511 
1908-9.] Dr Muir on the Theory of Jacobians. 
of n variables without any regard to possible connections with dynamics. 
Drawing attention at the outset to the analogy remarked on by Jacobi and 
Bertrand, he proposes to signalise it by denoting the determinant of f \ , f 2 , 
. . . , f n with respect to x 1 , x 2 x n by 
d(/i , f 2 , ■ ■ ■ > fn) ' 
’ • • • ) ^n) 
Further, he views the numerator and denominator here as standing for the 
determinants of Bertrand’s arrays of differences, remarking pointedly that 
the fraction indicated “is a real fraction, provided its numerator and 
denominator be interpreted in a manner exactly analogous to that in 
which the numerator and denominator of an ordinary total or partial 
differential -coefficient are interpreted.” 
Having thus explained his notation he proceeds to generalise the 
proposition that 
of r ?)Xy ^ 3 / r d%2 _j_ kf ?l j ^ q 
dxj df s dx 2 df s ‘ ' dx n df s 
according as r and s are equal or unequal. He recalls Jacobi’s theorem 
(Be determin. fund. §11) that if u x , u 2 , . . . , u m be functions of y li y 2 , 
. . . , y n ,n being greater than m, and the y’s be functions of x x , x 2 , . . . , x n , 
then 
d(u a iu a 9. . . . u am ) _ | 9(?< a i u a2 . . . u an} ) _ dji/pi i/p . ■ . i/p m ) [ 
d(x y ix y2 . . • Xym) 3(2//3iyj82 • • • V/3 m) d(x yl X y2 . . . X ym ) j 
where a x , a 2 , . . . , a and y x , y 2 , . . . , y m are fixed sets of m integers 
chosen from 1, 2, . . . , n , and /3 1 , /3 2 , . . . , /3 m as any such set whatever. 
Taking then what he considers to be a particular case of this, namely, 
S(?/al?/a2 ■ ■ . Vam) = y f d(Vall/a2 • • • J/am) _ djxpiXpl . . ■ Xp m ) \ 
d(y y iy y 2 , ■ • • y y m) l 3(^1^2 • • • ^m) 9(^yl?/y2 • • • // 7 m) * 
he points out that if the sets cq , a 2 , . . . , a m and y 1 , y 2 , . . . , y m be identical 
the determinant on the left is equal to 1, and that on the other hand if 
even one of the y’s be not included among the a’s the determinant will 
have a column of zeros and therefore be zero itself. The generalisation 
aimed at thus is, that If y x , y 2 , . . . , y n be functions of x x , x 2 , . . . , x n , 
and m be less than n, then 
j 9( ^/al ^a2 • • • Vam) 3(^1 Xpi • • • I _ | or 0 
0 I d(xpi Xp 2 . . . Xpm) d(y y 1 lj y 2 . . • y y m) ) 
according as the a set of m integers chosen from 1 , 2 , . . . , n is identical 
