Note on Bosanes Functions Resembling Jacobians. 159 
This is the same as to say that the determinant of the preceding 
paragraph continues to be equal to 0 when in place of its first column we 
substitute a repetition of one of the other columns. 
5. The foregoing results we may without fear of ambiguity write 
thus : 
and therefore in what follows we shall use only the like abbreviated 
forms. 
6. With the same notation as before 
For the first of these assertions is the same as 
^3 ^)^J^3 
~dx~dy 
Dx'c)'!/ 
~dx^y 
'dx^y 
= 0, 
'^X ^X^ 
which follows at once from Euler's theorem that 
X.G0I2 + ^.col. = {m — l).coli ; 
and the second assertion similarly follows from using 
ic.col3 + y.Q>o\^ = {m — l).col, 
on a similar zero determinant. 
7. Eeturning now to the equality %ii,v^ = 0, and differentiating with 
respect to x, we obtain 
