Retardations, and on the Theory of Foucault's Test. 175 

 take the limits to be expressed as +f, and (31) becomes 



+ 2 cos T sin , J " Si " «- rin <« + »*+ S! " (g ^ g ^- (32) 



I£ we apply this to f =oo to find what occurs when there is 

 no screening, we fall upon ambiguities, for (32) becomes 



2 sin T cos p {i7r±i7r} + 2 cos T sin p {2Si(*fl -^77 + ^}, 



the alternatives following the sign of — <f>, with exclusion 

 of the case <£ = #. If </> is finite, 2Si(<££) may be equated 

 to 7r, and we get 



I = 4tt 2 (1 or 0), 



according as — (j> is positive or negative. But if <j> — 

 absolutely, Si(<£f?) disappears, however great f may be; and 

 when cj) is small, 



I = 47r 2 cos 2 p + 4sinV{2Si(<£f)}^ 



in which the value of the second term is uncertain, unless 

 indeed siup = 0. 



It would seem that the difficulty depends upon the assumed 

 discontinuity of It when — 0. If the limits for be +a 

 (up to the present written as +0), what we have to consider 

 is 



f>C 



dd 



sin{T-R + (0 + *)f}], 



in which hitherto we have taken first the integration with 

 respect to 0. We propose now to take first the integration 

 with respect to £, introducing the factor e ±fL * to ensure con- 

 vergency. We get 



2 sin (T-B)£ rrt« (tf + *)f . <$- ^ffffl - (33) 



There remains the integration with respect to 0, of which 

 It is supposed to be a continuous function. As /jl tends to 

 vanish, the only values of which contribute are confined 

 more and more to the neighbourhood of — <£, so that 

 ultimately we may suppose to have this value in It. And 



J 



+ »_ ^e ^ Un - 1 ±±?_ Ua - 1 <t 



which is 7r, if cj> lies between +a, and if <f> lies outside 

 these limits, when fi is made vanishing small. The intensity 



