476 OENEnAL CASE OP THE 



It has not been hitherto noticed, at least to my knowledge, 

 and is the object I had in view in the present paper. 



It may be proper to observe that the equation (10) may be 

 easily modified so as to apply to the case in which the vibra- 

 ting point moves in a vertical plane. We have merely to put 

 5'=0 in the expressions represented by E and k. since then 

 the constant arbitrary quantity b' expressing the effect of the 

 impulse from wliich the conical motion arises will vanish, 

 and the vibrating point will descend to the horizontal plane 

 X. The values of K and k will become, in this case, 



and -— i- — , and the corresponding value 



of {t) will be exactly of the same form as that just consider- 

 ed. If we limit the expression to the factor K, we shall 

 have a paiticular case of the first expression found for (/), 

 the same as that already given for the common pendulum. 



With respect to the equation (lO), I shall only observe 

 that its integral may be found in a seiies bv means of the 

 integrals A, B, C, ^c. Ji', B', C\ ^c. after the factors 

 (l +ku), (i + k'u, (l — k"u), are expanded by the binomial 

 theorem. By using peculiar artifices, other series may also 

 be found ; but their coefficients are so complicated, as to de- 

 ter me from inserting their invesiigatiou. 



I shall conclude this paper by a computation of the value 

 of (t) for a particular value of k. This will give an idea of 

 the rapidity with which the quantities k,. k^, ^c. decrease, and 

 at the same time show the great superiority of the formula 

 (lO) over the series (9). 



The extreme value of which k is susceptible being unity, 



let us suppose this quantity greater than its mean value , 



or= — . Then we have k = — , k. = , a fraction of 



which the square and higher powers may be safely neglect- 



