123 



tl = _ JUL 4. AT (*' ~ *) • 

 dt* r + l I ' 



" In these equations the origin is the centre of the earth, 

 the positive axis of z is the axis of rotation of the earth di- 

 rected upwards; the positive axis of y is directed towards the 

 spectator, and the positive axis of x to the right hand ; x, y, z 

 denote the co-ordinates of the centre of oscillation of the pen- 

 dulum ; x', y, z the co-ordinates of the point of suspension ; 

 g is the attraction of the earth ; I the length of the pendulum, 

 and N is the tension of the string. 



" If equations (2) be transformed to the point of suspen- 

 sion as origin, the positive axis of z being vertically down- 

 wards, the positive axes of x and y being in the horizon, and 

 directed towards the east and north respectively, we shall ob- 

 tain the following : 



d?x Nx _. . . dy n1 ,, dz ,„ 



-rz + —r = 2k sin A-£ + 2k cosX -r- + k 2 x; (3) 



dt 2 I dt dt v ' 



dt 2+ l = ~^ 2rcos ^ sul ^~ 2 ^ sm ^37+^ 2sm ^(y sul ^+ 2 ' c osA); 



d?z N"z dx 



-^+-T-=y-A 2 rcos 2 A-2AcosA-^-+A 2 cosA(ysinA+2CosA). 



" Or supposing the axes of co-ordinates transformed to the 

 vertical and horizon of the actual spheroid, and supposing y to 

 denote gravity, we find 



d 2 x ^ r x „, . % dy „, . dz ,_ 



-7- + N T = 2k sin A -j- + 2k cos A -r- + k 2 x ; 

 at* I at at 



d 2 v it dx 



-^ + N - = - 2k sin A -=- + k 2 sin A (y sin A + z cosA); (4) 



d^z z dx 



— + N j = g - 2k cos A -=- + k 2 cos A (y sin A + z cos A). 



