Bending of Magnetic Needles on the Magnetic Dip. 211 



and we may confine ourselves to the case of the pointed 

 needles, for which the calculation is very simple. 



We take a pointed needle of unit thickness and half length 

 Z, and find in the first place the equation of its axis when it 

 is in a horizontal position and subject to its own gravity alone. 

 Let the axis of x be horizontal, and that of y downwards. 

 If a cross section of the needle is taken at a point P, the 

 bending-couple at the section is that of the weight of the 

 pointed end multiplied by the distance of the centre of gravity 

 of the end from the cross section, or 



v(l—x) [l—x) 



VP -*-■*■' 



In this equation v represents the width of the needle at 

 the point P. 



If a is the width at the origin, 



vl = a(b— x), 



and the bending-couple becomes 



If Y is Young's modulus of elasticity, the resistance to the 



bending will be Yy^. Substituting the value of v in terms 



of Zj the differential equation for the axis becomes 

 dy* 21? 

 dx^ = aW^ & 



Hence the curvature is constant and the needle takes up a 

 circular shape. The integral of equation (1), taking account 

 of the conditions holding at the origin, is : — 



The distance h between the centre of gravity and the axis 

 is now obtained in the usual manner. 



I yvdx p ( x*{l-x)dx 



t vdx I (l — x)dx 



J° Jo 



from which it follows that 



h- gpl * an 



Phil. Mag. S. 5. Vol. 31. No. 190. Mar. 1891. X 



