522 SCIENCE PROGRESS 



series about it. Then there is one singular point a on the 

 boundary of the region of convergence. But if a is a limit- 

 point there is more than one singular point on the boundary. 



A particular example is i/(i ^ z"-), for which the zeros of 

 R"(^) are the points z = cot rir lin + i), (r = 1,2. . . n). 



IV. If V{z), Q(2) are polynomials, Q(s) = ^;s^ + . . . + ^3, 

 b ^o,q:^2, andG(s) = P(s) e«'^', then the limit-points depend 

 only on q and on the two highest coefficients of Q, b and Z»i. 

 They are the points of two half-rays from z = - bjqb, dividmg 

 the plane into q equal angles, i.e. are parallel to the (vector) 

 roots oi bs!^ + i =0. 



An example of this, given by Markoff, is exp ( - 2'), the zeros 

 of the derivatives being the zeros of Hermite's polynomials, 

 which can be found as close as we like to any point of the real 



axis. 



See also Alander (Arkiv, 14, 1920, No. 23), who deals with 



integral functions of genera 2, 3, 4, 5- 



Weierstrass, in a letter to Schwarz in 1875 {Ges. Werke, ii, 

 p. 235), remarked : " Je mehr ich iiber die Prinzipien der 

 Funktionentheorie nachdenke— und ich tue dies unablassig — um 

 so fester wird meine Uberzeugung, dass diese auf dem Funda- 

 mente algebraischer Wahrheiten aufgebaut werden muss." 

 He himself carried this out in his lectures on Abel's Transcen- 

 dentals {Ges. Werke, iv). R. Konig {Math. Zs., 15, 1922, 26-65), 

 taking this as his motto, proceeds to develop the theory of 

 Riemann's transcendentals, which include Abel's as a particular 

 case, in particular as regards that part of Weierstrass 's lectures 

 which deals with the fundamental function }:l{xy, x'y'). The 

 fruitfulness of the theory shows itself in the discovery of a 

 complete series of " interchange " theorems, which include 

 those of Abel, Weierstrass, and Fuchs ; the development of 

 these is, however, postponed to a later paper. 



Among the literary remains of Gauss is a fragment on the 

 multiplication of lemniscate functions for the number 7 ; having 

 completed this, K. Schwering {Crelle, 152, 1922, 40-8) investi- 

 gates the more difficult problem of division. See also G. B. 

 Mathews {Proc. L.M.S., 14, 191 5, 464-6). 



Poincare and others have investigated the form of the 

 integral curves of the differential equation 



[kx + ly + f{x,y)]dy = [mx + ny + cf){x,y)]dx 

 near the origin, where f{x,y) and cf){x,y) are power series beginning 

 with terms at least of the second degree. If kn - Im ^ o, the 

 integral curves approximate to those of the simpler equation 



{kx + ly)dy - {mx + ny)dx, 

 and it is easy to obtain conditions for the origin to be a node 

 (Knotenpunkt, noeud), a saddle-point (Sattelpunkt,col), a wind- 



