264 ON THE DETERMINATION OF THE TOTAL 



since Ar/ 3 = Arn. Accordingly, the second term of the preceding 



equation becomes R cos u Aj = - ; and we have 



&R = 



We learn, then, that the probable error of the force varies 

 inversely as the sine of the angle of deflection ; and that it is 

 therefore requisite for accuracy that this angle should be con- 

 siderable. There is no difficulty in augmenting the angle of 

 deflection as much as we please in the first part of the process, in 

 which the magnet is deflected by a weight ; but in the second the 

 case is different, and with the slender needles to be employed as 

 deflectors, a large deflection can only be obtained by placing the 

 deflecting needle at a very short distance from the moveable one. 

 The most convenient arrangement appears to be to attach the 

 deflecting needle to the moveable arm of the divided circle which 

 carries the verniers, and at right angles to the wires of the 

 microscopes.* So attached, it must always be rendered perpen- 

 dicular to the deflected needle in the course of the observation, 

 although in a different plane. 



The form of the function denoted by 77, 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 r', and let /u and // denote the quantities of free 

 magnetism at these points, contained in the slices perpendicular 



* To obtain the value of AS by observation, we must substitute for / its value 

 given above. But when ij = 6, or when the needle is undeflected, / = mltAO ; 

 wherefore 



V 2 sin u ' 



In the instrument with which I made trial of this method, the length of the needles 

 was 3J inches, and the angle of deflection produced, in the position of the deflecting 

 needle here described, was 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 l'-6 ; and if four readings (which is a very usual number) 

 e taken, the probable error of their mem will be one-half of this. In this case, 

 therefore, A0 = 0'-8 ; and the probable error of the deduced force, computed by the 

 preceding formula, is AS - -0004 Jt. 



