394 
Transactions of the Boyal Society of South Afi'ica. 
and that finally " by putting the coefficients of the products ajf>j, separately 
equal to zero " there results 
"ihWh - ihiOk = 0, 
as desired. 
Even as thus epitomized the procedure is seen to be artificial and 
somewhat arbitrary : and I therefore propose to put on record a quite 
simple and more direct treatment, giving also as introduction the lemmas 
on w^hich both demonstrations rest. For the sake of economizing space n 
will be taken equal to 3. 
2. If Ui, U2, U3, U4 he homogeneous functions of the mth degree in x, y, 
z, and 
then 
U^V^ + U^V^ + U^V^ + U^V^ = 0. (a) 
This is the same as to say that 
~dx 
dy 
dz 
'du^ 
'dx 
^y 
'dx 
^2/ 
Yz' 
~dy 
^z 
which follows at once from Euler's theorem that 
ic.col2 + ^.col3 + ^^.col^ = m.coli. 
8. WitJi the same notation 
^ 
0.2: ^ 
5F^* 
(/3) 
This is the same as to say that the determinant of the preceding para- 
graph continues to vanish when in place of its first column we substitute 
a repetition of one of the other columns. 
