658 SCIENCE PROGRESS 



single direction ; and a large number of elementary theorems of differential 

 geometry become co-ordinated and expressed by means of properties of a 

 few simple functions. Of these we may mention the bilinear curvature, 

 wliich depends on two tangential directions and reduces to normal curvature 

 when these directions coincide ; and the Codazzi function, depending on 

 three directions and reducing to the cubic function of Laguerre when they 

 coincide ; it is so called because the equations of Codazzi can be interpreted 

 as asserting that it is a symmetrical function. The theory is used to prove 

 the relations between the cubic functions of Laguerre and Darboux, and to 

 obtain formulae for the twist of a family of surfaces and for the rates of 

 change of the two principal curvatures of a variable member of a family of 

 surfaces along an orthogonal trajectory. 



Prof. Neville has a faculty for inventing new names and symbols as well 

 as new methods ; his book, in consequence, is terribly difficult to read. Is 

 it necessary to introduce, not only an inverted kappa, but also inverted 

 affixes ? The decimal notation for paragraphs and equations may be 

 theoretically desirable, but if carried too far gives a repulsive appearance to 

 the page and leads to such absurdities as : 



" 4.11. That 



" 4.111. The bilinear curvature is a bilinear function . . . is obvious. .. ." 

 Or: 



" 4.12. Dupin's theorem, that 



" 4.121. At any ordinary point of a surface the sum of the normal curva- 

 tures in two directions at right angle is a constant, is shown by 2.321 to be 

 a case of 1.824." 



F. P. W. 



Methodes et Problemes de Theorie des Fonctions. Par E. Borel. (Pp. 

 ix -t- 148.] (Paris : Gauthier-Villars et Cie., 1922. Price frs. 12.) 



When M. Borel undertook twenty-five years ago the editorship of the well- 

 known series of monographs on the theory of functions it was with the idea 

 of writing eventually a treatise on the subject. This he has not managed to 

 do, and now, publishing the twenty-sixth volume, the ninth written by 

 himself, he recognises that he will never do it. " Je laisse done a de plus 

 jeunes le soin d'ecrire ce Traite, dont I'heure viendra bientot. La theorie 

 des fonctions a toujours ete, d'ailleurs, une science de jeunes, et il est probable 

 qu'elle la restera, car les qualites d'imagination abstraite qu'elle exige 

 paraissent etre le privilege de la jeunesse, tandis que les parties des mathe- 

 matiques qui touchent aux applications exigent peut-etre plus de maturite 

 d'esprit. Cet Ouvrage . . . sera done vraisemblablement mon dernier livre 

 de theorie des fonctions." 



The contents of the book are thus miscellaneous, and consist, for the 

 most part, of reprints of notes and memoirs which have not found a place 

 in the author's previous books, but which he thinks may be the starting-point 

 of new investigations. They deal with the theory of aggregates, with the 

 study of the simple operations which can be performed on functions and 

 which serve to define or to create them, with the theory of the growth of 

 functions and the part played by arbitrary constants. With this reproduc- 

 tion of earlier work goes a running commentary, in which the author con- 

 stantly recurs to a somewhat fanciful analogy between the theory of functions 

 and biology. Under the impulse of necessity and of natural curiosity man 

 has extended his biological knowledge beyond the rudiments which are 

 indispensable for agriculture ; he has catalogued and classified more and 

 more species, he has tried to perfect the species which he uses and to create 

 new varieties, he has studied the normal and pathological behaviour of the 

 difierent species, and come to examine more and more deeply their funda- 

 mental element, the cell, Simileirly, the theory of functions has developed 



