MATHEMATICS 9 



at by finding what is the effect on the solution if the curve 

 acquires a new double point. 



Thesame author {Rend. Palermo, 46, 1922, 105-116) gives a 

 simplification, suggested by remarks of Schlesinger, of the 

 algebraic proof of the existence theorem which is given in 

 the new Algebraische Geometrie, p. 334. He also shows that 

 the most general curve, of genus p> i , which contains at least 

 one linear series of freedom i and grade n<^p + i, depends 

 precisely on 3^-3-^ moduli, i being the index of specialty 

 of the series obtained from twice the given series. 



C. V. Hanumanta Rao {Proc. Camb. Phil. Soc, 20, 1921, 

 434-36) proves that, considering the several modes of genera- 

 tion of a bicircular quartic by means of circles, a varying circle 

 of one mode makes with two fixed circles of a second mode 

 angles with a constant sum. 



D. G. Taylor {Proc. Edin. Math. Soc, 39, 1921, 2-6) shows 

 that on a plane cubic curve three n-ads can be found, 

 every pair being in n-ple perspective from the points of the 

 third. 



Following Cayley, Bacharach and Gergonne, L. Casteel 

 {C.R., 173, 1921, 512-14) constructs plane cubics through nine 

 points by linear and quadratic constructions only. 



Two papers which make use of elliptic functions, and which 

 are both extremely detailed, are by J. Thomse {Jahresber. d. 

 deut. Math. Verein, 29, 1920, 183-236) on Cassini curves, and 

 by A. R. Forsyth {Q.J. Math., 49, 1921, 139-85) on the 

 developable surfaces through a couple of guiding curves in 

 different planes. 



The usual classification of collineations of space is by means 

 of the invariant factors of Weierstrass ; K. Kommerell {Jahres- 

 ber. d. deut. Math. Verein, 29, 1920, 1-27) invents a new classifi- 

 cation, by means of three invariants of the coefficients, which 

 is not affected by displacements before applying the collinea- 

 tion. 



E. Veneroni {Rend. Lombardo, 54, 1921, 166-74) discusses 

 congruences of conies, and L. Godeaux {Bull. de. I'acad. roy. de 

 Belgique, 7, 1921, 596-607) a linear congruence of twisted 

 cubics. 



C. Segre {Atti Torino, 56, 1921, 143-57) shows that if a 

 two-dimensional surface in space of four or more dimensions 

 has a double infinity of plane curves, then the}^ must be conies 

 and the surface must either be a ruled cubic or a surface of the 

 fourth order of Veronese in space of four or five dimensions. 

 He similarly reduces to a few classes surfaces with a double 

 infinity of ordinary twisted curves. 



W. Burnside {Proc. Camb. Phil. Soc, 20, 192 1, 437-41) 

 has a note on convex solids in higher space, and proves, for 



