482 Lord Rayleigh's Investigations in Optics. 



tions of it to be obtained by varying the focus. At the pri- 

 mary and secondary foci the point is represented by perpen- 

 dicular lines of small width, and at a particular intermediate 

 position by a circle of light, called the circle of least confusion. 

 In most cases the last representation would be the best; but 

 if the object under examination be itself a uniformly lumi- 

 nous line parallel to one or other of the focal lines, the best 

 result will evidently be obtained at the corresponding focus. 

 Under these circumstances the image is not prejudiced by the 

 astigmatism, and its perfection depends upon the amount of 

 aberration. In the case of a properly adjusted spectroscope 

 the slit is parallel to the edges of the prisms, and the spectrum 

 is seen with best definition at the primary focus. 



The aberration that we have now to consider is of higher 

 order than that which affects symmetrical pencils, and there- 

 fore, when it occurs, is presumably of greater importance. 

 Before calculating its amount in particular cases, it will be 

 convenient to consider the general character of the effects pro- 

 duced by it. The axis of the pencil being taken as axis of z, 

 let the equation of the wave-surface, to which all rays are 

 normal, be 



z=^ + ^~ / +xx* + j3x; 2 y + yxf + S/ + (1) 



The principal focal lengths are p and p r . In the case of sym- 

 metry p and p' are equal, and the coefficients of the terms of 

 the third order vanish. The aberration then depends upon 

 terms of the fourth order ; and even these are made to vanish 

 in the formulae for the object-glasses of telescopes by the selec- 

 tion of suitable curvatures. If this be effected, the outstand- 

 ing aberration will be of the sixth order ; whereas in the case 

 of unsym metrical pencils, even if we should succeed in de- 

 stroying the terms of the third order, there will still remain an 

 aberration of the fourth order. It follows that every effort 

 should be made to retain symmetry about the axis ; but in the 

 case of the spectroscope this is usually impossible. If we 

 could secure a perfect parallelism of the incident light, and 

 perfectly flat faces for our prisms, we should indeed get rid of 

 aberration, and at the same time render ourselves independent 

 of the adjustments of the spectroscope ; for it is evident that 

 no repetition of refractions at plane surfaces, however situated, 

 could disturb the original parallelism of the light. The fact 

 that most large spectroscopes are more or less sensitive to 

 maladjustment of the prisms proves either that the faces are 

 not flat, or that it is difficult to obtain a sufficiently accurate 

 adjustment of the collimator. We shall suppose that the faces 



