MATHEMATICS 349 



ences to subsequent proofs, is given in Dickson's History of the 

 Theory of Numbers , vol. ii ; this should now be supplemented 

 by a couple of notes and two long papers by E. T. Bell {Bull. 

 Amer. Math. Soc, 27, 192 1, 273, 330, and Trans. Amer. Math. 

 Soc, 22, 192 1, I, 198). 



An interesting article by L. E. Dickson {Bull. Atner. Math. 

 Soc, 27, 192 1, 312) points out that numerous writers, including 

 Gauss, have thought erroneously that when they have found 

 all rational solutions of a homogeneous equation they have 

 thereby found all integral solutions of the corresponding non- 

 homogeneous equation. For example, the equation X* + 

 5 y* = Z can easily be solved in rational numbers, but to deduce 

 therefrom the solution oi x^ + S y^ = ^^ in integers involves a 

 knowledge of all divisors of all numbers which can be put into 

 the form x* + 5 y*. He follows this up {ibid., 27, 1921, 353) by 

 developing a new method in Diophantine Analysis, based on 

 the theory of ideals. 



A more elementary paper by the same author {Amer. Math. 

 Monthly, 28, 1921, 244) deals with rational triangles and 

 quadrilaterals. 



G. H. Hardy and J. E. Littlewood {Proc. Lond. Math. Soc, 

 20, 1 92 1, 15) investigate the lattice-points {i.e. points whose co- 

 ordinates are both integral) of a right-angled triangle. 



With regard to the Riemann Zeta-function we may note a 

 paper by H. Weyl, whose versatility is wonderful, on the order 

 of ^(i + ti) as t tends to infinity {Math. Zs., 10, 1921, 88), and 

 one by Hardy and Littlewood {ibid., 10, 1921, 283) on the 

 number of zeros of ^(o- + it) on the critical line <y = \. 



Georges Humbert, whose death on January 22, 1921, was 

 announced in Nature for March 17, first became known by his 

 geometrical applications of the theory of functions. Of late, 

 however, the theory of algebraic numbers had engaged his 

 attention and a memoir on ternary Hermite forms in an imagi- 

 nary quadratic field has appeared since his death {C.R., 172, 

 1 92 1, 497, and Liouville, 4, 1921, 3). 



W. H. Young {Proc Roy. Soc, A. 99, 1921, 252) finds sets 

 of sufficient conditions for the transformation of the variables 

 in multiple integrals, by a method involving the notion of 

 " associated summabilities." 



lif{x,y) = f {x, yi, y^ • • - yh-i, • • .) is a finite, measurable 

 function of any number of variables we can define four partial 

 derivates with respect to x (upper right, upper left, lower 

 right, lower left), being the upper and lower limits of 

 {f{x + h,y)-f{x,y)}lh, for h>0 (right) and /? < O (left). 

 G. C. Young {Proc Lond. Math. Soc, 20, 1921, 182) proves 

 that the points at which one of the upper derivates, being finite, 

 is not equal to the lower derivate on the other side form a 



