Double Suspension Mirror. 4-79 



vertical component, say V, we shall regard as unaltered by 

 small motions of P and Q. The component H cos </> of H is 

 parallel to the knife-edge and does not tend to deflect the 

 beam. The component H sin <£ tends to increase this de- 

 flexion, while V tends to diminish it, the moment about the 

 knife-edge being HZ sin <£ — VI sin ty, where I is the length 

 of the pointer. 



It will be convenient to compare the effect on the sensi- 

 tiveness with that due to an imaginary alteration of the level of 

 the centre of gravity of the beam, which may be effected by 

 raising an imaginary weight to originally coincident with the 

 knife-edge, through a distance r along a vertical wire attached 

 to the beam. For a deflexion -^r the moment due to w will 

 be ivr sin o/r. Thus the effect of the tension will be equal to 

 that due to w provided that 



HZ sin $ — VZ sin *ty = wr sin yjr, 



or, since ^ and d> are small, 



Hty — VZ>Jr = wrf (1) 



But Q Q or h tan <£ is equal to Z tan \fr or say 



h<j> = l1r, (2) 



thus (1) becomes 



H/-VA = ^, (3) 



and we may say that for small values of </> the effect of the 

 fibre on the balance is the same as that which would be due 

 to a weight w placed at a distance r above the knife-edge, 

 r being given by (3). 



Since V is proportional to H and Z is a constant, we may 



HZ 2 

 write V = nZH: then putting- k 2 for the above condition 



, 10 



becomes 



h(r + nk 2 )=k 2 , (4) 



where n and k 2 are constants. It is represented in fig. 3 by 

 a rectangular hyperbola, whose centre is nk 2 below the axis 

 of h. The part of the curve below OA represents cases in 

 which the fibres are separated so far as to diverge upwards, 

 when they cause a reduction of sensitiveness instead of an 

 increase. But our conditions do not apply to these extreme 

 ca^es, since evidently H would not remain constant. 



dr HZ 2 



The variation of r with h is given by =- = — — . If 



2K2 



