178 SCIENCE PROGRESS 



See also, on first order equations, L. Koenigsberger (Sitz. 

 Heidelberg, 1921, Abh. 2 and 7). 



W. Sternberg {Math. Ann., 86, 1922, 140-53) writes on 

 the asymptotic integration of partial differential equations 

 involving a parameter. 



If in a problem of the maxima and minima of a function 

 of two variables with the condition that another function is 

 to have an assigned value we interchange the roles of the 

 two functions, it may happen that the solution is the same with 

 the interchange of maximum and minimum ; J. K. Whitte- 

 more {Amer. Journ. Math., 43, 1921, 271-90) examines this 

 reciprocity 



C. Carath^odory (Math. Ann., 85, 1922, 78-88) and M. 

 Picone {Rend. Lincei, 31, 1922, 46, 94) write on the calculus 

 of variations for multiple integrals. 



A. E. Western {Proa. Canib. Phil. S., 21, 1922, 108-9) 

 compares Hardy and Littlewood's conjectured formula for the 

 number of primes of the form n^ + i with the facts as given in 

 tables of Lt.-Col. A. Cunningham up to n = 15,000. 



G. Torelli {Rend. Napoli, 27, 1921, 262-8) and P. Stackel 

 and W. Weinreich {Abh. d. Heidelberg. Akad., 1922, Abh. 10) 

 write on the representation of an even number as the sum of 

 two primes. 



G. H. Hardy and J. E. Littlewood {Proa. L.M.S., 21, 

 1922, 39-74) study the zeta-function in the critical strip, 

 and obtain a new result in the order of the error term in the 

 divisor problem. 



S. Wigert {Proc. Camb. Phil. S., 21, 1922, 17-21), C. Siegel 

 {Math. Ann., 85, 1922, 123-8) and H. Hamburger {ibid., 

 129-40) also write on the Riemann zeta-function. 



Papers on Waring's problem are by H. Weyl {Gottingen 

 Nach., 1 92 1, 189-92), G. H. Hardy and J. E. Littlewood 

 {Math. Zs., 12, 1922, 161-88), E. Landau {ibid., 219-47) and 

 E. Kamke {ibid., 323-8). 



G. H. Hardy and J. E. Littlewood {Proc. Camb. Phil. S., 

 21, 1922, 1-5) have a note on the trigonometrical series asso- 

 ciated with the elliptic theta-functions ; E. T. Bell {Math. Zs., 

 13, 1922, 146-52) gives arithmetical equivalents for an identity 

 between theta-functions ; I. Schur {Gottingen Nach., 192 1, 

 147-53) writes on Gauss's sums, and G. Herglotz {Leipzig Ber., 

 73, 1 92 1, 271-6) on the last entry in Gauss's Tagebuch, con- 

 cerned with the connection between the theory of biquadratic 

 residues and the lemniscate functions. 



Connected with the theory of algebraic numbers are papers 

 by I. Schur {Sits. Berlin, 1922, 145-68 ; Math. Zs., 12, 1922, 

 95-113); G. Herglotz {ibid., 255-61); E. Hecke {Hamburg 

 Seminar, 1, 1922, 102-12) ; T. Nagel {ibid., 140-50); K. Hensel 



