in Theory and Practice. 409 



The first term gives the principal spectra as due to a grating- 

 space of — and number of lines nm as if the grating were 



perfect. The last term gives entirely new spectra due to the 

 grating-space a, and with lines of breadth due to a grating of 

 n lines and intensities equal to (b/ubv) 2 . 



Hence, when the tangent- screw is used on my machine for 

 14,436 lines to the inch, there will still be present weak 

 spectra due to the 14,436 spacing, although I should rule, 

 say, 400 lines to the millim. This I have practically observed 

 also. 



The same law holds as before that the relative intensity in 

 these subsidiary spectra varies as the square of the order of 

 the spectrum and the square of the deviation of the line or 

 lines from their true position. 



So sensitive is a dividing-engine to periodic disturbances, 

 that all the belts driving the machine must never revolve in 

 periods containing an aliquot number of lines of the grating ; 

 otherwise they are sure to make spectra due to their period. 



As a particular case of this section we have also to consider 



Periodic Errors of Ruling. — Theory of " Ghosts." 



In all dividing-engines the errors are apt to be periodic, 

 due to "drunken" screws, eccentric heads, imperfect bearings, 

 or other causes. We can then write 



y = n a + a L sin (^w) + a 2 sin (e 2 n), + &c. 



The quantities e u e 2 , &c. give the periods, and a lf a 2 , &c. 

 the amplitudes of the errors. We can then divide the integral 

 into two parts as before, an integral over the groove and 

 spaces and a summation with respect to the numbers. 



Cv" Cy"—y' 



J y' J 



It is possible to perform these operations exactly ; but it is 

 less complicated to make an approximation, and take y n — if = a, 

 a constant as it is very nearly in all gratings. Indeed the 

 error introduced is vanishingly small. The integral which 

 depends on the shape of the groove will then go outside the 

 summation sign, and we have to perform the summation 



Vg— ibfi ia n n+ a { sin e,» +a. 2 sm e 2 n+ &c. J- 



