174 Proceedings of the Royal Society of Edinburgh. [Sess. 
21. Next, there are no other integrals which are independent of the 
magnitudes a, f 3 , y, ■ • • If such an integral existed not involving these 
magnitudes, it would satisfy the relations 
ca^ 
8 'A ’ 81 , 
because of the equations ; the relations 
da,,^ 8 k. 
^ + 2-'^ = 0 
8 hy db. 
H ej> 
8<J2 Sf, ’ dm. 
because of the ? ^12 equations ; the relations 
'^day 8 g, ’ dhy 8 f, ’ dgy^dc, ’ diy dn, ’ 
because of the , ^^3 , 6 ^^ equations ; the relations 
5 = y] — — -d, ^ — h ^ — = d, 
oa^ cc^ ch^ cm^ cg^ on^ 
dl, 
dd^ 
because of the ^^4 , 77^4 , ^44 , ^44 equations ; the relations 
d(f) 
dll 
Lo ^=0 ^=0 -^=0 
„ ’ diK, ’ 0/2 ’ 0/»„ ’ 
because of the ^22 ^ V22 ’ ^22 > ^22 equations ; the relations 
dh^ dgo 
c>HH 
^dby df., 
M + 2^ = 0 
dfj dc, ’ 
M . M=o 
because of the ^23 ^ ^^23 ’ ^23 ^ ^23 equations ; the relations 
def) d(/) 
^ 4-^-0 
5 /q dl^ 
oM + M-o ^ + ^‘^ = 0 
^/>4 
+ 
5/4 dn^ ’ dd^ 
because of the ^24 » Vzi ’ ^21 ’ ^24 equations ; the relations 
= 0 
d(j) 
W: 
r=0 
d(f) 
dc.^ 
= 0 , 
d(j) 
dn 
- = 0 
because of the ^33 , 7733 , ^33 , ^33 equations ; the relations 
M + = 0 ^ + -M„o 2^ + ^ = 0 
# + 2|f = 0, 
dd^ 
-1^=0. -fi = o 
because of the ^34 , 7734 , ^34 , 6 .^^ equations ; and the relations 
del, ■ 
These relations can only be satisfied when each of the differential 
coefficients of <p vanishes ; consequently, no function cp, involving only the 
first derivatives, can exist. 
22. We therefore require twenty integrals involving the magnitudes 
a, y, . . . It has already been pointed out that these magnitudes are 
