8IMPLE PENDULUM. 471 



which may be written in the form, 



={tZ^).(i-.u^)P^,(i^ku), by putting 

 P±^-.xJ^u, and 2a + —^ l^J!^^, 



S 2 ' 2a— p — q S '2k 



By virtue of tliis assumption, we have 



vF=p=.^.(J^^)\^l ii^tn.ii +ku)i 



in which u= — 1 corresponds to x=p, and u=i to x=q. 



Substituting this expression in the equation (5), and ob- 

 serving that dx= — ^IZl{\ii^ we shall find 



^ lgip—q)y ^Ui-^^),[i^ku)\, and thence 

 d^ ak^ ' ^ 



With respect to the equation (6), we have 



= ~^.{i + k'2i).{i—k"u\ by assuming 

 ^tll=k{2a—^^)=k"^^, and accordinglr 



