540 The Eev. Dr. Llotd on the Determination of 



deflected needle in the course of the observation, although in a different 

 plane. 



The form of the function denoted by U, in this position, is easily found. 



Let the distances of any points of the axes of the deflecting and deflected 

 magnets from their respective centres be denoted by r and ?'', and let fi and fx 

 denote the quantities of free magnetism at these points, contained in the slices 

 perpendicular to the axes whose thicknesses are dr and dr'. Then the force 

 exerted by the former upon the latter is 



HfjJdr dr' 



? ' 



z denoting their mutual distance. The portion of this force contained in the 

 plane of the deflected magnet, and perpendicular to its axis, is 



fiix'dr dr' r 



z- z 



and the moment of this force to turn the magnet is obtained by multiplying by 

 r'. But 



D being the distance of the centres of the two magnets ; and accordingly the 

 total moment of the acting forces is 



[ivdr . \xr'dr' 



Expanding the denominator, and making, for abridgment, 



[lirdr =il/„ \iJ.rHr =M^, jfxr'dr ^ M^, &c., 



\fi.'r'dr' = M\, ln'r"dr'^M'„ \^'r"dr' = M'„ &c., 



in which the integrals are to be taken between the limits r = ±l,r' = ±l',l and I' 

 being half the lengths of the two magnets, this becomes 



24° 10'. But the probable error of a single reading of the inclination, obtained by repetition — the 

 needle being lifted off the agate planes between the successive readings — was \'-Q ; and \( four read- 

 ings (which is a very usual number) be taken, the probable error of their mean will be one-half of 

 this. In this case, therefore, A& = 0'-8; and the probable error of the deduced force, computed by 

 the preceding formula, is AiJ = -00040 R. 



ff( 



