CCJCIV EQUATIONS OP CONDITION. 



in which equations the signs are respectively correspondent. Thus, if the time if belong to a 

 comparison or mean of comparisons of the north limb, and the npper sign is consequently to 

 be taken, the upper sign will also hold in the other equation, which will belong to a comparison 

 of the south limb; and so in the reverse case. 



The sum and difference of the foregoing equations give, after halving 



+ - } { (t" - Ty - (f- T? I T i r, + -, (V + O + i (^"- ^') A* - i (*"-*0 >* 



Let us now, slightly varying the notation of 3, denote the middle time (<" + f) by t, and 

 the half-interval ^ (<" <') by r, and consider the unaccented symbols as pertaining to the instant t, 

 We shall then have,* omitting all terms of the third and higher powers 



3" = d,+ r A*. + i r A 9 *. <*.' = 5. - r A*. + i ^A 2 *. 



F = k + rZ>,i -f $ 



Jm" = (<*" <*) = (3 d) -f (J" ff) Jm' = (ffd) = (d d) 



(t"TJ t =(t 2 T )a-}-2r(< rj-r-T 2 (' Z T )=(<' J 7 ) 2 2r(< 



The substitution of these expressions in the preceding equations gives us 



0= T ft- 



in both which formulas the upper sign is to be taken if the north limb was first observed, and 

 the lower sign if the observation of the south limb preceded. 



By reason of the smallness of r, all consideration of many of the terms may be dispensed 

 with. Not only do all the terms of the second order become negligible, but the quantities 



r D t ^ \ p t -\- (}p + *o) ( iu the first, and T D,fc. <5 in the second equation are likewise inap- 



preciable. And we shall moreover find that y , the daily change of the correction to the tabular 

 declination, as also z (t T) which is of about the same order, are so small that their products 

 with r are never sensible 



The only terms of the first order remaining in the first equation and containing r will then 



be T D, p,= rD,^ or the variation of tabular semidiameter during the interval r. Re- 

 curring to the ephemerides, we find the maximum amount of this variation in one day to be 



0".06 for Mars I. 0".43 for Venus I. 



.05 for Mars II. .38 for Venus II. 



The terra Jm U strictly = I rf/n, the quantity f m denoting the distance of the thread used from the standard or 

 MTO thread of the movable diaphragm. But this coefficient U only employed in the present discussion, for instrnmenti 

 provided with a single movable thread. 



