RECENT ADVANCES IN SCIENCE 3 



vanish at a point x = a. The problem arose in relation to the 

 scattering of sound waves by a cone, and is also of interest 

 for the pure mathematician. Prof. Carslaw called attention to 

 the need for a proof of the fact that all the roots of this equation 

 are real and separate. Prof. Priestley now supplies a proof 

 derived in a very elegant manner from the theory of a Homo- 

 geneous Integral Equation. 



J. Hodgkinson, part 4, p. 268, continues the problem of 

 conformal representation of a curvilinear triangle, in some 

 aspects previously developed in an incomplete manner. 



W. P. Milne, part 4, p. 274, discusses the determinant sys- 

 tems of co-apolar triads on a cubic curve. The configuration 

 of these triads had already been determined by a method which 

 did not indicate the symmetry of its form, or any convenient 

 method of visualising its structure as a geometrical unit. The 

 paper now under notice establishes the symmetry of the 

 configuration. 



H. S. Carslaw, part 4, p. 291, discusses the diffraction of 

 waves by a wedge of any angle, in which the older method of 

 Sommerfeld and Carslaw has been superseded by the later 

 methods of Macdonald, Bromwich, and others. The author 

 shows that the solutions, which are already known, can never- 

 theless be obtained readily from the older method, 



W. H. Young, part 4, p. 307, publishes a part of a long 

 memoir on non-harmonic Fourier Series, which is concluded in 

 part 5. It is related somewhat closely to a memoir which was 

 noticed at some length in Science Progress, April 1920, and 

 we shall state only the main problem, which is the possibility 

 of expanding a function j{x) in a series of the form 



'St-o. ^nCOs{k 4- n)x 



where k is not a whole number. Conditions for the existence 

 of such an expansion, together with the equations for deter- 

 mining the coefficients, are discussed with some completeness, 

 and some important applications to the series of Bessel functions 

 follow. 



W. H. Young, part 5, p. 339, discusses the formula for an 

 area, familiar in elementary integral calculus, and submits it 

 to an exhaustive examination. It has been usual to suppose 

 the curve, whose area is discussed, to be a simple Jordan curve, 

 or a curve defined by a continuous (i, i) correspondence with 

 a straight fine or circle. It thus possesses an " inside " and an 

 " outside," the former being a simply-connected region. The 

 area can be defined as the common limit of the areas of two 

 sets of polygons, one set inscribed and one set circumscribed, 

 whose sides increase in number. This limit is the content of 



