CONCERNING TERRESTRIAL MAGNETISM. 241 



Also, from (52.), 



2(^^b)=^^^^^±^i^^^^I^ (,,,) 



y 



Insert (55.) in (53.) ; then there results, after combining (53.) and (54.) and slightly 

 reducing 



{x + a)r,^-{x-a)rf^ f - ^^ + g, + g,,^ - g,g^,' • • • • K^^-) 



which is a rectangular equation to the curve which is traced out by N. It separates 

 at once into the two components 



3/ = (57.) 



^/ - ^/^ _ y , ±gu-^^ . . 



{x + a)r^^-{x-a)r'^ f - oe^ + g, + g,,^ - g,gi,' ^ '^ 



which last is readily reduced to 



{y^ +(x + ay^a-{-g,.a-{-g,} r/ = {3^2 -{. (^ - a)'^ - a ■-- g,.a- g,,} rf. (59.) 



Squaring both sides, and restoring the values of r, and r^^ it becomes an equation 



of the ninth order ; in which, arranging according to powers of x or 3/, the coeflScients 



become very complex, and altogether unmanageable by any of the usual methods. 



It is of the form 



(^-^, + ^.)3/' + A^« + By + C3/2 + D=:0, .... (60.) 



where A, B, C, D are functions of x, which in all cases render the terms not 

 higher than of the ninth degree. 



XVII. — Though we cannot completely discuss the course of the curve and the 

 character of its singular points by means of this equation, we may yet learn some 

 particulars of its general features with considerable facility ; and as they will be of 

 great use to us in our future inquiries, we shall insert them here. 



1 . Since y appears only in even powers, the curve is composed of pairs of branches, 

 such that the branches in each pair are equal and symmetrically disposed with respect 

 to the axis oi x. 



2. Since x appears of an odd degree, there will be at least one real value of x for 

 every value of y, whether y be positive or negative. There will at least be one pair 

 of equal and symmetrical branches, and these branches will be infinite ones. 



3. Since (y"^) appears of th^ fourth degree, there may possibly be four values of y"^ 

 for every specific value of <r ; but there cannot possibly be an odd number. Of these 

 four possible roots any number may be minus, and the corresponding values of y 

 itself be still impossible or imaginary. But by art. 2, there must be at least one pair 

 of real values of y, there must be at least two real values of y-, one of which must 

 be -f-j for every value whatever of <r. 



4. Also four is the greatest number of pairs of symmetrical branches that can exist. 



5. If the system be made to revolve round the axis of symmetry (that of x), it will 



MDCCCXXXV. 2 I 



