194 
Proceedings of Royal Society of Edinburgh. [sess. 
Now the numerator here reduces to one term, viz., 
0F df\ djf 
dx * dx 1 dx n 5 
and the denominator in similar fashion to 
0F 0Fj dff 
¥ ’ ¥ l * ’ ’ ’ ¥n ’ 
as Jacobi might briefly have justified by a reference to Prop. Ill of 
§ 5 of his “ De formatione et proprietatibus Determinantium ”• — 
this proposition being that which concerns a determinant whose 
elements on one side of the ‘diagonal,’ as it afterwards came to be 
called, all vanish. Further, the factors of the reduced numerator 
are equal to 
and those of the reduced denominator to 
1 , 1 , 1 ,-. 
Our final result thus is 
y + ¥ m ¥i 
J — J ~dx dx 1 
where the brackets on the right are meant as a reminder that f i is 
there expressed in terms of/,/ x , ... ,/*_!, x ir x i+1 , . . . , x n . 
The last section (§ 19) is occupied with a theorem of the Integral 
Calculus, viz., 
+ d l. s A 
dx dx 1 
• dx • dx 1 • • - • dx n t 
it being noted that the cases where the number of variables to be 
changed are 2 and 3 had been already dealt with by Euler and 
