OF MAGNETIC DECLINATION. 



35^ 



from which, if e x be the value of the error in the observed declination expressed in 

 seconds of arc, we have 



_ sin a cos ^ (B 1 + /3 3 ) sin \ ((3 1 — /3 3 ) 

 1 sin 1" cos J (d t — 8 3 ) 



2dly, If errors occur in the values of 4- alone. 



Let ^ = 90° - ^ , 4 3 = 90° - £ 3 



Then supposing /? to have its correct value 



sin a sin /3 



sin X = 



cos ^ 



. , sin a sin /3 



sin d>„ = =— L 



cos l z 



and putting e 2 for the error in seconds in the value of the observed declination, 



€ = sin a sin (3 sin j (^ + g„) sin fr (^ - g 8 ) . 

 sin 1" cos £j cos £ 3 cos J ($ x — $ 3 ) 



Finally, if e be the error in declination arising from simultaneous errors in the 

 values of (3 and 4, since the whole error produced by imperfect inversion will 

 generally be small, we shall have, with sufficient accuracy, 



e = e L + <f 2 



14. To give some idea of the extent of the errors which may be expected to arise 

 from inaccurate inversion, I will assume the case of a magnet which, when accu- 

 rately inverted, gives a difference of 20' in the theodolite readings; and also, that 

 in the erect position, 4=90° and (3=45°. The following table shows the errors 

 in declination due to imperfect inversion of such a magnet, corresponding to the 

 tabulated errors 7 and £ in the values of ft and 4", respectively, after inversion, 

 these errors being supposed to occur separately. 



7 



2° 



4° 



6° 



8° 



10° 



6 1 



-10"-3 



-20"-2 



-29"-7 



-38"-8 



-47"-5 



I 



2° 



4° 



6° 



8° 



10° 



f 2 



-0"-2 



-0"-7 



-l"-7 



-2"- 9 



-4"-6 



It thus appears that errors in (3 affect the accuracy of observations of declina- 

 tion much more than those in 4, a result which might have been anticipated ; 

 and also, that an error in (3 of only 2° would, in the assumed case, cause an 

 error in the observed declination exceeding 10", — a quantity quite appreciable 

 in such observations. 



VOL. XXT. PART II. 5 D 



