on i he Mariner's Compass. 449 



■^Urftclions will be as mi and hk = {im')', which, putting the 

 angle N O C equal «, is as cos a and sin a. In the same way it 

 appears that the dift'erence of the attractions pv, pvf is as cos a 

 and — sin a : the attractions therefore perpendicular to N S 

 destroy each other; whilst in the direction of that line tiiey are 

 equal to a constant quantity ; and as NS is the direction oV the 

 dip, it h manifest that this distrihution of the fluid is agreeable 

 *o the law that we have laid down. 



Let cf; be the angle which the conjugate diameter of any j)oint 

 in the surface makes with the meridian ; t the thickness "of the 

 -shell at the ^ivcn point ; and T the polar thickness. Then from 

 the foregoing denion-rtration we readily derive the following ana- 



. Jytical expression for the tliickness of'the shell at any point of 



.. tlie surface, viz. / = T. sin ^. 



It would be forei^jn to our present purpose to enter snv further 

 into the investigation of particular cases; all that we have pro- 

 posed to iuvestigats may be derived either from what has pre- 

 ceded, or from tke following general theorem, which it will be 

 our next object to demonstrate. 



Ill any maiS (./iron, whether regular or irreguhir, ift , t , t 



le the respective thicknesses rf the mugvetic fluid at any point, 

 when the dip is direcud according to three rectangji/ar co-ordi- 

 nates a , a^, a^; then the thickness t at the same point, when the 



dip is in a direction that makes with a , a , a, the angles a , a , and 



? ., will le expressed ly the formula 



t — t cos a 4- / cos a -{- 1 cos a . 



The truth of this theoreu) may be readily established; for, con- 

 ceiving the force in the direction of the dip to be unitv, the forces 

 in the directions a, a^ and o^ will be respectively cos a ', cos a , an<l 



cos a. But it is manifest that the thickness at any point is, ceteris 

 paribus, as the disturbing force; and it therefore follows, that if 

 the force in the direction a were alone concerned, the thickness 

 at the given point would be /_ c&s a^ : moreover, the fluici .Jij,r,ji- 



buted according to this law would produce an attraction in the 

 direction a^^ equal to cos a ^ ; whilst at right angles to that di- 

 rection its effect v.'ould become zero. 



Hence, as a similar observation applies to the distributions of the 

 fluids by the hclions of the forces cos a^ and cos a , we may infer 

 that if these 'ttirfi;. developments had place at the same time, the 

 ertects they woiihi. produce in the directions a , a and a woul<l 



be respectively cos o , cm a^ and cos a. But the forces' cos a ^ 

 Vol. 55. No. 2f)G. Jitnr 1^20. ' T t ..os 



