Dr. G. J. Stoney on Microscopic Vision. 337 



constituent disturbances are resolvable. The number of 

 these undulations may be reduced wherever any of them 

 travel in the same direction, since any number of undulations 

 of plane waves of wave-length \ travelling in the same direction 

 may be combined into a single undulation of plane waves 

 travelling in that direction. Hence the total disturbance is 

 resolvable into undulations of uniform plane waves, only one 

 of which for each value of X travels in each direction. 



7. This valuable optical theorem bears a remarkably close 

 analogy to Fourier's Theorem for the expansion of an immense 

 class of functions. Thus by Fourier's Theorem a portion of 



curve mn along with equidistant repetitions of the same to 

 the left and right may be expanded in the form 



^Ao + AiCos f-A 2 cos2 f- . . . 



Qj (X 



-_ . 2lTiV -r, . „ 2tTX 



-f JBi sin 1- J3o sm 2 f- . . . 



a a 



in which the values of the constants A , A 1? A 2 , &c, B b B 2 , 

 &c, depend on what direction has been selecied for the line 

 over which the repetitions are to be disposed, and on what 

 interval has been chosen for a (a being mm f , the spacing of 

 the curves from one another) . So in our optical theorem, the 

 plane waves into which the light emitted by a point p in the 

 objective field is to be resolved will depend on what plane has 

 been chosen for the objective plane, and on the intervals at 

 which p, p r , p" , &c, are to occur in that plane, as well as on 

 whether the lines joining them lie (as we have placed them 

 above) at right angles to one another, or in other available 

 positions. However, just as in a Fourier's expansion the 

 original curve is always correctly represented whatever 

 assumption w T e may have made as regards the orientation of 

 the axis of x and the length of the line a, and it is only the 

 situation of its replicas which is affected by this choice ; so 

 under our theorem the light in front of the objective field is 

 always adequately resolved whatever selection we may have 

 made as regards the optional matters (provided the conditions 

 laid down in the footnote on p. 336 are observed), and it is 

 only where its replicas are to be regarded as situated that is 

 affected by that choice. Moreover, when once we have made 

 Phil. Mag. S. 5. Vol. 42. No. 257. Oct. 1896. 2 B 



