1 80 SCIENCE PROGRESS 



£. Picard (Compt. Rend. 191 6, 163, 284-9) has a note on 

 certain sub-groups of hyperfuchsian groups corresponding to 

 ternary quadratic forms with conjugate undetermined quan- 

 tities. Connected with this is his note (ibid. 317-9) on func- 

 tions of two complex variables which remain invariable by 

 the substitutions of a discontinuous group. Allied questions 

 in certain groups of substitutions are dealt with by G. Julia 

 (ibid. 599-600, 691-4; 191 7, 164, 32-5). 



W. H. Young and (Mrs.) Grace Chisholm Young (Proc. 

 Lond. Math. Soc. 191 8, 16, xii-xiv, 337-51), developing various 

 suggestions in their former work on the theory of functions of 

 real variables, give an account of some new theorems in that 

 part of the theory of sets of points in space of any number 

 of dimensions which corresponds to the classification of the 

 limiting points of a set on the straight line into those which 

 are limits on one side only, or on both sides (descriptive pro- 

 perty), as well as into those which are, and those which are 

 not, the end-points of intervals containing only a sub-set of 

 the given set of content zero (metrical property). These 

 theorems are of importance when we try to pass from a single 

 variable to two or more variables, and lead to the confirmation 

 of a surmise of Young (1909) that the set of the first category, 

 of exceptional singularities of a function of any number of 

 variables has content zero. The same authors (ibid, xiv- 

 xv ; 71, 1 -1 6) apply the new theorems to the investigation of 

 the symmetry which must in general hold below the limits of 

 a function at a point, and the discussion of the distribution 

 of the points at which this symmetry is more or less imper- 

 fect. Analogy with physical relations suggested the applica- 

 tion of the word " crystalline " to such symmetrical relations. 

 H. B. Fine (Annals of Math. 191 8, 19, 172-3) gives a 

 simpler and less general substitute for " Duhamel's theorem ' 

 than that of G. A. Bliss (1914), and which, like that of Bliss 

 and unlike the particular case treated by Huntington, shows 

 directly the limit of the sum in question. The theorem is 

 that, if F(x) is the product of continuous functions, the sum 

 formed as if for a definite integral for this product, where the 

 variable in each of the continuous functions may have a dif- 

 ferent value in an interval of subdivision, is the integral of 

 F(x) : 



limS/itf'O/^ri) • • • h = limXF(6)Af. 



