652 Mb. stokes, ON THE FORMATION OF THE CENTRAL SPOT 



If we determine in succession the angles 6, ^, rj from the equations cot = /u, tan ^= q, tan ij = 

 sin 2 6 tan 2 T, we have p^^ = 1 - p^ = -1^ versin 2 r;. The expression for the intensity may be adapted 

 to numerical computation in the same way for any angle of incidence, except that or must be 

 determined by (2) or (6) instead of (11), and q by (5) instead of (12). 



18. When light is incident at the critical angle, which I shall denote by 7, the expression for 

 the intensity takes the form - . Putting for shortness y'^u^sin'^i - 1 = w, we have ultimately 



= 1 w, tan = = : = / „ , (b = fi i 



\ /i cos J v /" - 1 



and we get in the limit 



O"- =—7771 T- (13). 



according as the light is polarized in or perpendicularly to the plane of incidence. The same for- 

 mulae may be obtained from the expression given at page 304 of Airy's Tract, which gives the inten- 

 sity when i < 7, and which like (4) takes the form — when i becomes equal to 7, in which case 

 e becomes equal to — 1. 



19. When i becomes equal to 7, the infinite series of Art. 11 ceases to be convergent : in fact, its 

 several terms become ultimately equal to each other, while at the same time the coefficient by which 

 the series is multiplied vanishes, so that the whole takes the form x co . The same remark applies 

 to the series at page .TO3 of Airy's Tract. If we had included the coefficient in each term of the 

 series, we' should have got series which ceased to be convergent at the same time that their several 

 terms vanished. Now the sum of such a series may depend altogether on the point of view in which 

 it is regarded as a limit. Take for example the convergent infinite series 



2xsmy 

 f(.v, y) = .1? sin y + lie^sin 3y + 1 a: sm 5y + ... = 1 tan , , 



where « is less than 1, and may be supposed positive. When x becomes 1 and y vanishes 

 f(x,y) becomes indeterminate, and its limiting value depends altogether upon the order in which we 

 suppose X and y to receive their limiting values, or more generally upon the arbitrary relation which 

 we conceive imposed upon the otherwise independent variables jc and y as they approach their limit- 

 ing values together. Thus, if we suppose y first to vanish, and then x to become 1, we have 



TT 



/(.r,y) = 0; but if we suppose x first to become 1, and then y to vanish, /(.r,y) becomes ± -, +or- 



according as y vanishes positively or negatively. Hence in the case of such a series a mode of 

 approximating to the value of te or y, which in general was pefectly legitimate, might become inad- 

 missible in the extreme case in which ir = l, or nearly =1. Consequently, in the case of Newton's 

 Rino-s when j ~ 7 is extremely small, it is no longer safe to neglect the defect of parallelism of the 

 surfaces. Nevertheless, inasmuch as the expression (4), which applies to the case in which i>y, 

 and the ordinary expression which applies when i<y, alter continuously as i alters, and agree with (13) 

 when i = 7, we may employ the latter expression in so far as the phenomenon to be explained alters 

 continuously as i alters. Consequently we may apply the expression (13) to the central spot when 

 i = 7, or nearly =7, at least if we do not push the expression beyond values of D corresponding to 

 the limits of the central spot as seen at other angles of incidence. To explain however the precise 



