214 The Rev. Dr. Lloyd on determining the Total Intensity 



When the plane of motion of the needle coincides with the 

 magnetic meridian, or « = 0, this becomes 



Rsin(0->/)=™U; (5.) 



which gives the ratio of the earth's magnetic force to the mag- 

 netic moment of the needle, when U is known, and the angles 

 and */ given by observation. The coefficients p and q, in 

 the value of U, may be obtained (as in the ordinary method) 

 by observing the angles of deflection, 0— */> at different di- 

 stances ; it is probable, however, that their values may be 

 inferred, a priori, from the lengths of the needles, with as 

 much accuracy as is attainable in observations of this nature. 

 When the plane of motion is perpendicular to the magnetic 

 meridian, or a. = 90°, 



Y cosri'=zmU ; (6.) 



which gives, in like manner, the ratio of the vertical compo- 

 nent to the magnetic moment of the needle. 



The total force is determined absolutely by means of the 

 two observations in the plane of the meridian : for, multiply- 

 ing the equations (2.) (5.), m disappears, and we have 



WrU 

 Rg= wru 



sin u sin u' 

 in which the angles of deflection, Q— >j, — >j', are denoted for 

 abridgement by u and u'. Again, dividing the former of these 

 equations by the latter, 



2 Wr sin u' ' 

 U sinu 



The equations (3.) (6.) furnish, in like manner, a similar 

 value of the vertical component of the force. 



In order to determine the probable error in the resulting 

 value of the force, arising from the errors of the observed 

 angles, u and «', we have to observe that the moveable needle 

 is acted on, in each case, by two forces, one of which is the 

 moment of the earth's magnetic force wR sin u, while the other 

 is constant. Hence, in any position, the directive force is 



F=wRsin u— G. 



Let u denote the value of u, corresponding to F=0, or to the 

 case of equilibrium ; then /raRsintt =G, and 



F=mR(sinw— sinz/ ). 

 Let 11 = ^ + Au , Au being a small angle, — or, in other words, 

 let the needle be displaced by a small amount from the posi- 

 tion of equilibrium, — and let the force brought into play by 

 the displacement be just balanced by friction ; then 

 /"= mR cos UqAuq, 



