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NOTE ON EOSANES' FUNCTIONS EESEMBLING JACOBIANS. 
By Thomas Muir, LL.D. 
(Received December 29, 1913.) 
1. The expressions in question are exemplified by 
' ^^u D^n 'dHo j 
'd^V ~d^V 'd^V 
l)x^ ^x^y ^y^ 
"hx^ ^x~dy "dy^ 
Manifestly they differ from Jacobians in that the number of variables 
is always tivo : on the other hand, a resemblance exists in that the 
order of the determinant is the same as the number of functions to be 
differentiated. In Rosanes' paper" no other order than the third is dealt 
with ; he had, however, the properties of higher orders in his mind, as he 
was avowedly following a paper of Clebsch's on a property of Jacobians, 
where the order was unrestricted.! When the order is the second, the 
Rosanian and Jacobian are evidently identical. 
2. The basic functions being f^, f^, f^, and the independent variables 
x^, x^, Rosanes wrote his determinant in the form 
f\' f" 
fv fv /;- 
/',■ /r- 
In what follows we shall write instead simply B>(f^,f^, f^). 
Only one property is investigated by him, namely, the analogue to 
Clebsch's theorem regarding the second set of Jacobians derivable from 
* EosANES, J. Ueber Functionen, welche ein den Functionaldeterminanten analoges 
Verhalten zeigen. Crelle's Journ., Ixxv. pp. 166-171. 
t A greater number of variables than two was also thought of (see p. 171). 
p3^ 'd3n ()3V ^3W\ 
I t)ic3 ^x^~dy ~dx^y'' %3 |, etc. 
