134 



SCIENCE. 



[N. S. Vol. XVI. No. 395. 



On a Class of Real Functions to tvhich 

 Taylor's Theorem does not apply; and 



On a Class of Transcendental Functions 



ivith Line-Singularities: Professor John 



A. EiSTAND, Tliiel College. 



In the first paper a class of real functions 

 to which Taylor's theorem does not apply 

 was discussed. Examples of such functions 

 were given and the non-identity of the ex- 

 pansion with the function expanded was 

 shown. 



In the second paper a new type of trans- 

 cendental functions fulfilling certain condi- 

 tions within and on the unit-circle was dis- 

 cussed. These conditions are : The function 

 together with all its derivatives is finite 

 and continuous within as ivell as on the 

 unit circle; which is a singular line for 

 the function. The form of the functions 

 is as follows : 



/(^)- 



:n 



z—(\+av)e'- 



where 



2— (1 -h ftOe^"*"" 

 lim ov = 0, lim6,. = 0, 



a is an incommensurable number, and 



. ...+ ...+ J_ r {a.-K)e>-^^^^ 1-1. 

 + + ^r-1 L3-(l-t-60e-"-"J 



On a General Method of Subdividing the 

 Surface of a Sphere into Congruent 

 Parts: Mr. Haeold C. Goddaed, Amherst 

 College. 



The problem was incidental to the prac- 

 tical problem of constructing a steel sphere 

 one hundred feet in diameter, in connection 

 with a new method of mounting a tele- 

 scope, as outlined in an article in the Amer- 

 ican Journal of Science for June, 1902, by 

 Professor David P. Todd, of Amhei-st Col- 

 lege. 



If a regular dodecaedron be inscribed in 

 a sphere, planes determined by the center 

 and each edge of the dodecaedron cut out 



on the sphere twelve equal regular spher- 

 ical pentagons. If the vertices of each pen- 

 tagon be connected with its center by arcs 

 of great circles the surface of the sphere 

 is divided into sixty congruent isosceles 

 spherical triangles, whose angles are deter- 

 mined as 60° J 60° and 72°. 



A Possible New Laiv in the Theory of Elas- 

 ticity: Professor J. Buekitt Webb, 

 Stevens Institute of Technology. 

 Owing to the absence of Professor Webb 

 at the time this paper was called for, it was 

 presented only in abstract. The law re- 

 ferred to in the title is: "If the forcible 

 change of the distance between two points 

 in an elastic sytem changes the distance of 

 two other points by a certain amount, then 

 the same force applied to alter the distance 

 of the two other points will change the dis- 

 tance of the first two points by the same 

 amount." 



On Extracting Roots of Nunibers iy Sub- 

 traction: Dr. Aetemus Maetin, Wash- 

 ington, D. C. 



A paper on 'Evolution by Subtraction' 

 was published in the Philosophical Maga- 

 zine for September, 1880, communicated by 

 the Rev. F. H. Hummell, who ascribed the 

 method to his friend and neighbor, the Rev. 

 W. B. Cole. The rule given in Mr. Hum- 

 mell 's paper for finding the square root of 

 any number is : 



From any square number subtract the 

 even numbers in succession, beginning with 

 2, until the remainder is less than the next 

 even number to be subtracted. This re- 

 mainder will be the square root sought. 



The statement of the rule may be simpli- 

 fied as follows : 



For the nth subtrahend add 2 to the pre- 

 ceding subtrahend. The last remainder 

 will be the square root sought. 



Or: For the nth subtrahend, multiply n 

 by 2 ; the last remainder will be the square 

 root sought. 



