516 Dr. F. W. Aston ami Mr. R. H. Fowler on 



§ 3. The linearity of the mass- scale. 



It was observed that in the most important part of the 

 plate the mass-scale was nearly linear — more precisely that 

 NF was proportional to m over a wide range. Equation (2) 

 enables us to explain this, and in fact to prove that such 

 linearity must always occur near (£ = 4#, which agrees exactly 

 with experience. For if we write (in/m > )i = z i we have 



NJB> 

 injm 



l + 2z tan W 



z 2 {2z~ (z 2 ~l) tan 26}' 



An approximately linear scale of the observed nature will 

 occur where 



dm \m/m J dz \m/?w I 



On differentiation and simplification we find that 



d, /NF//A _ 2(^ tan 26> - 1) {(3£ 2 -l) tan 2<9^(> 2 - 3)} 

 dz\mlm-J ~ z z \2z- (* 2 -l) tan 2d} 2 



which vanishes when 1/z —tan 26, i. e. when 

 tan J<£= tan 26. 



Thus the mass-scale Mill be approximately linear near $ = 4:6. 

 Actual numerical calculation by (2) shows that the approxi- 

 mation to linearity should be (as was observed) very close. 

 For the actual apparatus 6 = j\ radian, tan 2$ = 0*168. 



ISTFfe 

 Values of — -~- in arbitrary units are given in the following 



table : — 



m/m . 



NF/p 



m/m - 



139 



m/m . 



~NF/p 

 on/ m Q ' 



44 



32 



138 



42 



138 



30 



138 



40 



138 



28 



139 



38 



137 



26 



140 



36 



137 



24 



141 



34 



137 



22 



143 



The value of m/m corresponding to <£ = 40 is given by 

 m /m=tan 2 i0=tan 2 2(9 = 0*0272, 

 m/m = 36'7. 



