352 SCIENCE PROGRESS 



tends to infinity ; and, secondly, that the sequences, found by 

 Young, which compounded with an arbitrary Fourier series 

 S (a„ cos nx + b^ sin nx) give a Fourier series S\„ {a„ cos nx + bn 

 sin nx) are the only sequences with this property. 



H. Steinhaus {Proc. Lond. Math. Soc, 20, 1921, 123) gives 

 the first example of a thoroughly divergent orthogonal develop- 

 ment, i.e. he shows how to define an orthogonal, normalised and 

 complete sequence of functions 4> (x), integrable (L), together 

 with their squares, in the interval {a, b), so that a suitable 

 integrable function / (x) can be found, whose Fourier-like ex- 



pansion ^ ^i ^*' I / (/) </>i (/) dt is divergent for every value of x 



in {a, b). 



G. N. Watson {Proc. Lond. Math. Soc, 20, 192 1, 189) 

 connects the product of two hypergeometric functions with 

 hypergeometric functions of two variables of Appell's fourth type. 



P. Humbert {C.R., 173, 1921, 217) investigates a formula of 

 multiplication for Kummer's function ^ (a, 7, x) which is one 

 of the confluent hypergeometric functions ; he also develops 

 (Proc. Roy. Soc. Edin., 41, 1921, i), the theory of the confluent 

 hypergeometric functions of two variables, which is related to 

 the potential equation in space of four dimensions. 



H. Faxen (Arkiv for Mat., 15, 1921, No. 13) obtains series 

 for the integrals 



[ e-*('±'~'') tvdt and | g-^^'i'"'^) ivlogt. dt, 



Jy Jy 



which include the Hankel-Bessel functions, the integral-loga- 

 rithm, and the error-function as special cases. 



S. Chandra Dhar {Tohoku Math. Journ., 19, 1921, 175) 

 obtains series for the solutions of Mathieu's equation of the 

 second kind in a different form from those obtained by E. L. 

 Ince. 



Geometry. — Connected with a plane cubic curve are two 

 curves of class three, called by Cayley the Pippian and the 

 Quippian (the former is now usually called the Cayleyan). 

 W. P. Milne and D. G. Taylor {Proc. Lond. Math. Soc, 20, 

 1 92 1, 10 1 ) show how the pencil of class-cubics given by these 

 two curves may be defined geometrically from apolar relations 

 with regard to the pencil of cubics through the intersections of 

 the given cubic and its Hessian. W. P. Milne {ibid., 20, 1921, 

 107) also generalises properties of corresponding points on the 

 Hessian of a plane cubic, regarding such points as degenerate 

 conies apolar to the net of polar conies of a given cubic. 



K. W. Rutgers {Proc Amst. Acad., 23, 1921, 797) examines 

 the number of degenerations possible in linear systems of plane 

 cubics. 



