RECENT ADVANCES IN SCIENCE 367 



some theorems concerning linear differential equations with 

 irregular singularities, preparatory to collecting his researches 

 in a larger treatise. 



Just as the notion of a derivate of a function suggests that 

 of differential equation, so the notion of the saltus function 

 (or oscillation function) leads to that of " saltus equation " ; 

 H. Blumberg, Amer. Journ. of Math., xli (1919), pp. 183-190, 

 investigates the complete solutions of a number of such saltus 

 equations of various simple types. 



A. L. Nelson, Amer. Journ. of Math., xli (1919), pp. 123-132, 

 computes a set of semi-invariants of systems of partial differ- 

 ential equations ; the results have applications to projective 

 differential geometry. 



E. H. Neville, Quarterly Journal, xlviii (1918), pp. 136-141, 

 gives proofs of formulae connected with moving axes with 

 variable angles ; the formulae have been announced by him 

 without proof at the Fifth International Congress in 191 2. 



C. D. Rice, Amer. Journ. of Math., xli (1919), pp. 165-182, 

 examines the invariants of differential geometry by the use 

 of vector forms. The analysis is much more simple and com- 

 pact than the familiar analysis involving Cartesian co-ordinates, 

 which is to be found in Forsyth's Differential Geometry. 



A. Egnell has examined, in a thesis on Infinitesimal Vec- 

 torial Geometry, asymptotic directions in vectorial fields, 

 and has proved that there are, in general, two asymptotic 

 directions. He now investigates, Comptes Rendus, 168 (191 9), 

 pp. 1263-265, the exceptional case, pointed out to him by 

 C. Guichand, in which the asymptotic directions are deter- 

 minate. Another solution of Egnell's problem is given by 

 R. Gamier, Comptes Rendus, 169 (1919), pp. 324-326. 1 



L. Silberstein, Phil. Mag., (6), xxxviii (1919), pp. 1 15-143, 

 extends the ideas in his ' Projective Vector Algebra ' (London, 

 1 919) to the projective definition of the scalar product and the 

 vector product of two vectors. 



E. Bompiani, Comptes Rendus, 168 (1919), pp. 755-757, in- 

 vestigates quasi-asymptotic curves on surfaces in n-dimen- 

 sional space. His results form a generalisation of Koenig's 

 theorem that the projection on a plane of the asymptotic lines 

 of a surface forms a conjugate net with equal invariants. 



W. Sensenig, Amer. Journ. of Math., xli (1919), pp. 1 1 1-122, 

 contributes to the Invariant Theory of Involutions of Conies 

 by deriving in terms of the system of two conies the complete 

 simultaneous system of the involution of the pencil of conies 

 through their four common points and their harmonic conic ; 

 the work follows on the researches of Gordan connected with 

 invariants of two conies and the researches of Baker and 

 Ciamberlini on three conies. 



