190 NOTES ON MAGNETISM. 



to one another. The quadrilateral CD EN' is a parallelogram, 

 EN' is equal to Z>(7, and consequently 



tan COD : tsmN'OE :: EO : OD :: 2 : 1. 



But COD = 6 and N'OE=^- 0, 



since CON' is a right angle. Consequently 



tan = 2 cot 0, 

 or tan0 = V2. 



The corresponding value of 3 cos 2 6 is therefore unity ; and 

 consequently we have, in this case, 



(3) = . Here D (and therefore E) coincides with 0, 



while ON' lies in the axis OS'. Consequently N r is at 0, and 

 the needle is therefore again perpendicular to the magnet. 

 In this case 



T MM' 

 : 



The first and third cases were noticed in the last number of the 

 Journal. In the second, the value of L is a mean proportional 

 between what is in the other two cases, in the last of which it 

 is a minimum maximorum. 



If we were required, for a given position of (7, to find the 

 position in which the needle would be in equilibrium, or the 

 moment L equal to zero, we might have recourse to Gauss's 

 construction already mentioned; for if the needle be placed 

 along the line in which the magnet tends to attract or repel (7, 

 as the dimensions of the needle are small, every element 

 would approximately be attracted or repelled along this line, 

 and therefore the total action would be destroyed by the 

 resistance of C. 



Thus there are always two directions for every position of C'; 

 one of maximum moment and the other of equilibrium : these 

 two directions are at right angles to one another. 



