48 n. PARTICULAR MOTIONS OF A POINT. 



We may develop K in a series, for since 



JL 1 7 Q Q , , *- ' ** T A A , JL ' O ' O 



we have 

 K 



= Afy> + yF Csin^dtlj + ^W Csin 







Now since 



71 



1-3-5 . . . 2w 1 



sm 







If a be the maximum value of #, for which ^ = > A; = sin ? 

 and the period is given by 



rt\ or o 



65) 2'=2 



This is the formula which is used to correct our result 56) for finite 

 oscillations. If cc is 1 the correction is less than one part in fifty 

 thousand, and if a = 5 it is less than one in two thousand. 



23. Spherical Pendulum. Let us now return to equations 53) 

 and 54), which we will write 



66^ t = + I -^L, 



cldz 



> 



where <&(z) = 2(l 2 2 ) (gz -f li) c 2 . 



As the integrals are real (P(V) must be positive for all values 

 of z that occur in the motion. 



Substituting successively for , oo, ?, # , + I we find 



Accordingly the polynomial 3>(z) has three real roots. If we call 

 these a, /3, 7 in the order of magnitude, they lie so that 



Fig. 13 is the graph of <&(2) as ordinate, with 2 as abscissa. 



