1901-2.] Dr Muir ou tlie Theory of Jacobians. 
175 
Binet’s multiplication-theorem is equal to 0 whenp<?z; is equal 
to the product of two determinants 
y+ d L. d A .... :M . y + d ± . d ±i ..... d -h 
— d<f> d<f>i 00 n dx dxj dx n 
when p = n; and, when p>n , is equal to a sum of such products, 
viz., 
3/l . . 
0/n 
Y +00 9 0i 
fyn 1 
0<£ i 
dcfrn 
^ “ dx dx 1 
dx n ) 
where the different terms included under the S are got by taking 
all the different sets of n + 1 </>’s from the p + 1 available. This 
tripartite result Jacobi carefully enunciates at length in the form 
of three propositions. He notes, too, that the first is practically a 
result already obtained, because the functions in that case are not 
independent ; that the second has for its analogue or ultimate case 
the theorem 
df _ df dy . 
dx dy dx ’ 
and similarly that the third is the extension of a result actually 
used in the proof, viz., 
df _ df dc/> df d(f>i df dcf> p 
dx 0</> dx S (/q dx ^ d<f> p dx 
He even enunciates formally a variant of the second proposition, 
calling the variant “ Proposition iv,” viz., If f , f x ,...., f n be 
functions of y , y x , . . • , y n , and it be possible to express both the f } s 
and the j’s in terms of n + 1 other quantities x, x l5 , , . , x n} then 
V + M . ... 
Y + — ^ ^ ~ ^ Xl ^ Xn 
^ ~ d V ’ tyi _ y < _ byf 
^ ~ dx " dx l dx n 
The analogue also is again referred to in the form 
df 
df _ dx 
dy ~ dy 7 
and the special case, already twice obtained, where f=x,f 1 = x l ; 
