96 



W, so that in the altered form of the integral m is positive, there 

 results 



W=2-i(3m)-U-V I 1- — + 1-5.7.11 -,., 1 . 

 v } I 1.729 1.2(729)2 J 



By means of these expressions, W may be calculated with great 

 facility when m is at all large. The author has given a table of the 

 roots of the equation W=0, from the second to the fiftieth inclusively, 

 calculated by a formula derived from the former of the above expres- 

 sions. This formula was not sufficiently convergent to give the first 

 root to more than three places of decimals ; but this root may be 

 obtained more accurately from Mr. Airy's table. 



The method by which the author has treated the integral W ap- 

 pears to be of very general application, and he has further exem- 

 plified it by applying it to the infinite series 



-) — +... = —/ cos (a; cos 6)c?9, 



1 — 



2~ 2 



which occurs in a great many physical investigations, as well as to 

 the integral which occurs in investigating the diffraction produced 

 by a screen with a small circular aperture, placed in front of the 

 object-glass of a telescope through which a luminous point is viewed. 



Curvature of Imperfectly Elastic Beams. By Homersham Cox, 

 B.A. Jesus College. 



The equation to the curve of an elastic deflected beam is usually 

 deduced from the assumption, — 1, that the longitudinal compression 

 or extension of an elastic filament is proportional to the compressing 

 or extending force ; 2, that for equal extension and compression the 

 compressing and extending forces are equal to each other. 



These hypotheses are not quite correct in practice. All substances 

 appear to be subject to a defect of elasticity, i. e. their elastic forces 

 of restitution increase in a somewhat less degree than in proportion 

 to the extension or compression. If the forces be taken as functions 

 of the latter quantities expressed by a converging series of their 

 ascending integral powers, the terms after the third may in general be 

 neglected as of inconsiderable magnitude. If then e be the exten- 

 sion of a uniform rod of a unit of length and a unit of sectional area, 

 the longitudinal force producing that extension is 



ae + l3e"~ + ft'e 3 , 

 where a, (3, /3' are empirical constants. 



Similarly, if c be the compression of a similar rod, the force pro- 

 ducing that compression is 



yc-\-cc l -r $'c 3 , 

 where y, 8, b' are three other empirical constants. 



These formula? are to be applied to a uniform beam of rectangular 

 section, resting on horizontal supports and slightly deflected at its 

 centre. For this purpose, the compression and extension of every 

 filament of the beam are expressed in terms of the radius of curva- 

 ture and the distance from the neutral axis. Analytical expressions 

 are thus obtained for the elastic forces developed in any transverse 

 section of the beam ; and the position of the neutral axis is obtained 



