RECENT ADVANCES IN SCIENCE 7 



simple verification. Many interesting general theorems on 

 harmonic functions are developed. 



The same Journal contains valuable accounts of the work of 

 D. Hilbert and G. Darboux. 



G. Fubini {Rendiconti di Palermo , xliii, i, p. i) discusses the 

 fundamentals of projective differential geometry. The funda- 

 mental differential equations are deduced in curvilinear co- 

 ordinates by a purely projective method. 



C. Bonomi (p. 46), in the same number, describes the theory 

 of a special type of hyperelliptic surface. 



F. Gerbaldi (p. 78) treats the continued fractions of Halphen 

 in relation with (2,2) correspondences and the Poncelet polygons. 



P. Nalli (p. 105) makes an interesting contribution to the 

 theory of integral equations with a symmetrical kernel k{s, t). 



A. Palatini publishes \u Rendiconti di Palermo, xliii, i, p. 192, 

 two important contributions. The first, on the fundamentals 

 of the absolute differential calculus, in the sense associated 

 with the recent Einstein theor}^, discusses invariant and co- 

 variant systems, with their addition and multiplication, derives 

 the covariants of a mixed system, and an interesting integral 

 formula. The second proceeds to the invariantive deduction 

 of the gravitational equation from Hamilton's principle. 



A. F. Dufton, Proc. Roy. Soc, A, xc, describes a model 

 made for the drawing of conies. The principle used is that the 

 conic is the polar reciprocal of a circle. This has not been 

 used before, and the author has found a simple mechanism 

 which gives a real practical solution of a very old problem. 



J. W. Nicholson, Proc. Roy. Soc, A, xc, 1920, in a paper 

 on the lateral vibrations of sharply pointed bars, continues 

 his investigations of 191 7. The vibrations of bars of cir- 

 cular cross-section formed by the revolution of the curve 

 y = Ax"* about the axis of x are now completely discussed 

 for values of n between o and i , and for the isolated value n = 2. 

 The problem is interesting to the pure mathematician, since 

 for its general solution, for all values of n, functions which are 

 generalisations of the Bessel functions are needed. The prop- 

 erties of these functions are not yet adequately worked out. 



ASTRONOMY. By H. Spencer- Jones, M.A., B.Sc, The Royal Observa- 

 tory, Greenwich. 

 The Secular Acceleration of the Moon.— It has long been known 

 that, in order to make the theory of the moon's motion agree 

 both with the modern observations and with ancient and 

 mediaeval observations of ecHpses and occupations, it is neces- 

 sary to introduce into the theory certain empirical or quasi- 

 empirical terms. These consist of certain periodic terms of 

 long period with the addition of a term which indicates that 



