CONCERNING TERRESTRIAL MAGNETISM. 



^ sin Ui, sin ^„ — sin «y sin /S^ sin a^^ sin fi„ cos a^ — sin u, sin /S^ cos Un 



cos «yy — COS a-i cos ay^ — cos a, 



sin «yy cos j8yy — sin ay cos /3y sin «yy cos /3y, COS a, — sin ay cos fi, cos ayy 



•^ cos «yy — cos ay COS ayy — cos ay 



223 



• (1.) 



• (2.) 



III. — Let M, N, P be the centres of three 

 dipping-needles at known positions on the 

 surface of the earth, and denote the poles by 

 T and U. Then the needles will arrange 

 themselves so as that each shall be in a plane 

 passing through T U ; and hence each needle 

 prolonged will cut the magnetic axis T U in 

 some point, as A, B, C, respectively. 



Take any point, O, in T U, and refer all the 

 points to this origin ; denote the several distances O U, O T, O C, O B, O A, by m, t, 

 c, h, a respectively ; the angles MAO, N B O, and P C O, by A, B, C ; and the di- 

 stances M A, N B, P C, by/, g, h. 



Then we have 



MT2=(a- #)2~ 2/(a- OcosA+/2 



M U2= (a - w)2 _ 2/ (a - u) cos A -\-f^ 



'NT^=z{b—ty--2g{b-t) cos B -f- ^2 



NU2= (b — uy — 2g{b-u)cosB+g^ 



P T2 = (c - ^)2 ~ 2 A (c - t) cos C + ¥ 



P U2 = (c — 2^)2 — 2 A (c — w) cos C 4- h\ 

 Again, by the properties of a needle subjected to the action of the magnetic poles 

 T and U, we have (those needles being small in comparison with its distance from 

 those poles,) the following proportions : 



TA : AU : : TM^ : M U^ ^ 



TB : BU :: TN3 : NU3 !> (4.) 



T C : C U : : T P3 : P U3. J 



Inserting in (4.) the values of the lines T A, &c. given in (3.), we get the three 



equations 



a- t _ r (g -if -2 f{a - t) cos A +/' ") 

 a — u (^ (a — «)^ — '2,J\a — u) cos A +J^j 



h — t {{b -ty-Qg{b - t)co?iB + g^ 



> 



(3.) 



b- 





{b — uy — 2g{b 

 {c-tf-Q.h{c 



\ 



u) cos B + g^ J 



t) cos C 4- A' 



(c _ m)8 _ 2 A (c - u) cos C -I- A* 



}' 



(5.) 

 (6.) 

 (7.) 



MDCCCXXXV. 



2 G 



