238 MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



Represent now the perpendiculars by p, and p^^, and then, by the common formula 

 for the inclination of the tangent to the radius-vector, we have 



Hence 



or 



^ - _ ^^'^^/ -A - ImtI 



or again, since ^ = -^^', it becomes finally 



F^ sin d,d0,-{- F,i sin ^^ d d„ = 0, 



the integral of which, as before, is 



F^ cos 0, + F^, cos d^i = 2 cos |3'. 



If now we take F, + F^^ = 0, or F^ = — F^^, we shall have our formula simplified, 

 at the same time that we adopt the hypothesis which seems best to accord with all 

 we yet know of the disposition of the forces in the artificial and also in a natural 

 magnet ; and hence it is the most appropriate assumption we can make respecting 

 the constitution of the terrestrial magnet itself. We shall thus have 



cos 0, — cos 0^1 = 2 cos |3, 

 where cos |3 = F^ cos j3 ; or, since one of these angles is external to the triangle N T U 

 formed by the magnet and its polar distances from N, we may substitute instead of 0^, 

 its supplement, and then the last equation will take the form 



cos 0, -\- COS dif = 2 cos (3 (45.) 



This property was originally given by Professor Playfair in Professor Robison's 

 article "Magnetism," published in the First Supplement to the Encyclopaedia Britan- 

 nica. See also Robison's Mechanical Philosophy, vol. iv. p. 350. Professor Robison, 

 from one or two passages in his writings, seems to have entertained some idea that 

 these curves could be rendered available to an explanation of the phenomena of ter- 

 restrial magnetism ; but I do not recollect that either he or any one else has sug- 

 gested how this was to be accomplished, nor, much less, attempted to actually ac- 

 complish it by such means. 



XV. — If, still considering the axis of the magnet as the axis of cT, we conceive the 

 magnetic curve to revolve about that axis so as to describe a surface of revolution, 

 its equation obviously is obtained by putting 3/^ ^ z^ instead of y^ in the equation of 

 the generating curve. That is, the magnetic surface is expressed by the equation 

 a + a X — a c_ . 



vy T^^TT^-T^)^ ^i/^ + z^ + {^- af ~" a V •; 



In this equation of the magnetic surface, a, or half the length of the magnet, is 

 constant, and c is arbitrary, giving different curves according as the value of that 

 parameter is varied. 



