268 proceedings: phii^osophicai^ socmTY 



of investigations will be found in treatises and papers by eminent 

 authors, but the derived expressions either stop at the gravitation 

 potential and intensity components, or but special cases of magnetiza- 

 tion are treated. Furthermore the published expressions for general 

 cases are often needlessly complex or they contain errors of one kind 

 or another which in some instances have been repeated by later authors. 

 Hence, the attempt was made to derive the desired expressions in the 

 simplest manner possible for practical application. Certain war prob- 

 lems gave added zest to this attempt. 



According to Poisson, who first solved the problem of induced mag- 

 netism in an elUpsoid placed in a uniform magnetic field, if V be the 

 gravitation potential at the point (x, y, z) of a body of uniform density 



W 

 p, then — — - is the magnetic potential of the same body umformly 



magnetized in the direction x with the intensity A = p. Similarly 

 with regard to any other direction of uniform magnetization. If the 

 imiform magnetization results from magnetic induction, the mag- 

 netizing field at all points in the interior of the body will be uniform. 

 So that if the external magnetizing field is uniform, the magnetic field 

 resulting from the magnetization will also be uniform for all points in 

 the interior of the body. 



The ellipsoid is the only body for which — is a linear function of the 



coordinates x, y, z in the interior, and V, accordingly, a quadratic 

 function of the coordinates. Poisson's method can, therefore, be ap- 

 plied to the case of the ellipsoid. 



Hence ii A, B and C be the intensities of magnetization parallel 

 to the three axes of the ellipsoid, and X', V and Z' the components of 

 .gravitational intensity due to a homogeneous ellipsoid of uniform den- 

 sity p = I, the magnetic potential due to the ellipsoid at any external 

 point as resulting from the induced magnetization will be 



V = AX' - BY' - CZ' (i) 



As defined by Thomson and Tait^ "an elliptic homoeoid is an in- 

 finitely thin shell bounded by two concentric similar elUpsoidal surfaces." 

 The total intensity produced by such a shell at points within the hollow 

 interior is zero, and at external points anywhere infinitely near the 

 homoeoid it is perpendicular to the surface, directed inward and equal 

 to 47rp/, where p is the constant density of the homogeneous mass and 

 t is the thickness of the shell at the point for which the intensity is 

 sought. 2 Since, furthermore, any two confocal homoeoids of equal 

 masses produce the same intensity at all points external to both, we 

 have in general that the total intensity produced by a homogeneous elliptic 

 homoeoid at an external point {x, y, z) is equal to 4irpt, p being the constant 

 density and t the thickness of the elliptic homoeoid at the point {%, y, z), 



^ Thomson and Tait's Natural Philosophy. Pt. 2: 43, footnote 2. 

 "^ Idem. Pt. 2: paragraphs 519-525. 



