meridian. This position, I soon perceived, was less advanta- 

 geous than that described above ; and it was accordingly aban- 

 doned in all the instruments subsequently constructed for other 

 observatories. In the Dublin Observatory, however, I pre- 

 ferred the continuance of the less perfect arrangement to the 

 interruption of the series of observations consequent upon its 

 alteration ; and I have been thus compelled to re-discuss the 

 theory of the instrument, which is rendered much more com- 

 plicated in this form. 



" It was natural to suppose that the moment of the force 

 exerted by the iron bar upon the suspended magnet was of the 

 form M U cos u, U and u denoting as before ; so that the 

 equation of equilibrium should be, simply, 

 U = X tan u. 



Differentiating, and dividing by Y = X tan 9, and denoting (as 

 before) the change of U produced by a small change of the 

 vertical component of the earth's force by VS Y, we have 



gr ( 2 s , sx\ 



-==r =p[ sec- u ou + tan u -=— I , 



in which p = V' 1 cotan 6. 



"This formula, however, is only an approximate one. The 

 expression for the moment of the force exerted by the iron bar 

 upon the suspended magnet, in this case, is not, strictly, of the 

 form supposed above ; and it becomes necessary to seek the 

 exact form, and to examine in what manner the formula of re- 

 duction is modified. 



' ' Let CP be the freely suspended 

 horizontal magnet ; C'F the verti- 

 cal iron bar ; and O the point in 

 which it intersects the horizontal 

 plane passing through CP. Also, 

 let dm and dm denote the elements 

 of free magnetism at P and P', and q I 

 p the distance PP. Then the 

 force exerted by dm on dm is - 



