OF ARTS AND SCIENCES. 



195 



lab] z=n Z sin (s ^ d) cos (s^^ 6)=0 



[hb] =n' Z cos2 (b^ d)=zl m ri" 



[an'] = 2 sin (£ =F ^) i i -S [sin' (e =F ^) + « cos^ (s ^ ^)] — 



M = n 2 cos (e :f ^) U i2 [sin2 (e q: e) ] + n cos2 (£ qp (9)] — 



2~^ 



As the value of [ab] comes out zero, we now evidently have 



[hn] 



^ [aa] 



8x = 



m 



U60) 



(61) 



If m is an even number, there will be ^ m pairs of values of the angle 

 £ ^ ^, and in every case the two angles which compose the pair will 

 differ from each other by 180°. If R and H are the radii vectores of 

 the two points composing a pair, and if the summation is extended only 

 through ^ m points, then putting 



RR' 



n = l(Ii-E')[l + —+(n- 1) cos-^ (e T 6)] (62) 

 we have 



2 ^T«=0 

 dy = — Z sin (s ^: d) JD 



e=f d=Tr 



8x=—Zcos(s^:d)D 



€^ e=ir 



(63) 



To simplify these expressions still further, we remark that n is always 

 very near unity, not becoming so small as 0.950 until a zenith distance 

 of 85'' is reached. Assuming n = 1 is equivalent to supposing the 

 apparent sun to be replaced by an artificial one of perfectly circular 

 outline, whose area is the same as, and whose centre coincides with the 

 centre of gravity of, the apparent sun. As this assumption does not 

 affect the values of di/ and 8x, we adopt it ; and then equation (62) 

 reduces to 



D=(R — B') 



(64) 



