224 



MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



Fig. 2. 



intersection of the plane B K C with the horizontal plane, and let C B be the line 

 along which the dipping-needle disposes itself. 



Join CO, OB, (O being the centre of the circle and, of consequence, of the sphere) : 

 then the arc C K B measures the angle COB, which is twice the angle E C B, or twice 

 the dip of the free needle. This arc, then, is known from observations at several 

 particular places on the earth's surface. 



Next, let the spherical triangle ABC denote that whose vertex 

 A is the geographical pole of the earth ; C the place of observation ; 

 and the angle B that determined as above from observations 

 made at C : and let the angle A C B denote the observed variation 

 of the horizontal needle at C. Then we have the sides A C, C B, 

 and included angle A C B, from which to determine the colatitude 

 A B and polar angle B A C. 



We have therefore the polar spherical coordinates of the point B, 

 the polar distance A B at once, and the polar angle by adding BAG 

 to the longitude of C with its proper sign. 



I shall designate the coordinates of C and B by a^ /3y and a^^ jS^^ respectively, as is 

 done in my paper on Spherical Geometry in the twelfth volume of the Edinburgh 

 Transactions ; a denoting the polar distance and j8 the longitude of the point. 



II. — Given the dip, variation, and geographical coordinates of the place of observation, 

 to express the equations of the line in which the dipping-needle disposes itself. 



Let a, b, c, and a^, h,, c^, denote the coordinates of two points in space : then the equa- 

 tions of the straight line through them are 



X = 





a„ c, — a,c 



/"/i 



hn — If, 

 y = — 



•7 r.. — r. 



Z — 



Cii c, 



Cn — Cl 



But in the present case the points a^ b, c, and a^^ b,, c,, are on the surface of the 

 sphere ; and if we consider the axis of the sphere to be the axis of z, the intersection 

 of the meridian with the equator to be the axis of «/, and that of the meridian at right 

 angles to it with the equator to be the axis of tr, then a^ (3^ and a,^ |3^, being, as before 

 stated, the coordinates of the extremities of the chord in which the dipping-needle 

 disposes itself, we shall have, for determining the equations of the needle, the follow- 

 ing values of the constants : 



Ci = r cos a^ 



6^ = r sin a^ cos 3, 



a^ = r sin u, sin ^, 



:^i = r cos ocii 



bji = r sin a^, cos /3^^ 

 a^i = r sin a^^ sin fi„. 



Hence the equations of the needle take the following forms : 



