192 C. J. MERFIELD. 



the complete corrections to the elements would then take the form 

 (da)) 2 = d(a + A' (de) 2 

 (d&) 2 = d& + B' (de) z etc., etc. 

 upon introducing the values of v into equation (2) then it would 

 be found that (de) 9 = 10~ 6 x 0-58 



This correction having been made, and the values of (d(o) 2 (d&) s 

 etc. computed, it was found that the residuals were practically 

 the same, the resulting value of [vv] being 109*70. 



Probable Errors etc. — The probable errors of the quantities c?o>, 

 d&, etc., have been obtained from the following formulae 



= 0-6745 |M 

 V m- fi 



r r 



r x = /= . r = 7==- etc. 



vPx yp 7 



in which m equals the number of equations of condition and /x the 

 number of unknowns. 



For Elements IV. it will be found that 

 Log. r = 0-34593. 



In obtaining the values of p x p y etc., the order of elimination 

 has been reversed, so that these quantities were deduced by the 

 usual formulae. 



Concluding Eemarks. — The reversal of the order of elimination 

 of the unknowns from the normal equations acted as a check upon 

 this part of the calculation, a further check being obtained by 

 placing the deduced values of the unknowns in the normal equa- 

 tions, the checks in both cases being completely satisfied. In 

 reducing the preliminary equations, the usual controls were 

 maintained, the several sums being fully verified. The residuals 

 of the several elements being compared, it will be observed, that 

 assuming no correction is required to the eccentricity, then the 

 sum of the squares of the residuals has been reduced from 10 4 x 15*5 

 approximately, to 10 4 x 0*4 ; taking de into account reduces this 

 sum to 10 4 x 0-010812. 



The observations appear to be well represented by the Elements 

 IV., and there appears to be every indication that the orbit of 

 this comet is an hyperbola. i^s previously mentioned, the 

 purturbations produced by the earth will be small, and should the 

 definitive discussion of the orbit elements be undertaken, it will be 

 found that the Elements IV. will require no material correction. 



