in Mathematical Reasoning. 329 
and ~. That combination is in this mode of expressing it left 
arbitrary and undefined. The same function / may at the same 
time be a function of 
x? 4-11 2 xt + a°7+1 
Davia a+ 1? a ; 
or of a thousand other quantities; all which circumstances al- 
though deducible from the original expression, are not presented 
to the eye, because in the consequences which it is proposed to 
deduce, they are entirely immaterial. 
If two circumstances in the nature of the function are jointly 
the ground on which any of its properties depend, they may be 
separated from the rest and made prominent by several methods. 
Thus the index x subjoined to a function 
¥(@, Y)n 
may be defined to mean that it is homogeneous with respect to 
x and y and of the dimensions 2 : these two circumstances are 
the causes on which the truth of the following property depends. 
xt being substituted for 7 
Y(@, et), = a(t, Wn. 
In all our attempts at mathematical generalization, it is of 
great importance to discover and distinguish these immediate 
causes of successful operations; in almost all cases they lead us 
at once to the highest point of generality, and very frequently 
contribute in no inconsiderable degree to simplify the processes 
of the investigation. This advantage so peculiar to algebraic signs, 
has been remarked by M. Degerando, from whose writings I have 
derived much satisfaction by observing the support which many 
of those views that I had taken previous to my acquaintance with 
them, received from the reflections of that distinguished philoso- 
