TRANSACTIONS OF SECTION A. 561 



4. Tlie Determination of Successive High Primes. 

 By Lieut. -Colonel A. Cunningham, R.E., and H. J. Woodall. 



5. Algehraic Curves on Kummer's \6-nodal Quartic Surface. 

 By R. W. H. T. Hudson. 



The chief difficulty in studying algebraic curves traced on a surface is due to 

 the fact that auch a curve is in general not the complete, but only a partial, inter- 

 section with another surface, and) is, therefore, not representable by only one 

 algebraic point equation in addition to that of the first surface. When the 

 surface can be represented parametrically by means of known functions, a single 

 equation in the two parameters is sufficient to determine the curve, and its proper- 

 ties follow from the known properties of the functions. M. Georges Humbert has 

 proved that every algebraic curve on Kummer's surface can be represented by 

 equating to zero a certain kind of theta function.' When certain fundamental 

 theorems concerning theta functions have been proved, many important geo- 

 metrical theorems follow immediately. There are two objections to this pro- 

 cedure. The geometrician who has but a slight acquaintance with transcendental 

 analysis finds the excursion into the realms of hypei--elliptic functions troublesome, 

 and, on the other hand, the arithmetician has doubts as to whether the said 

 fundamental theorems have ever been rigorously proved. The purpose of the 

 present paper is to remove both these objections. 



The matrices 



and 



are obviously orthogonal ; they are as familiar as quaternions. The rotos of the 

 first are transformable into each other by the operators of a group, and, similarly, 

 the columns of the second are related to another group of four operators, iso» 

 morphic with the former. The product of these matrices is an orthogonal matrix 

 of sixteen linear forms, containing in itself the whole theory of Kummer's con- 

 figuration of sixteen points and sixteen planes ; it is associated with a group of 

 sixteen members, the product of the two preceding groups, which explains the 

 existence of the configuration. 



The theorem which, it is suggested, may replace Humbert's is as follows : 

 Every alyebraic curve on Kummer's su7-face is representable by an equation — homo- 

 geneous and possessing a certain liind of isobarity in the square-roots of these six- 

 teen linear forms. If such an equation is squared, the result can be rationalised 

 by the aid of the equation of the surface ; whence follows the well-known theorem 

 that along every algebraic curve an algebraic surface can be inscribed ; other 

 theorems follow in a like elementary and algebraic manner. 



SUB-SKCTION ASTKONOMY AND METEOROLOGY. 

 The following Papers and Report were read : — 



1. Emploie de V Hygrometre h Cheveu au lieu du Psychromeire. 

 By HoFRATH J. M. Pernter. 



' LionviUe'a Jutir. ser, 4, vol. ix. (1898). 

 1903. o o 



