AS EXHIBITING THE ROTATION OF THE EARTH. 17 



and the moment of the force thereby developed is equal to -- (the moment of its 



quantitj' of motion). This moment for the pendulum in any latitude ?„, is ur" cos" 'A, 



of which the differential coefficient, taken with reference to 2 as a function of /, is 



7*} 

 — 2h/-^ sin 7^ cos X--, which is the moment of the force required. Dividing by the 



. d7. 

 radius of rotation, r cos A, we have -\-2nr sin 'a ,j for the exin'cssion of the force 



which (acting positively in the direction of the axis of a-) is to be taken with the 

 plus sign. 



In the system of rectangular co-ordinates which I am using "^ corresponds to the 

 velocity expressed by r ^^ ; and substituting it therefor, we have for a disturbing 

 force in the direction of the axis of x the expression 



%i sin 7!\^ 



lu almost precisely the same way it may be shown that a body falling towards 



the centre of the earth with a velocity - " will have the moment of its quantity of 



. ... „ <iz . . 



motion dmnnished by 2;;/ cos" ^if gi^'i^^g i"!**^ to the force 



2n cos A — 

 dt 



The sum of tlicsc two expressions constitutes the disturbing force Xof Poisson, 



and is to be added to the second member of the first of equations (1), and these 



equations become' 



d'x . Nj: ^ . d// , ^ dz 



(3) ^+Y = -2MsmX^ 



drz , Nz ^ ^d!x 



-wH — T^^<] — ■in cos X ,-- 

 dt'~ I -^ dt 



' There are really other disturbing forces (comparatively slight iiiflecd) than the X Y and Z of 

 Poisson (ei.[uation 2), as a])pears from th(! following considerations : — 



Draw a lino through the origin of co-ordinates parallel to the axis of the earth, and project the 

 moving body on the plane of y z. The distance of the projection from the line will be y sin >. + 2 cos j., 

 the distance of the body from the plane of yz being. r: hence there will Ije a centrifugal force 

 relatively to this line, due to the earth's rotation, tending to increase the ordiuatcs a: y z by its 

 components 



n^ a; 



«= sin \ (>/ sin %+z cos \) 

 n' cos A (y sin x-\-z cos x) 

 With these expressions added, respectively, to the second members of equations (H), they correspond 

 to those found in Carniidiael (f'alcnl. of Operations), who quotes from GaHjraitli and Houghton 

 (Proc. R Irish Acad., 1851). They express forces of the second order in minuteness, compared 

 with those expressed by equations (2), and, insensible in their effects, are neglected in all di.scu.ssioiis. 

 They arc noticed here only to recognize their existence and to show their origin. 

 3 January, 1873. 



