78 
PROFESSOR DONKIN ON THE 
differential coefficients employed, by referring to the hypotheses on which they are 
taken, and which I shall denote as follows : — 
Hyp. I. — The 2 n variables x x , x 2 , ... y 1} y 2 , ... expressed as functions of a„ a 2 , ... b x , 
b 2 , ... and t. 
Hyp. II. — The 2 n constants a 15 a 2 , ... b x , b 2 , ... expressed as functions of x x , x 2 , ... y x , 
y 2 , ... and t. 
Hyp. III. — The n variables yi,y 2 > ...y n expressed as functions of the n variables 
x l} x 2 , ... x n , the n constants a 2 , ... a n , and t (as by equations (5.)). 
Hyp. IV.— The n constants b x , b 2 , ...b n expressed as functions of the n variables 
x i} ... x n , the n constants ... a n , and t (as by equations (1 1.)). 
6. Differentiating totally the equation (11.), 
dX 
d(l i ' 
■h 
with respect to t, we obtain (observing that by virtue of the conditions (5.)), 
=o 
daidt' da,i 1 'da i 2 ''“'dcti n 
^ where &c. are taken on Hyp. III., art. 5.^. 
Now let (Z) be a function of x l9 ... x n , t. a x , ... a n , defined by the equation 
= 
(15.) 
the above equation then becomes 
_i Jy? r ‘ 
da, — dai 1 ^ daf 2 ^ dai n ' 
If this equation be multiplied by , and the result on each side summed with respect 
to i, it will be seen that the coefficients of x\, x 2 , See. on the second side all vanish 
except that of xj, which reduces itself to 1 (see art. 1, equations (1.) (2.)); so that 
we have 
d( Z) da x d(Z) da q d(Z) da n , 
da x dyj 1 da 2 dyj' 1 da n dyj r 
Now the expression on the left of this equation is equivalent to 
dZ 
dy- 
if by Z (without brackets) we denote the result of substituting for a x , a 2 , ... a n in (Z), 
their values in terms of all the variables {Hyp. II.), so that Z is a function of the 
variables only. We have then, finally (writing i instead of;), 
i dZ 
x i=~r 
dyi 
Again, we have {Hyp. III.) 
(16.) 
