977 



bourhood of a magnet, which agree with those of Poisson ; and 

 secondly, formulas for the resultant action experienced by any finite 

 portion of magnetized matter placed in a " field of force," either 

 given, or determined by the preceding formula from a specification 

 of the magnets to which it is due, by which the mathematical treat- 

 ment, according to one method, of the problem which forms the 

 subject of the chapter is completed. The chapter is concluded with 

 the statement of a method of expressing the mutual action between 

 two magnets by means of the differential coefficients of a function of 

 their relative position, which is of importance chiefly because the 

 principles on which it is founded lead to a new field of investiga- 

 tion in the theory of magnetism, having for subject the " mechanical 

 value of magnetic distributions," 



The fifth chapter contains, in the first place, explanations of the 

 principal properties of the peculiar distributions to which the author 

 has given the names solejioidal and lamellar. A solenoidal distri- 

 bution may be briefly defined as one of which the polarity, or the 

 representative imaginary magnetic matter is entirely superficial ; and 

 a lamellar distribution, as one of which the representative galvanism, 

 or the resultant equivalent electrical currents, are entirely superficial. 

 If a, /3, y be the components of the intensity of magnetization at any 

 point X, ?/, of a magnet, the condition that the distribution of mag- 

 netism may be solenoidal is expressed by the equation 



da ^djS dy__^ 

 dx dy dz ^ 



and, again, the condition that the distribution may be lamellar is 

 expressed by the three equations 



dz dy ^ dx dz ' dy dx 



In the concluding part of Chap. V. three new methods of ana» 

 lysing the action of a magnet, suggested by the consideration of these 

 special forms of distribution, and constituting, v/ith Poisson's method 

 mentioned above, a system of four expressions for the magnetic force 

 connected with one another by certain analogies, are given. One 

 of these (Poisson's) expresses the force at any point in terms of 

 double integrals for the surface (the components of the force due to 

 the superficial polarity) ; and triple integrals for the whole interior^ 



involving ^^ + 4^ + -7-^5 ^^^d vanishing when this vanishes, i. e, 

 dx dy dz 



when the distribution is solenoidal. The analogue of this expresses 

 the force in terms of double integrals for the suri'ace (the components 

 of the force due to the superficial representative galvanism) and 

 triple integrals for the whole interior, involving 



d[D dy dy da da d (3 

 dz dy dx dz dy dx 



and vanishing when these vanish, i. e. when the distribution is 

 lamellar. Of the two remaining methods, one is confined to solenoi- 



3 



