690 



SCIENCE. 



[N. S. Vol. XVn. No. 435. 



series values of y, all the roots of the equa- 

 tion f{y)=0, real and imaginary, are 

 found in convergent series involving only 

 the coefficients of f{y) ^0. 



The same device of introducing a factor 

 X which is eventually made unity makes 

 it possible to obtain by a direct application 

 of Maclaurin's series all the expansions 

 which hitherto have been obtained by La- 

 grange's series and Laplace's series. 



The Mechanical Construction and Use of 

 Logarithms: Mr. Charles E. Brooks, of 

 Baltimore. (Introduced by Professor 

 George F. Barker.) 

 In this paper is described a simple in- 

 strument for constructing the logarithmic 

 spiral with great accuracy. The device will 

 be useful for drawing the curve in the 

 class room; it may be used also for the 

 preparation of tables of logarithms of all 

 the possible systems, or for the mechanical 

 solution of arithmetical problems. 



The machine consists of a screw pivoted 

 so that it may be rotated, but will remain 

 parallel to the paper. A wheel is threaded 

 to the screw and rests with its circumfer- 

 ence on the paper. As the screw is rotated, 

 the wheel rolls on the paper, but this rolling 

 makes it travel along the screw. The track 

 of the rolling wheel is, therefore, a spiral. 



To show that this spiral is the logarithmic 

 curve, consider the equation of motion of 

 the center of the wheel, which has the same 

 motion as the point which draws the curve. 

 Let OA (see figure) be the screw, pivoted 

 at O; let B be the wheel; C its center. 

 Call C the point pd measuring with as 

 origin and any line OP as axis. Let the 

 pitch of the screw be p, and the radius of 

 the wheel be r. 



As increases an amount Af), c moves 

 through an arc cc' equal pJO. At the 

 same time the wheel turns through an arc 

 p/io, so the angular motion around OA is 



pAd 

 r 



Under the influence of the thread on OA it 

 is moved along OA a distance 



But this distance is Ap^ so we have 



Ap: 



and in the limit, 



^ = ^p 



Integrating, 



6p _-p 



p^ce'-e 



That is, 6 is the logarithm of p. 



The Theory of Assemblages and the In- 

 tegration of Discontinuous Functions: 

 Professor I. J. Schwatt, of Philadel- 

 phia. (Introduced by Professor C. L. 

 Doolittle.) 



An historic review of the state of the 

 theory of continuous and discontinuous 

 functions prior to the creation of the theory 

 of assemblages by Bolzano and Cantor is 

 first given. It is then shown how the theory 

 of assemblages has served to make this part 

 of the theory of functions more clear and 

 definite. The question of the content of a 

 mass of points, distributed along a line, is 

 discussed ; the more important principles of 

 the theory of assemblages are given, and 

 applications of these principles to the in- 

 tegration of discontinuous functions are 

 made. 



The Franklin Papers in the Library of the 

 American Philosophical Society: Mr. J. 

 G. Rosengarten, of Philadelphia. 

 In the collection of this society there are 

 some seventy large folio volumes of ' Frank- 

 lin Papers.' Franklin left all his papers 

 to his grandson, William Temple Franklin, 



