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Transactions. 



of force on the counterpoise side for any angle A. Let W = the weight 



on the scale-pan side : then 



L X sin A X C 



= the required distance from the 



fulcrum at which the scale-pan must be suspended for any given angle A. 

 Because the angle A must be in proportion to W, let W = nk : then 



L X sin A X C sin A LC ,-, • j j- - r ^i j- i 



7 = —. — X — = the reqmred distance from the fulcrum at 



which the scale-pan must be suspended for any given angle A. Hence it 

 is evident that in a gravity lever balance, in order to make the angle of 

 deflection (A) of the long arm proportional to the weight suspended from 

 the short arm throughout the whole length of the quadrant, whilst the 

 virtual length of the long arm varies as sin A, the virtual length of the 

 short arm must vary as sin A /A. 



The above relation is fixed for all balances of this class, but L, C, and n 

 may have any values according to the desired dimensions of the balance. 

 In the example given they are 140, 6-428^, and i respectively. The value 

 of w is I because the number of ounces to be indicated by any angle of 

 deflection happens to be one-half of the number of degrees in the angle. 

 In this or in any other case n will equal 90 divided by the total number 

 of units of weight (including the Aveight of the scale-pan) intended to be 

 the full range of the balance. 



In the following table, column (1) gives A, the angle of deflection from 

 the vertical of the long arm, in degTees. Column (2) gives sin A. The 

 virtual length of the long arm varies with the angle A as the figures in this 

 column. Column (3) gives sin A /A. The virtual length of the short arm, 

 with the angle A, must vary as the figures in this column. Column (4) 

 shows the application of column (3) to the particular example given, in 

 which LC/n is equal to 1800. It gives, in millimetres, the various dis- 

 tances from the fidcrum at which the scale-pan must be suspended accord- 

 ing to the angle A given in each case. ' 



Sine A 



"a 



(3.) 

 0-01745 

 0-01743 

 0-01736 

 0-01710 

 0-01667 

 0-01607 

 0-01532 

 0-01443 

 0-01342 

 0-01231 

 0-01111 



(4.) 

 31-41 

 31-37 

 31-30 

 30-80 

 30-00 

 29-00 

 27-50 

 26-00 

 24-00 

 22-20 

 20-00 



From the figures in column (3) a curve can be plotted which will give 

 correctly the shape of the required cam (see fig. 3). 



Taking the point F. which represents the fulcrum of the balance, as 

 centre, describe the quadrant marked F, 0°, 90°. Draw eight intermediate 

 radii, all at equal distances of 10 degrees, and number them accordingly. 

 On each radius numbered as in column (1) of table, mark off the length 

 indicated in column (3) according to any convenient scale of units. For 

 instance, on the radius for 10 degrees mark ofi 173-6 mm., on the radius 



