12 On the Laws of Terreslrial Magnetism 



This formula rc()Tiires that we know how to calculate X^ Let 

 A E be the tcnestrial e<|uator (li-^. 3) ; N E' tlic magnetic equa- 

 tor (su])|)ose<l in like niauuer to he a threat circle), and i\i the 

 given place on the globe, having for its longitude AE = /, anti 

 tor its geograpliic latitude l\iE = A. If from tiiis point we draw 

 the arc of a great circle i\I E' perpendicular to the magnetic 

 equator, this arc will express the magnetic latitude of M. Now 

 as we know the longitude AN, or a, of the node of the magnetic 

 equator: expressing it by r/, we sliail haveNE = /— ». Thus 

 in the spheric triiiugle MNE, right angled at E, we know the 

 two sides NE, ME: we may therefore compute the hvpotheuu^e 

 MN, or H, and the angle N, by these fornuihe : 



cos H = cos \ C'ji {I— a) and tang N = 



sin 1^/ — (/)' 



The angle N being thus known, we add to it the angle of in- 

 clination I of the two ecpiators, and we get the angle MNE'. 

 Then in the triangle MNE', the arc ME', or A'= tlie magnetic 

 latitude of the point M will be obtained by the formula 



sin a'= sin II. sin (N + I). 



Now let us compute these for Paris. 



Here the longitude Z=:o ; the latitude A = 48'' 50' 14" ; NE 

 or I — a will be (54' 26'; that is to say, equal to the longitude of 

 the eastern node of the magnetic equator. With these data we 



ftnd 



11 = 73" 29' 10"; N = 51'' 44' 10''; and A' = 59= 20' 10". 

 Lastly, with this value of a', computing i-\-X', and i, we find 



i + a'=132" 49' 20"; and consequently ? = 73" 29' 10", 



This is therefore the dip of the magnetic needle at Paris, 

 according to oiu- formula: direct experiments give it about 

 70°. 



Our formula, therefore, gives a verv sinqilc relation between 

 the observed dips near the magnetic equator. Indeed, in this 

 case / and A arc very small quantities. Bv contining ourselves 

 to their first powers, we may consider cos 2a as equal to^l ; and 

 we may substitute for tang (?"4-A'), and for sin 2a, tlic arcs 

 which correspond to them. Then the formula is reduced to 



7: = 2a'; 

 that is to say, each dip is double the corresponding magnetic 

 latitude. This property is found to be completelv substantiated 

 in all the observations made at a liltle distance from f/ie mag' 

 netic equator, between the longitudinal limits where it is ap- 

 parently circular. 



For example : At Tompenda in Peru, M. Humboldt observed 



the 



