MAGNETISM. 



85 



Fig. 74. 



force of the earth e. Let I be the angle 

 of the dip ENH, and d the complement 

 of that angle, or ENV. 



Then by the properties of the triangles 

 NVE, orNEH, we have 



v = e cos. d; and h=e sin. d. 

 (326.) The force v, being vertical, 

 acts wholly in the plane of the needle's 

 motion, SANB : but the force h is out 

 of that plane, and must be decomposed 

 into two others; the one, HP, perpendi- 

 cular to that vertical plane, and which we 

 shall cally; the other NP, which we 

 shall call x, directed horizontally in 

 that plane. The angle HNP is equal 

 to the deviation of the vertical plane 

 SANB from the magnetic meridian; 

 let us call this angle a . 

 We shall thus have 



y - h sin. a ; and x - h cos. a ; 

 or, substituting for h its value as ex- 

 pressed in the former equation, and 

 joining the value of v, we have, 

 v = e, cos. d. 

 y = e, sin. d. sin. #. 

 x-e, sin. d, cos.a. 

 Of these, the force y is destroyed, 

 being resisted by the axis of motion ; 

 and Ihe forces v and x are those only 

 which are effective in giving motion to 

 the needle. Let R express the re- 

 sultant of these forces, and <p the angle 

 which it makes with a vertical line. We 

 shall have 





and tang. 0= , 

 v 



or, substituting for x and v their respec- 

 tive values, as above found. 

 R = ecos. d *] 1 + tang. 2 t/, cos. 2 a; 



tang. (p=tang. d, cos. a . 

 (327.) From these equations many 



important consequences may be de- 

 rived. 



In the first place we may deduce, that 

 the intensity of the force 'R diminishes 

 as the angle a increases ; or in other 

 words, as the plane of motion deviates 

 more from that of the magnetic meri- 

 dian. It is greatest when these planes 

 coincide, being then equal to e; it is 

 least when they are at right angles to 

 one another, for then a =90, and cos. 

 a = o, whence 



R=e. cos. d. 



(328.) The direction of the resultant, 

 and consequently the position into 

 which it brings the needle, also vary 

 in the different azimuths in which it is 

 placed. In proportion as the angle a 

 increases, the cosine of that angle dimi- 

 nishes ; and therefore the tangent of the 

 angle 0, which expresses the angle the 

 resultant makes with a vertical line, 

 also diminishes. Hence, in proportion 

 as the plane of motion comes nearer to 

 a position perpendicular to the magnetic 

 meridian, the position of the needle will 

 approach more nearly to the vertical 

 position ; and it is exactly vertical when 

 its plane of motion has arrived at that 

 situation. This has been already no- 

 ticed as affording a method of deter- 

 mining the position of the magnetic) 

 meridian, independently of the hori- 

 zontal needle ( 313). 



(329.) We may deduce also the for- 

 mula given in 314, from the foregoing 

 equations ; for when the two azimuths 

 in which the observations are made 

 differ by 90 degrees, the tangents of 

 in the two cases will be respectively 

 tang. <p' = tang, d cos. a 

 tang. <p" = tang, d sin. a . 

 By taking the squares of each term of 

 these equations, and adding them, we 

 obtain 



tang. \l = tang. V + tan g- 2 $"; 

 which, when , ' and S" express the 

 angles of the dip, or the complements 

 of d, <p, and (p", respectively, become 



cot. 2 S = cot *' + cot. 2 S". 

 (330.) The same formula is derivable 

 more simply from the following consi- 

 derations : 



Let XD, Jig. 75, be the line of dip 

 in the magnetic meridian; and let 

 XVFA, XVGB, be the two vertical 

 planes at right angles to each other. 

 From D, draw the lines DF and DG, 

 perpendicular to these planes ; and also 

 FV and GV, perpendicular to XV. The 

 lines XF and XGwill be the positions of 

 the needles in these planes, according 



