PRESIDENTIAL ADDRESS. Sal 
circumstance that the group of interchanges among four quantities which leave 
unaltered the product of their six differences is exactly similar to the group of 
rotations of a regular tetrahedron whose centre is fixed, when its corners are 
interchanged among themselves. Then I mention the historical fact that the 
problem of ascertaining when that well-known linear differential equation called 
the hypergeometric equation has all its solutions expressible in finite terms as 
algebraic functions, was first solved in connection with a group of similar kind. 
For any linear differential equation it is of primary importance to consider the 
group of interchanges of its solutions when the independent variable, starting 
from an arbitrary point, makes all possible excursions, returning to its initial 
value. And it is in connection with this consideration that one justification arises 
for the view that the equation can be solved by expressing both the independent 
and dependent variables as single-valued functions of another variable. There 
is, however, a theory of groups different from those so far referred to, in which 
the variables can change continuously; this alone is most extensive, as may be 
judged from one of its lesser applications, the familiar theory of the invariants 
of quantics. Moreover, perhaps the most masterly of the analytical discussions 
of the theory of geometry has been carried through as a particular application of 
the theory of such groups. 
The Theory of Algebrace Functions. 
If the theory of groups illustrates how a unifying plan works in mathematics 
beneath bewildering detail, the next matter I refer to well shows what a wealth, 
what a grandeur, of thought may spring from what seem slight beginnings. Our 
ordinary integral calculus is well-nigh powerless when the result of integration is 
not expressible by algebraic or logarithmic functions. The attempt to extend the 
possibilities of integration to the case when the function to be integrated involves 
the square root of a polynomial of the fourth order, led first, after many efforts, 
among which Legendre’s devotion of forty years was part, to the theory of 
doubly-periodic functions. To-day this is much simpler than _ ordinary 
trigonometry, and, even apart from its applications, it is quite incredible that it 
_ should ever again pass from being among the treasures of civilised man. Then, 
at first in uncouth form, but now clothed with delicate beauty, came the theory 
of general algebraical integrals, of which the influence is spread far and wide; 
and with it all that is systematic in the theory of plane curves, and all that is 
associated with the conception of a Riemann surface. After this came the theory 
of multiply-periodic functions of any number of variables, which, though still 
very far indeed from being complete, has, I have always felt, a majesty of 
conception which is unique. Quite recently the ideas evolved in the previous 
history have prompted a vast general theory of the classification of algebraical 
surfaces according to their essential properties, which is opening endless new 
vistas of thought. 
Theory of Functions of Complex Variables: Differential Equations. 
But the theory has also been prolific in general principles for functions of 
complex variables. Of greater theories, the problem of automorphic functions 
alone is a vast continent still largely undeveloped, and there is the incidental 
problem of the possibilities of geometry of position in any number of dimensions, 
so important in so many ways: But, in fact, a large proportion of the more 
familiar general principles, taught to-day as theory of functions, have been 
elaborated under the stimulus of the foregoing theory. Besides this, however, 
all that precision of logical statement of which I spoke at the beginning is of 
paramount necessity here. What exactly is meant by a curve of integration, 
what character can the limiting points of a region of existence of a function 
possess, how even best to define a function of a complex variable, these are but 
some obvious cases of difficulties which are very real and pressing to-day. And 
then there are the problems of the theory of differential equations. About these 
I am at a loss what to say. We give a name to the subject, as if it were one 
subject, and I deal with it in the fewest words. But our whole physical 
outlook is based on the belief that the problems of Nature are expressible by 
differential equations ; and our knowledge of even the possibilities of the solutions 
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