226 MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



IV. — Given the equations of the dipping-needle at three given places on the surface 



of the earth to find the magnetic poles. 



Let M, Nj P be the places on the surface of the earth, and denote the needles as 

 follows, viz. 



MA by x=^ a' z + a' and y = Vz-\-^' (8.) 



NB by a: = «"2; + a" and i/ = ft"% + /3" (9.) 



P C by a: = «'" z + a'" and y = V" z + (3'", (10.) 



and denote the magnetic axis itself (T U) by 



0? = as; + a and 3/ = ft ;s -j- |8. . (11.) 



Then, since the line (11.) intersects the lines (8.), (9.), (10.), we have the three 

 equations of condition 



(a'~a) (fe'-I) = (i3'-^) (a'-a) (12.) 



(a" -a) (fe" - I) = (i3" - ^) (a" - a) ........ (13.) 



(a'" -a) (5'" - «0 = (l3"' - ^) («'" - «) (14.) 



Taking now as the unknown coordinates of the magnetic poles, T and U, the sym- 

 bols a^ h, c, and a^^ b^ c^, we have a b, a p, given functions of a, b, c, and a^^ b,, c^^ ; and 

 hence we have in the equations just given three equations for the discovery of these 

 six quantities which determine the poles. The three other requisite equations are 

 thus derived ; 



By means of (11.) combined separately and successively with (8.), (9.), (10.), we 

 can find the coordinates of the points A, B, C in terms of a, b, c, and a^, b,, c^^ and given 

 quantities a! a' V |3', &c. ; and the coordinates of T and U are themselves a^ b^ c, and 

 ^// ^11 ^ir ^^ hence have the several quantities a—^t,b~t,c—t,a — u, b — Uj and 

 c — urn terms of a^ b, c, and a^^ b,, c,,. Also the distances M A, N B, PC, that is,y,^, h, 

 are also given in terms of the coordinates of M and A, N and B, P and C respec- 

 tively, and hence in terms of a, b^ c^ and a,, b,, c,, and given quantities. And lastly, the 

 equations of the lines T U and MA, N B, PC being given in terms of a^ b^ c^ and 

 a^i b^j c^f and known quantities, the cosine of the inclinations of T U to each of them, 

 that is, of the angles A, B, C, are given so as to involve no quantities but known ones 

 and the coordinates of the magnetic poles. It hence follows, that all the terms which 

 enter into the composition of the equations (5.), (6.), (7.), are functions of the coor- 

 dinates of the poles and of given quantities. The three remaining requisite equations 

 for the actual determination of the magnetic poles are furnished, then, by those equa- 

 tions marked (5.), (6.), and (7.) ; the whole six equations which we have laid down 

 being each, obviously, independent of the others under every combination. 



V. — ^The preceding processes show that the determination of the magnetic poles, 

 their duality being admitted, as well as the equality of their intensity, can be effected 

 from three observations of the magnetic needle as to dip and azimuth ; and hence 



