68 
ON THE THEORY OF DETERMINANTS. 
[12 
If the two sides of this equation are multiplied by the two sides of the equation 
(2), written under the form 
«= 1, (27), 
ot, ¡3,... 
/3', 
the second side is reduced to 
JF a£ + (3r)..., a'f + /3V--> 
x, tc , . , 
x’, . , K , 
= -JF. K n - 1 .U 
VFU = Jf.(KU) n ~\ U ... 
and hence 
Similarly 
(28), 
(29) , 
(30) . 
FJU = JF.(KU) n ~ 2 .U (31); 
also combining these with the equations (22), (23), 
JFU FVU U 
KFU~ K'lU KU 
(32). 
It may be remarked here that if U, V are functions connected by the equation 
FU = cFV, or VU = c r iV, (33), 
i 
then in general U = c n ~ 1 V (34). 
To prove this, observing that the first of the equations (33) may be written 
i 
FU=F(c n ~ i V) (35), 
i 
we have . FU = 7 . F (c n_1 V) (36), 
i_ _L_ 
or JF. {Kt7) n_2 77= JF [K (c n ~ l V)] n ~ 2 c n ~ 1 V (37). 
If neither J, F nor (KU) vanish, this equation is of the form 
U=kV (38), 
k n ~ l = c (39), 
whence substituting in (33),