CONCERNING TERRESTRIAL MAGNETISM. 243 



and putting also for abbreviation (and at the same time retaining the homogeneity 

 of the equations into which these terms enter,) 



a+g,-a + g„ = k'^ 



we have (59.) converted into 



(r;^- h^)r,^-(r,;^-¥)r^ = 0; (61) 



or, arranging them in reference to r^^ it is 



"^if — r^ L h^ ' ^if — — rf -h^ (62.) 



For r,, write r + s(r^'-^h^Y ^^^ ^^^ equation (62.) becomes 



which is a cubic equation wanting the second term, and which for accommodation 

 to the usual notation may be written for the moment thus : 



r3 — 3 c r = 2 </. x 



Then we have 



^^"^ ""~ Ax%r{rl-h^f V64.) 



Now in the case before us, putting g^= — (a-\- a) and^^^ = (a + ««), which, since 

 the poles of the terrestrial magnet being either within or upon the surface of the earth 

 is always the case in nature, we have 



h'' = a- g ,. a-g ,==-a,{2a-{- fl,)l 



A:2 = a-\-g, .a-\-g^,= —a,{2a-\- ajj' 



and which equations, since a, a^, and a^ are essentially -j-, are themselves essen- 

 tially —. These values of A^ and k'^ inserted in (30.) render the whole value of 

 (^ -\- d"^ essentially +, whatever the value of r, may be. There is hence one pair of 

 symmetrical branches indicated by this method also, as in the former. But in addi- 

 tion to this we learn at once that there is ow/y one such pair ; since when the root is 

 given by Cardan's formula, (which is the case here,) that root is the only real one*. 



9. The conclusion is now established, that there is only one sheet of the tangential 

 surface compatible with the actual condition of the terrestrial magnet, and hence only 

 one line of intersection between it and the earth's surface ; or, in other words, the 

 magnetic equator is one isolated and continuous line on the surface of the earth. 



10. Also, since from other considerations it can be shown that the two poles are 



* Lagrange's test might have been employed instead of this ; but as that appears to be something more 

 laborious, I have preferred the present one. 



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