352 



SCIENCE 



[N. S. Vol. XXXVIII. No. 976 



one who wislies to form some idea of the 

 richness of the landscape offered by pure 

 mathematics might do worse than make 

 himself acquainted with this comparatively 

 small district of it. But the theory of 

 groups has other applications. It may be 

 interesting to refer to the circumstance 

 that the group of interchanges among four 

 quantities which leave unaltered the prod- 

 uct of their six differences is exactly sim- 

 ilar to the group of rotations of a regular 

 tetrahedron whose center is fixed, when its 

 corners are interchanged among them- 

 selves. Then I mention the historical fact 

 that the problem of ascertaining when that 

 well-known linear differential equation 

 called the hypergeometrie equation has all 

 its solutions expressible in finite terms as 

 algebraic functions, was first solved in con- 

 nection with a group of similar kind. For 

 any linear differential equation it is of pri- 

 mary importance to consider the group of 

 interchanges of its solutions when the inde- 

 pendent variable, starting from an arbi- 

 trary 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 vari- 

 able. There is, however, a theory of 

 groups different from those so far referred 

 to, in which the variables can change con- 

 tinuously ; this alone is most extensive, as 

 may be judged from one of its lesser appli- 

 cations, -the familiar theory of the invari- 

 ants 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 ALGEBRAIC FUNCTIONS 



If the theory of groups illustrates how a 



unifying plan works in mathematics be- 

 neath 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 ordi- 

 nary integral calculus is well-nigh power- 

 less when the result of integration is not 

 expressible by algebraic or logarithmic 

 functions. The attempt to extend the pos- 

 sibilities 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 in- 

 credible that it should ever again pass from 

 being among the treasures of civilized man. 

 Then, at first in uncouth form, but now 

 clothed with delicate beauty, came the the- 

 ory 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 asso- 

 ciated with the conception of a Eiemann 

 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 es- 

 sential properties, which is opening endless 

 new vistas of thought. 



THEORY OF FUNCTIONS OF COMPLEX VARI- 

 ABLES: DIFFERENTIAL EQUATIONS 



But the theory has also been prolific in 

 general principles for functions of complex 

 variables. Of greater theories, the prob- 

 lem of automorphic functions alone is a 



