1901-2.] Dr Muir on the Theory of Jacobians. 
165 
'Sp _J_ Of bfm ^ + bf m+ 1 ^ 5/m+ 2 bf n 
^ ~dx 0% 0ar m " _ 0^ m+1 0^ m+2 ' ' ‘ ' dx n ' 
The important proposition regarding a vanishing functional 
determinant is then dealt with (§ 6), viz., the proposition “ func- 
tionum a se non independentium evanescere Determinans, function es 
quarum Determinans evanescat non esse a se independentes.” The 
proof of the first part of it opens with the assertion that since the 
functions are not mutually independent, there must exist an 
equation 
n(/,/i = 0 
such that on substituting for /, / l5 . . . , f n their expressions in 
terms of x , x x , . . . , x n we shall obtain an identity. From this 
by differentiating separately with respect to x, x ± , . . . , x n there 
is obtained the set of equations 
dfdu dfdn m dfnf n 
dx 9/ dx 0/j 3^i '0/ M ’ 
^ an an .... n 
3/ 9^1 3/ x * ’ 0« 1 ‘ 0/n ’ 
A 0/ 0n 0/, 3 n 0/ n 0n 
~”0^n 0/ + bx n ' 0/l + + dx T ‘df n ' 
Then it is recalled that in a set of linear equations 
+ a 12 ^ 2 + • • • + a ln x n = 0 
a nl X Y + + * * • + a m x n—Q 
the determinant of the coefficients must vanish unless all the 
unknowns vanish. And as the vanishing of 
an 3 n an 
9/ ’ 9/r " ' ’ ’ bf n 
would imply that the expression n(/, f v ... , f n ) was free of 
/, /i > • • • , f n , the conclusion is reached that the determinant of 
the coefficients of these differential-quotients must vanish, i.e ., that 
the functional determinant 
+ Qf_ t 
^ ~ dx a^ 
bfn 
dx n 
= 0 . 
