97 



from the integrals of these expressions by the principle, that the sum 

 of all the horizontal forces above is equal to the sum of all the hori- 

 zontal forces below the neutral axis. 



Next, the sums of the moments of the elastic forces about the neu- 

 tral axis are obtained ; and the sums are equated to the moment 

 about that axis of the pressure (P) of the fulcrum, the latter moment 

 being the product of half the deflecting weight by the distances (x) 

 of the fulcrum from the point of the neutral axis here considered. 

 This equation involves the radius of curvature, and is solved with 

 respect to the reciprocal of that quantity. It is to be observed, that 

 this equation, and also the preceding one determining the neutral 

 axis, are not of such a form as to admit of direct solution, and are 

 therefore solved by an ordinary method of approximation. 



The reciprocal of the radius of curvature of a point (x, y) of a 

 curve is equal to (the second differential of y with respect to x)-r- 

 (a quantity which becomes equal to unity when, as here, the incli- 

 nation to the axis of x of the tangent at any point of the curve is 

 comparatively very small). 



Making the substitution indicated, and integrating twice the equa- 

 tion last obtained, we obtained finally for the equation to the neutral 

 line of a rectangular beam of vertical depth d, and horizontal breadth 

 fx,, and length 2a, 



_ xx 3 bxW (2i 3 -cV¥ _/*a 2 fe% 3 (25 2 — c)a 4 K 3 \ 

 V 2.3 3.4 4.5 \2 ~" 3~ 4 J' 



where 



b— | rf(j3-r.$ay- 2 Xl + a4r- 4 )" 9 

 c=|^.(H- a 4y-4)-3(/3' + 5'afy-*) 



*=--|-(l + a iy-*) 2 . 



ju, a 3 a 



If, according to the ordinary hypotheses of perfect elasticity, we 



put a=y and neglect terms depending on |8, |S', 8, 8', this equation 



to the elastic curve coincides with that given by Poisson and others. 



If we put x=a, the value of the deflection at the centre of the 



beam is 



xa3 _ bKW (2b°- — c)y?a ; > 

 T ~1~ + 5 ' 



Whence it may be seen that the deflection is greater than it would 

 be if the elasticity were perfect. 



On the Knowledge of Body and Space. By H. Wedgwood, M.A. 



No part of the great metaphysical problem chalked out by Locke 

 has been more assiduously laboured, and none has attained a less 

 satisfactory solution, than that which relates to the origin of the 

 idea of space and its subordinate conceptions, figure, position, mag- 

 nitude. 



It was seen that the exercise of the muscular frame must somehow 

 be instrumental in making us acquainted with the material and ex- 



