EQUATIONS OF CONDITION. CCX111 



9. EQUATIONS OF CONDITION. 



We have now the means of forming our equations of condition for each series, as proposed in 

 3 ; and the coefficients of the unknown quantities in each equation of the several sets, being 

 computed as there described and arranged in tabular form, are given in the present section ; 

 the equations being numbered for convenience of reference to the observations from which they 

 are respectively derived. The values of a and d being always unity, it is of course unnecessary 

 to include them in the tables, for no confusion or embarrassment can arise on account of their 

 signs, inasmuch as a is always positive, and d has the same sign as 'e. The quantity s denotes, 

 as usual, the sum of all the coefficients, andj> the weight, computed as hereafter to be explained. 

 The unknown quantity v, which is the correction to the adopted value of a revolution of the 

 micrometer-screw, has only been introduced in the equations derived from observations at 

 Santiago and the Cape, these being the only ones, excepting the Washington series, in which 

 the number of comparisons and fullness of detail render the determination practicable with 

 sufficient accuracy. At Washington, there were a number of declination-threads upon the 

 diaphragm carried by the micrometer-screw, so that the interval actually traversed in measuring 

 a difference of declination was in every case small. 



To the groups of equations of condition, as given by the comparison of each observation with 

 the place of the planet derived from computation, are to be added still other series resulting 

 from the comparison of the measurements of the respective limbs, and capable of serving to 

 some extent as aids toward freeing our equations from the influence of the two unknown 

 quantities which affect the apparent diameters. These unknown quantities, which have been 

 denoted by the letters t and u, measure, respectively, the error of the normal semidiameter 

 added to such irradiation as may be inversely proportional to the distance, and that portion 

 of the irradiation which is peculiar to the observer and the instrument, with which last is 

 inseparably merged the personal equation of a limb-pointing and any error in the assumed 

 thickness of the threads. 



Each measurement of a diameter furnishes an equation containing these two unknown 

 quantities t and u ; and, iu addition to the direct measurements, each of those observations, 

 which consist of comparisons of the limbs of the planet, affords a measurement of the diameter. 

 We thus obtain a large number of additional equations which may be directly incorporated with 

 the others. They comprise two classes : the one consisting of direct measurements of diameters, 

 for which the differential refraction is utterly insensible; and the other, which includes the great 

 majority of cases, being affected by the motion of the planet during the interval, often very con- 

 siderable, between the means of these poin lings for the different limbs, and also by the change 

 of the refraction during the same interval. 



To the regular equations of condition are appended these additional ones, arranged like the 

 others in tabular form, the two classes being separately given. 



To form the equations of condition for these last mentioned cases, let us retain the former 

 notation,* affixing one or two accents to the symbols, according as they refer to the first or second 

 observation. 



For the first observed limb, we shall then ha,ve 



0= d.' -, Po k*. 3' + x + y (i'T) + 1 (f T)* (t\ + 2) 2? (V f *.) + Jm'.f W* 



and for the second 



0=5." =F ^77 />. *"-. ff' + * + y (t" T) + z (t" T 7 ) 2 =F (t n + g) =F 33 (V + ) + Jm".itV'i* 



To avoid unnecessary complication, the factors 20 and 1000, which are combined with y and z in the numerical solution, 

 are here disregarded. 



