AS EXHIBITING THE ROTATION OF THE EARTH. 17 



and the moment of the force then 1>\ developed is equal to (the moment of its 



quantity of motion). This moment for the pendulum in any latitude ?., is nr* cos* ?., 

 of which the differential coeth'cient, taken with reference to >. as a function of /, is 



'2ni a siii 2, cos >. ", which is the moment of the force required. Dividing by the 

 radius of rotation, r cos/., we ha\e | ','//; sin /. . for the expression of the force 



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

 plus sign. 



In the system of rectangular co-ordinates \\ hich I am using ( ^ corresponds to the 

 velocity expn-srd l.y r _-. an d substituting it therefor, we have for a disturbing 

 force in the direction of the axis of x the expression 



2,< sin >.';' 



ii 



In 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 



7 



motion diminished by 2r cos 2 X , , giving rise to the force 



2 cos /.' " 



The sum of these two expressions constitutes the disturbing force X of Poisson, 

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

 equations become 1 



<* X , #*_.. _:_ J'J , o dz 



(V 



j-tn sm JlVj -f-Bn cos 



Ny_ dx 



d*z , Nz fix 



= - 2n cos * 



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

 Poisson (equation 2), ns appears from the following considerations : 



IM-UW a line through the origin of co-ordinates parallel to the axis of the earth, and project the 

 nn.ving body on the plane of y t. The distance of the projection from the line will be y sin * + z cos x, 

 tin- distance of the body from the plane of y z being x: hence there will be a centrifugal force 

 relatively to this line, due to the earth's rotation, tending to increase tin: ordiuates x y z by iU 

 components 



n'x 



n* sin \ (y sin x-f* cos x) 



n* cos x (y sin x+z cos x) 



With those expressions added, respectively, to the second members of equations (3), they correspond 

 to those found in Carmichacl (falciil. of Operations), who quotes from Galbraith and Houghton 

 (Proc. U 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 discussion*. 

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



3 January. 1678. 



