Jakuabt 31, 1908] 



SCIENCE 



165 



of integral equations to the isoperimetric 

 problem of the calculus of variation. 



13. Professor Halsted considered the 

 essence of inherent order, its foundation, 

 genesis, and the uses of inherently ordered 

 series in the attribution of factitious order 

 to primarily unordered assemblages. He 

 also investigated the betweenness relation, 

 both linear and of more than one dimen- 

 sion. 



14. The method of approximation de- 

 veloped by Poincare for obtaining the 

 terms of higher order of the perturbative 

 function depends uniquely upon certain 

 singularities of this function. With suit- 

 able numerical assumptions the case of 

 small eccentricities and a small inclination 

 of the orbits is carried out in detail by Dr. 

 Eeed, and the results to be found by vary- 

 ing the elements are indicated. 



15. The paper by Professor Eisland con- 

 tains a treatment of all the types of trans- 

 lation-surfaces which are determined by 

 a unicursal quartic in the plane at infinity. 

 The most important result is: To a uni- 

 cursal quartic with three real double points 

 correspond triply infinite types of transla- 

 tion-surfaces of a particular form. This 

 paper is a continuation of one recently 

 published in the American Journal of 

 Mathematics by the same author under the 

 title "On a Certain Class of Algebraic 

 Translation-surfaces." It will appear in 

 the same journal. 



16. Dr. Ranum proved the following 

 theorems: If 5f is a singular matrix not 

 belonging to any group, then if the num- 

 ber of zero roots of its characteristic equa- 

 tion is s, there always exists a positive 

 integer m which does not exceed s, such that 

 S"' belongs to some group. If m is the 

 lowest integer having this property then 

 among the invariant factors of the char- 

 acteristic determinant of 8 corresponding 

 to zero roots there is at least one of order 

 m and none of higher order. 



17. There is a discrepency between the 

 mathematical theory involved in the funda- 

 mental calculations of life, insurance con- 

 tingencies and the policy contract itself 

 which leaves some latitude and consequent 

 confusion. The object of Professor Beck- 

 ett's paper was to set forth this discrep- 

 ancy and to find a solution that can be 

 checked from the other items required to be 

 computed and exhibited in the report of a 

 company. 



18. Surfaces characterized by having 

 isothermal asymptotic lines and isothermal 

 lines of curvature Dr. Young has called 

 asymptotie-isothermic. In the present 

 paper he considers special classes of such 

 surfaces and thereby completes along cer- 

 tain lines a paper which he read before the 

 Chicago Section of the American Mathe- 

 matical Society several months earlier. 



19. The term division has two distinct 

 meanings in elementary mathematics. Ac- 

 cording to one of these it implies the opera- 

 tion of finding an integral quotient and an 

 integral remainder, while according to the 

 other it implies the finding of a number 

 which, multiplied into the divisor, produces 

 the dividend. While only the latter is the 

 inverse of multiplication, yet it is cus- 

 tomary to speak of division as an inverse 

 operation without specifying which of the 

 two commonly accepted definitions of the 

 term is meant. A very common definition 

 of multiplication is the performing upon 

 the multiplicand the same operation as that 

 which is performed upon unity to get the 

 multiplier. The vagueness of this defini- 

 tion follows directly from the fact that 4 

 may be obtained by doubling unity and 

 squaring the result, yet multiplying by 4 

 does not generally mean doubling the mul- 

 tiplicand and squaring the result. Such 

 vague definitions are contrary to the very 

 essence of mathematics and hence should 

 be avoided. 



Especial stress was laid by Professor 



