524 SCIENCE PROGRESS 



consist, in order, of i, p^+^y then prime numbers up to^%^.,, and 

 only after this include composite numbers. They appear to be 

 more tractable than the primes themselves. The first part of 

 the last paper quoted contains results on the expression of an 

 even number as the sum or difference of such gap-numbers ; 

 the second part contains guesses suggested thereby for primes, 

 which are tested experimentally by calculation. The}^ include 

 asymptotic formulae for the number of prime pairs of difference 

 2^ of which the first lies in the interval 2n, i, and for the num- 

 ber of Goldbach representations. This work should be com- 

 pared with similar empirical work by N. M, Shah and B. M. 

 Wilson {Proc. Camb. Phil. Soc, 19, 191 8, 228-54) ; see also a 

 paper by R, Haussner {Jahresber. d. Math. Ver., 31, 1922, 

 115-24). 



E. Kamke [Crelle, 152, 1922, 30-2) has a note on the simul- 

 taneous decomposition of integers into first and nth. powers. 



B.M.Wilson {Proc. Z.Af.S., 21, 1922, 235-55 ; Proc. Camb. 

 Phil. Soc, 21, 1922, 140-9) proves and extends some theorems 

 of Ramanujan connected with the sums of the rth powers of 

 the divisors of an integer n. 



R. Sturm {Crelle, 152, 1922, 90-8 and 99-108) has a couple 

 of interesting papers dealing with geometrical problems of 

 maxima and minima on the lines of his book Maxima und 

 Minim.a in der elementaren Geometrie (Leipzig, 19 10), which is 

 perhaps not sufficiently well known. He investigates in the first 

 the greatest tetrahedron, given the areas of the four faces ; and in 

 the second the two problems — (i) Given two lines and a point in 

 a plane, to find the line through the point for which the segment 

 cut off is a minimum ; {2) given three planes and a point (or a 

 line), to find the plane of the sheaf (or pencil) which is cut by 

 the given planes in the triangle of minimum area. Neither of 

 these problems can be reduced to one of the second degree, i.e. 

 they cannot be solved by ruler and compasses. 



The problem of the catenary in the calculus of variations 

 can be generalised if we suppose that, one end remaining fixed, 

 the other Xo, y'o is variable in a vertical plane, the length of 

 the strmg between the points being not a fixed quantity /, 

 but I— 4^{Xo,yo),4> being a given function. Queen Dido's Pro- 

 blem, of the maximum area enclosed between a string of given 

 length and a given boundary, can be generalised in a similar 

 way (G. Weyl, Crelle, 152, 1922, 76-98). 



Geometry. — A recent number of the Proceedings of the Cam- 

 bridge Philosophical Society contains three papers by pupils 

 of Prof. H. F. Baker. Miss H. G. Telling {Proc. Camb. Phil. 

 Soc, 21, 1922, 249-61) gives a geometrical treatment of the 

 theory of apolar quadrics, Re3^e's papers on the subject being 

 partly analytical. Given a fundamental quadric envelope S, self- 



