544 SCIENCE PROGRESS 



The first number of the fourth volume of The Rice Institute 

 Pamphlet (January 191 7) contains two lectures by Emile 

 Borel on " Aggregates of Zero Measure " (pp. 1-21) and 

 " Monogenic Uniform Non-Analytic Functions " (pp. 22-52), 

 and two lectures by Vito Volterra on " The Generalisation of 

 Analytic Functions " (pp. 53-101) and " On the Theory of 

 Waves and Green's Method " (pp. 102-17) I a ^ delivered at 

 the inauguration of the Rice Institute at Houston, Texas. 

 The aggregates mentioned in the first lecture of Borel play a 

 very important part in the theory of functions of a real and 

 of a complex variable, and a way of comparing such aggregates 

 among themselves is given. Of the results of the work de- 

 scribed in his second lecture, Borel says (p. 51) : " The results 

 we are establishing suppress the absolutely sharp demarca- 

 tion established by Weierstrass's theory between real analytic 

 functions and real non-analytic functions " ; and then insists 

 on the importance of this from the point of view of the relations 

 between mathematics and physics. Volterra 's generalisation 

 is that which has already been the subject of several investiga- 

 tions of his, but here it is his purpose to consider the general 

 case in some detail, beginning with the first foundations. Thus 

 he considers his " functions of hyperspaces," which represent 

 extensions of the functions of curves that he has already 

 treated several times, and extends to these functions the funda- 

 mental concepts of continuity and differentiation. Proceeding 

 in this way he finally develops the operations of differentiation 

 and integration and the extension of Cauchy's theorem in com- 

 plete generality. 



In connection with Van Vleck's interesting remark (cf. 

 Science Progress, 191 7, 12, 366) that a Lebesgue integral is 

 expressible as one of Stieltjes by a fairly simple transformation, 

 the lecture by G. A. Bliss {Bull. Amer. Math. Soc. 191 7, 24, 

 1-47) on Lebesgue integrals and the chief results of the works 

 of other mathematicians on analogous subjects will be found 

 very useful. Bliss points out, with Volterra, that a rapid 

 development is taking place in our notions of infinite processes, 

 examples of which are the definite integral limit, the solution 

 of integral equations, and the transition from functions of a 

 finite number of variables to functions of lines, and gives a 

 full account of the work on integration of Borel, Lebesgue, 

 Radon (191 3), de la Vallee Poussin, and others. 



