916 

 6abc 2abc 



+ S (2) 



where S represents the series i^' of (1). 



Formula (l) evidentlj' holds good tbr O <^ 2 <^ c. 



From the potential F determined in this way we can find 

 as above the potential energy of a dipole with the components 

 p'x, p'>/, p'z, when the latter is placed in a point (x, y, z) of the field 

 caused by dipoles p_„ py, p^ in the corners of the lattice. 



The expressions 



/ d*V d*V \ 



-y'^p''d^^-"-^^''^p'^^'"^-'^d^y-^---) 



will represent this energy. 



From (2) follows for the derivatives of second order occurring 

 in this expression 



(3) 



3. In order to examine generally the stability of the system 

 described in (2), w'e mnst study the behaviour of the qnadric form, 

 representing the potential energy as function of the variables deter- 

 mining the direction of all dipoles. The diflicnity of this pioblem 

 does not lie so much in the great number of variables, as in the 

 impossibility to form a single series, in which the variables relating 

 to neighbouring magnets, follow each other closely. 



This difficulty does not present itself in the case when there are 

 dipoles placed on one line at mutually equal distances. There the 

 stability may easily be examined in the well-known fashion with 

 the help of a determinant. We shall mention a few of the results, 

 as they may guide us in the case that has our attention. If all 

 magnets are directed by a field parallel to the line, the system is 

 still stable if the field is abolished. If we apply a slowly increasing 

 field, contrary to the magnetisation, there may be indicated a definite 

 group of small deviations of the dipoles, for which the system first 

 becomes unstable. These displacements are such that the magnets 

 lie in one plane and alternately will make angles -|- (f and — r/) 

 with the direction of the field. The coercitive force found for this 

 displacement of the magnets is only one third of the force found 

 from the supposition, that all magnets turn parallelly. 



