PURE MATHEMATICS 35i 



the brilliant young mathematician of Trinity College, Cam- 

 bridge, who was killed in the war ; he accompanies it {ibid., 

 114-33) by elucidatory remarks and references, and by tran- 

 scripts of four letters written 191 4-1 5. The paper begins with 

 a consideration of a particular birational transformation of 

 space by means of the cubic surfaces through four lines ; this 

 had already been dealt with by Hesse, Cremona, Noether, 

 and Cayley. Mr. Wakeford then gives two theorems, due in 

 the first place to Cremona, namely, that two twisted cubics 

 of general position have ten common chords, and that six 

 lines of general position are chords of six twisted cubics. The 

 relations of these six cubics are then developed with application 

 to Cayley's problem of the condition for seven lines to lie on a 

 quartic surface. Prof. Baker points out the intimate con- 

 nection between this theory and the theory of apolarity investi- 

 gated by Reye. Finally, a new proof is given of Mr. Grace's 

 theorem that if six lines have a common transversal and we 

 take in the double six determined by any five of them the line 

 which does not meet these five, then the six lines so obtained 

 have a common transversal. 



Using the method of defining curves in space which is 

 developed by Hensel in his arithmetical theory of algebraic 

 functions, H. W. E. Jung {Math. Zs., 13, 1922, 189-201) brings 

 the formulae which serve to determine the stationary elements 

 of curves and developables into a form suitable for application 

 to curves on surfaces and to developables circumscribed to a 

 surface. In two later papers {ibid., 13, 1922, 202-16 ; 14, 

 1922, 1-34) he applies these formulae to the plane sections and 

 tangent cones and to the cuspidal and inflexional curves of an 

 algebraic surface. 



H. von Koch was the first to consider the curve defined in 

 the following way : A segment AB of a straight line is trisected 

 at C and E, and an equilateral triangle CDE is constructed on 

 CE. The segments AC, CD, DE, EB, are then treated in 

 precisely the same way as was AB, and so on indefinitely. The 

 vertices D, etc., of the triangles then define a curve, which can 

 be shown to be a Jordan curve {i.e. its points can be put into 

 correspondence with the points of a line) and to have no double 

 point. F. Apt {Math. Zs., 13, 1922, 217-22) now shows that 

 the curve is completely without tangents in a wider sense 

 than was contemplated by von Koch ; it can have no corners ; 

 i.e. if we consider separately the chords joining any point Po 

 to following points P and to preceding points P, in neither case 

 have they a limiting position as P approaches Pq. 



Y. Tanaka {Tohoku Math. J., 21, 1922, 1-2) gives a new 

 proof of a theorem due to W. W. Taylor that the centres of 

 the six conies touching five out of six lines lie on a conic, and 



