd dn^’ 
four m number. Moreover, these four may reduce to two; for the first may be 
