an instrument for measuring the Density of the Air. 85 



its appendages hang, their weight being w' ; let g be the 

 centre of gravity of the beam at which its weight w acts; let 

 cb = r ! , cg=r. Drawing the straight lines bed and eg, let the 

 angle dcg = oc; also drawing the horizontal line McN, let the 

 angle Med (= angle Nc&) = 0. 

 Then in equilibrium, 



wxcM=w' xcN 9 

 or 



w xr cos (a + 0) =tt/ x r 1 cos 0, 



/ w'.r' \ 



.*. tan = cot a I 1 — I ; 



V w.r.cosu/ 



let 



where # will be a small fraction, positive or negative ; we have 



- 



1 



— 2sin s 



2 



-(i + 



*) 





2 sin 



2 



a 

 .cos - 







— 



tan-. 



s 



as 







in a 







X 



= -2 sin 2 



2 ; 





Here = when 



also for a given value + of x, changes very rapidly for cor- 

 responding changes in the value of a when this latter is very 

 small : or the sensibility increases very quickly when the 

 points b, c 9 and g are brought very nearly into one straight 

 line. 



Again, let w and w' be the apparent weights in air, and let 

 w 1 and w 2 be the absolute weights; so that w=w 1 (l—m) 9 

 w' = w^(l—n), where by the laws of hydrostatics mw l is the 

 weight of the air displaced by the beam supposed of homo- 

 geneous materials from which it differs very slightly, and nw 2 

 that displaced by the globe and appendages; 



specific gravity of air a 



or m= ^ s-j- — ^-t—t = T say; 



specific gravity or the beam b 



specific gravity of air a 



2i __ l p ~ — say» 



specific gravity of globe and appendages c 



