THE FUTURE OF MATHEMATICS—POINCARE. 7 
an unexpected obstacle has been encountered which has sometimes 
been overcome by special artifices, but which so far has more often 
remained insurmountable. We must therefore search for facts from 
first principles to make clear to us this difference between functions 
of one variable and those containing several. We should look first 
more closely at the devices which have brought success in certain 
cases to see what they may have in common. Why is a conformal 
representation more often impossible in the domain of four dimen- 
sions and what shall we substitute for it? Does not the true general- 
ization of functions of one variable come in the harmonic functions 
of four variables of which the real parts of the functions of two 
variables are only particular cases? Can we make use of what we 
know of algebraic or rational functions in the study of transcendental 
functions of several variables? Or, in other words, in what sense 
may we say that the transcendental functions of two variables are 
to transcendental functions of one variable as rational functions of 
two variables are to rational functions of one variable? 
It is true that if z=f (wv, y) we can, whatever the function 7 may 
be, express w, y, 2, respectively, as uniform functions of two auxiliary 
variables, or, to employ an expression which has become common for 
this process, can we make uniform the functions of two variables 
as we do those of one? I limit myself to the setting of the problem, 
the solution of which may perhaps come in the future. 
Vil. THE THEORY OF GROUPS. 
The theory of groups is an extensive subject upon which there 
is much to be said. There are many kinds of groups, and whatever 
classification may be adopted we will always find new groups which 
will not fit it. J wish to limit myself and will speak here only of the 
continuous groups of Lie and the discontinuous ones of Galois, both 
of which we are now wont to classify as groups of finite order, al- 
though the term does not apply to both groups in the same sense. 
In the theory of the groups of Lie we are guided by a special 
analogy; a finite transformation is the result of the combination of 
an infinity of infinitessimal transformations. The simplest case 
is that where the infinitessimal transformation is equivalent to the 
multiplication by 1-++e, where « is very small. The repetition of 
these transformations gives rise to the exponential function; that was 
Neper’s method of procedure. We know that an exponential func- 
tion can be expressed by a very simple and very convergent series, 
and analogy should then show us what path to follow. Moreover, 
that analogy may be expressed by a special symbolism upon which 
you will excuse me from dwelling. We are already well advanced 
along this path, thanks to Lie, Killing, and Cartan; it remains only 
