1901-2.] Dr Muir on the Theory of Jacobians. 163 
The dependence or independence of equations is the next pre- 
liminary subject (pp. 323-325), the starting-point being the 
definition of an identical equation as one in which every term is 
destructive of another, and from which, therefore, it is impossible 
to express one of the involved quantities in terms of the rest. On 
this the definition of mutually independent equations is made to 
hang, such equations being defined as those of which no one at 
the outset is an identical equation nor can be transformed into 
an identical equation by aid of the others. Then taking m+ 1 
equations, 
u = 0 , u Y = 0 , . . . . , u m = 0 
involving n + 1 quantities x, x 1} . . . , x n he contemplates the possi- 
bility of solving u = 0 for x in terms of x lt x 2i ... , x n and the sub- 
stitution of the expression in place of x in the remaining equations. 
The latter equations as altered he supposes to be dealt with in the 
same way, and the process continued until k + 1 quantities have 
been eliminated and m-h equations left involving x k+l , x k+2 , . . . , 
x n . Reasoning from this, he concludes that a number of given 
equations are mutually independent or not according as by their 
help the same number of involved quantities can or can not be 
■expressed in terms of the remaining quantities. In this connec- 
tion he does not omit to draw attention to the existence of ex- 
ceptional cases, such as that in which two of the quantities, x h , x k , 
say, occur indeed in all the equations, but always in the form x h + 
x k \ and this leads him, for the sake of greater definiteness, to 
introduce the qualifying phrase £ with respect to certain quantities 7 
in using the expression ‘ mutually independent. 7 His words are — 
“Aequationes u = 0, u 1 = 0, . . ., u m — 0 quibus totidem 
quantitates x, x l3 . . . , x m quas involvunt, determinantur, 
harum quantitatum respectu dico a se independentes. 77 
From the independence of equations he naturally passes (§ 4) to 
the independence of functions , with the remark that exactly similar 
propositions are found to hold in regard to the latter, — a state- 
ment which it is not hard to believe when we recall that any 
function, x 2 + y 2 - 4 xy say, may be denoted by a functional 
symbol, / say, the equation f =x 2 + y 2 - 4 xy thus resulting ; and 
that any non-identical equation connecting two or more quantities 
