some Problems of the Mass-Spectrograph. 517 



§ 4. Practical use of the mass-scale. 



Let us denote the distance of an observed image from the 

 fiducial spot by D. Then D and NF differ by some constant 

 k — about 5*4 cm. in the existing spectrograph. Equation (2) 

 shows that in all cases the relation between D and m has the 

 form 



D=/(m/m ) (3) 



where /is a function in which all the coefficients, p, k, and 

 tan 20 are geometrical constants ; the fields affect m and 

 m alone. In the actual apparatus, which was not quite 

 rigid, p, k, and 26 probably altered slightly each time 

 the apparatus was assembled, but otherwise were strictly 

 constant. 



I he only assumption made in the actual analysis of the 

 plates was the following : — If T) 1 and D 2 are the distances 

 from the fiducial spot of any two points on the plate and m 1 

 and m 2 the corresponding masses, for given values of D 1 and 

 V2 the ratio m 1 /m 2 will be the same in every photograph taken 

 with the same setting up of the apparatus. This is a direct 

 consequence of (3). For so long as the apparatus be not 

 subjected to a new set of stresses, the function / will remain 

 unaltered. In any one photograph we have 



D 1 =f(m 1 /m ), D 2 =f(m 2 /m ); 



and in any other (with different fields) 



Pi=/(wh7W)j ~D 2 =f{m 2 '/m '). 



But if mj = a.m it follows at once that we must have m^ = am,, 

 m 2 ' = am 2 , and therefore in all cases 



mi/m 2 ' = mi/m 2 , (4) 



which is the hypothesis mentioned. The argument is more 

 general than equation (2), as an equation of the form (3) 

 must hold even when actual fields are considered instead of 

 the idealized fields of § 2. The practical accuracy of the 

 theoretical relations (3) and (4) was amply verified by the fact 

 that the construction of a consistent calibration curve — that 

 is, the evaluation of the function/ — was possible for all photo- 

 graphs taken with one setting up of the apparatus. Owing 

 to the fact that/ is very nearly linear, the calibration curve 

 results in applying only a small correction to the observed D 

 to make it proportional to m. The certainty of the results is 

 therefore greatly increased. 



