PURE MATHEMATICS 523 



ing-point (Strudelpunkt, foyer), or a vortex-point (Wirbelpunkt, 

 centre). A summary of these results, with references, is given 

 by H. Liebmann in the Encyklopddie Math. Wiss., Ill, D,8, 507. 

 O. Perron {Math. Zs., 31, 1922, 121-46) investigates the rather 

 more general problem in which. f{x ,y) , ^ {x,y) are not assumed to 

 be power series, but merely to be small relatively to the linear 

 terms. The integration can no longer be performed by means 

 of Poincare's series, but the same types arise, with the addition 

 of one new one, in which there are infinitely many closed integral 

 curves and also integral curves which wind infinitely many 

 times about the origin, having two closed integral curves as 

 asymptotic lines. In the present memoir, however, he confines 

 himself to three cases, for which (i) all the curves have a 

 common tangent at the origin, except one which has a different 

 tangent ; (2) one curve passes in each direction through the 

 origin; (3) all the curves without exception have a common 

 tangent at the origin and there is one which is crossed by all 

 the others. 



If a linear differential equation can be put into the form 



it will have in general a series solution 



^-^ +<^(a + i)'' ^<^(a + i)0(a + 2)^ + ' ' •' 

 where the initial exponent <z is a root of the indicial equation 

 </)(a) = o. T. W. Chaundy (Proc.L. M.S., 21, 1922, 214-34) applies 

 a similar method to partial differential equations of the type 



b'{4x- *^<4' • • •) - ^'"*''- • • ^(^-^z *'^,' • • •)] "= °- 



He obtams a solution which has as many arbitrary elements 

 as we ought to expect in the general solution, and which may 

 be made to coincide with the general solution in those cases in 

 which this has been otherwise obtained. He is not able to 

 show, however, that the solutions obtained are actually in all 

 cases general solutions. The method is expounded by the con- 

 sideration of particular examples. 



Progress has been made towards Goldbach's theorem on the 

 representation of an even number as the sum or difference of 

 two odd primes in a series of memoirs by P. Stackel [Sitz, 

 Heidelberg, 1917, Ab. 15; 1918, Ab. 2, 14; 1922, Ab. 10 (with 

 W. Weinreich)]. He makes use of " gap-numbers," which 

 arise in the process of finding primes by means of the sieve of 

 Eratosthenes. If p^ be the rth odd prime {p^ = 3), then the 

 gap-numbers of the rth rank are all the odd numbers which 

 are prime to p^, />,... pr, including i, but not these p's ; they 



