396 



SCIENCE 



[N. S. Vol. II. No. 39. 



al Magazine. In the absence of Dr. Martin 

 his paper was read by the Secretary. 



Semi-invariants of a ternary quantic sat- 

 isfy some, but not all, of the six differential 

 equations which characterize invariants. 

 Professor McMahon"s paper shows how to 

 distinguish between those semi-invariants 

 which are the sources of covariants and 

 those which are the sources of semi- 

 covariants, and gives a simple method of 

 deriving from the latter all the coefficients 

 of the semi-covariant, or of a semi-contra- 

 variant if desired. A systematic geometri- 

 cal interpretation of these three kinds of 

 semi-concomitants is presented and appears 

 to be particularly useful in cartesian co- 

 ordinates. This paper is intended for the 

 Annals of Mathematics. 



In the absence of Professor Echols his 

 two papers were presented by the Secretary. 

 The first one was the second section of an 

 essay ' On the DifiFerell and Integrell Calcu- 

 lus,' the preceding section having been read 

 at the May meeting of the Society. Pro- 

 fessor Echols approaches the infinitesimal 

 calculus from the calculus of finite differ- 

 ences. He considers the latter, however, 

 in a greatly generalized form. A dififerell 

 is defined to be the limit of a ratio whose 

 terms are the n-th differences of the func- 

 tion and the independent variable. An 

 integrell is a differell of a negative index. 

 The applications presented consisted chiefly 

 in the expansion of functions in terms of 

 differells and integrells. In the second 

 paper some of the more novel results were 

 translated into the language of ordinary 

 calculus. 



Mr. Lambert attempted to show that 

 functions of which Weierstrass's deriva- 

 tiveless function is the type may be so con- 

 sidered geometrically that their curves will 

 have determinate tangents. In order to 

 secure agreement of the analytic result 

 with the geometric he found it necessary to 

 replace Weierstrass's limitation of /;, 



n increasing indefinitely, and s being any 

 finite quantity. 



In Professor Moore's paper a tactical con- 

 figuration is established which exactly de- 

 fines Jordan's linear substitution group 

 when taken fractionally. This configura- 

 tion is self-reciprocal. The properties of 

 this configuration and of other allied con- 

 figurations similarly related to certain im- 

 portant subgroups of the main group are 

 developed. Professor Moore's paper will 

 appear in the Bulletin of the American Mathe- 

 matical Society. It was presented at the 

 meeting by the Secretary. 



Professor Baker's paper contained a de- 

 tailed discussion of the character of the 

 operations of algebra and the theory of 

 imaginary quantities. It will be published 

 in the American Journal of Mathematics. 



In Professor Van Yleck's paper an ag- 

 gregate of regular linear differential equa- 

 tions of the second order is taken, each of 

 which has a polynomial solution. The re- 

 quirement is made that these equations 

 shall have a common group arising from 

 their four common branch points; in Rie- 

 mann's phraseology, that they shall be re- 

 lated. This requirement necessitates the 

 introduction of one accessorj^ branch point 

 into each equation. These accessoiy points 

 do not, however, give rise to anj' substitu- 

 tions of the group. The paper outlines a 

 method bj' which the position of the ac- 

 cessory points may be investigated, as well 

 as the distribution of the roots of the vari- 

 ous polynomials between the four (real) 

 branch points and in the imaginarj' domain. 

 One result is the determination of the distri- 

 bution of the roots of all polynomials satis- 

 fjdng differential equations of the second or- 

 der with exactlj' four branch points which 

 with their exponent difiTerences are given. 



Mr. Roberts' paper gives the differential 

 equations of certain systems of conies in a 



