322 Mr. Galbraith on the Deviation of a Falling Body 



Hence the difference of the effects of centrifugal force in 

 the case of the earth being a spheroid instead of a sphere is 

 very small. The effect of the centrifugal force at the equator 

 being about -^^-^ of that of gravit}', it decreases on approach- 

 ing the poles upon the whole, though the direction becomes 

 most favourable at 45° for thowing a body towards the poles, 

 thus making it deviate slightly from a due east course. 



Let P E/; Q be a section of the earth considered a sphere, 

 Vp the polar axis, and E Q the equator ; 

 then OQ: a b :: the centrifugal force at 

 Q : to the centrifugal force at b. But 

 O Q : a b : : radius : cosine of the lati- 

 tude, or the centrifugal force at the 

 equator is to the centrifugal force at any 

 latitude as radius is to the cosine of that 

 latitude. 



Again, Ob: ab ::b c: bd = - — ^V— • As O i is constant, 



o ' Ob 



h d varies as abxb c. But b c varies as ab; therefore b d, that 

 part of the centrifugal acting in opposition to gravity, varies 

 as a b', or as the square of the cosine of the latitude. In like 



manner O b: O a: : be : cd = — ^ — ; and since O 6 is con- 

 stant, c d varies as O a x b c. Bat be varies as a b, or as the 

 cosine of the latitude, and O « as the sine ; consequently that 

 part of the centrifugal force, at right angles to the direction of 

 gravity, tending to move the body nearer the poles than the 

 point directly under that from which it was dropt, varies as 

 the product of the sine into the cosine of the latitude. 



Let X be the sine, then -/ 1 —x' is the cosine, therefore 

 cd will be a maximum when x{\—x-y is a maximum, or 

 when the latitude is 4-5°, and then sin x cos = 0*5 or \. But 



\ X -u^ = -^^ ; therefore the maximum effect of centrifugal 

 force to throw the body towards the poles is only — — of the 



effect of gravity. Hence if A denote the deviation directly 

 eastward arising from the earth's rotation combined with the 

 action of gravity, then the deviation northward or southward 

 from the effects of centrifugal force will be expressed by 



A xy X sinA X cos A (a) 



in which K is the latitude. 



Let d 7= A X /■ X sin A X cos A, then by the composition of 



forces D = v/ A' + t/-, as the triangle so formed, having the 

 hypothenuse the diagonal of the parallelogram, is a right-an- 

 gled 



