76 
PROFESSOR DONKIN ON THE 
in which the brackets indicate that x x , x 2 , ... x n are to be expressed in terms of 
y x , y 2 , ...y M so that Y may be a function of the latter variables only. It is easy to 
show a posteriori that the expression (8.) verifies the equations (6.), but I pass on to 
some further considerations. (See note at the end of Section II.) 
3. Suppose the function X involves explicitly, besides the variables x„ x 2 , &c., any 
other quantity/?, so that the expressions (a^), (x 2 ), &c. (or the values of x x , x 2 , ... in 
terms of y x ,y 2 , &c.) will also involve p explicitly, and we shall ha^e 
d{X)_dX,dX d(xj dX d(x 2 ) 
dp dp' dx j dp dxQ dp ' ” * 
dX , 
-~dp+V' 
d(x^) 
dp 
Now, differentiating the equation (8.) with respect to p (so far as it contains p 
explicitly), we obtain 
dj_ d(X) d{x x ) d(x 2 ) 
dp dp 'J 1 dp 'J 2 dp 
which the equation above written reduces simply to 
rfX rfY 
dp ' dp 
(9.) 
In the particular case in which X is a homogeneous function of x x , x 2 , ... x n , and of 
m dimensions with respect to those variables, the equations (8.) and (9.) become 
Y = (m — 1)(X) t 
?+<.-.>¥ J «*» 
• TTt 
and it is easily seen that Y is also homogeneous and of dimensions in y x ,y 2 , ...y n . 
4. The theorems (8.) and (9.) are cases of more general ones which are easily proved 
in a perfectly similar way, and which I shall therefore only enunciate. If, by means 
of the equations (5.), art. 2, we express a set of n out of the 2 n variables, consisting 
of r x's and n—ry s, of which no two indices are the same, for example, 
x x , x 2 , ... x r ,y r+ 15 . . ., y n (a.) 
in terms of the remaining n variables, 
3^15 5 • • • Vrl X'r+ 1? • • •} ^ n ? (f3.) 
then, taking Q= — (X) + (a; 1 )y 1 + (a: 2 )y a + ... + {x r )y r 
(in which the brackets indicate that the variables of the set (a.) are to be expressed 
in terms of those of the set ((3.), so that Q is a function of the latter set), we shall have 
dQ f • n . 
-^-=x i from i=0 to i = r, 
dQ 
dxj 
— y j ivomj=r-\-\ to j=n, 
