The Rev. R. R. Anstice on the Motion of a Free Pendulum. 385 



lowest point ; this we will call K : another variable, produced by 

 the motion, which call N. The whole force therefore = K + N. 

 Our equations of motion therefore are 



^/(R/4- ^) = ~^/' + (K + N) (/- ^) 



<?/(Rm+7/) = -^m' + (K + N)(m-^) )>. . (2.) 



<//(Rw+^) = -y+(K + N)(?i- ~) 

 Also, as before. 



/ m 



w= constant 



} 



(3.) 



b being the angular velocity of the earth ; and our equations 

 become 





(4. 



Now these equations must be satisfied when there is no oscil- 

 lation, and the particle remains in (apparent) rest at its lowest 

 point. In which case 



Consequently we must have 



0=-^Z'+(K + R62)/ - 



0=-^m'+(K + R6V > (^0 



0=—ffn' + Kn 



If in these equations we bring the term involving g to the 

 other side, square and add, attending to the relations 



/2^wi2-f-w2=Z'2 + m'H?i'2=l, 

 we find 



^2=:.(K + R62)2(l-n2)^KV, . , . (6.) 



from which quadratic K may be determined in terms of known 

 constants. 



From the same equations we may also find /', m' and n', in 

 terms of /, m and constants. But this is not necessary for what 

 follows. 



Now if we subtract each of equations (5.) from the correspond-' 



