NeilIi. — On the Deflection of the Plumb-line. 581 



Eliminating cot x We obtain 



sin 2b — mr sin 6 + nr cos 6 = o (3) 



Wherein m has the value —^ — cos — , and n the value — ^— - sin — , 



be 2 be 2 



or 2 sin 6 — mr sin + w = o, since (9 is small. 



.-. sin = - 5 (4) 



2 — mr w 



The amount of the deflection is r sin 6, and is towards the Equator, since 

 is positive. 



To transform the equation to rectangular co-ordinates we substitute in (3) 

 the values of the variable in terms of x = r cos 6,y = r sin 6, and x 2 + y 2 = r 2 , 

 and we have 



nx? — myx 2 + 2xy + nxy 2 — my 3 = o (5) 



When B and C are the foci of an ellipse the triangle ABC is isosceles 

 if A is at the extremity of the minor axis. 



Then m = T cos — and n = o. 

 b 2 



Substituting these values of the constants in equation (5) we obtain 



o 2 A 2 A . 



2 xy - j- cos — yxr - ^ cos — f = o. 



Therefore y = o (6) 



. _ / b A\ ih AV 

 and y- + [x - - sec —J = I - sec —J (7) 



Equation (6), y = o, represents the axis of X, and for a plummet 

 suspended at the Pole of the earth the axis of X is the vertical, and 

 consequently there is no deviation from the vertical when the plummet is 

 suspended from the Pole of the earth considered as an oblate spheroid. 



6 4 

 Equation (7) represents a circle whose radius is - sec — , and is therefore 



the circle circumscribing the triangle ABC. ^ 



b A 



The co-ordinates of the centre are y = o. x =- sec — . 



y 2 2 



When the point A coincides with the extremity of the major axis of the 

 ellipse the angle A vanishes, and we find 



b + c , 



m = -^= — and >t = o. 

 be 



These values of the constants substituted in equation (5) reduce it to the 

 following two results :■ — 



y = o (8) 



^(-^'-(^ , ■■••» 



The equation (8) represents the axis of X, and is the solution of the 

 problem when the plummet is suspended at the Equator, from which we 

 infer that for places situated on the Equator there is no deviation of the 

 plumb-line from the vertical. 



