1901-2.] Dr Muir on the Theory of Jacobians. 
151 
The Theory of Jacobians in the Historical Order of its 
Development up to 1841. By Thomas Muir, LL.D. 
(Read January 20, 1902.) 
It is not improbable that determinants in which the number of 
a row is distinguished by differentiation with respect to a definite 
variable, and in which the number of a column is distinguished by 
a particular function set for differentiation, may have appeared 
long before the time of Cauchy and Jacobi, the likelihood probably 
being the greater the fewer the number of functions and variables 
involved. There can be little doubt, for example, that expressions 
like 
du dv du dv 
dx dy dy dx 
may be found repeatedly in the writings of mathematicians belong- 
ing to the eighteenth century. It would appear, however, that the 
first who got beyond the second order, and clearly associated the 
expressions with determinants, was Cauchy. 
Cauchy (1815). 
[Theorie de la propagation des ondes a la surface d’un fluide 
pesant d’une profondeur indefinie. Mem. presentes par divers 
savants a V Acad. roy. des Sci. de I’ Inst, de France . . . . I. 
(1827) : or (Euvres V ser. I. pp. 5-318.] 
Cauchy was a competitor for the prize for mathematical analysis 
in the ‘concours’ of 1815, and gained the prize. His work, 
however, like others belonging to that interesting political period, 
was not printed until long afterwards. In the form which it takes 
in the collected works the essay proper extends to only 108 pages, 
the remaining 210 being occupied with notes : this was probably 
due to the circumstances under which the paper was first written. 
In the same way is explained the writer’s action in referring in it 
to himself by name, the object being to preserve his anonymity. 
