to be applied to the Length of a Pendulum. 357 



Putting the arcs for the sines we have these simultaneous equations 



d'e d°-<b 



df - F ^^ " '^^• 



If we multiply the second equation by m and add it to the first, 

 we have 



r i k" k- ak' 



JLet = p, or 7/1 = — p : 



a r r ^ 



,, d-9 , / k" ak" \ d'cb ^ , ^ 



then «!^ + (^ + 7 - — ?^ (^= -^ff/^e-g-Ci -«/>)«^. 



This will be integrable, if 



r k" k" \ - ap 



- + — p = '- : 



a ar r ' ap 



(\ r r\ r 



a ak- k/ ak' 



Let jj and p" be the two roots of this equation, which are both 

 possible and positive, m and m" the corresponding values of m. 

 Then the differential equation, upon substituting these values, 

 takes the forms 



Their solution gives 



+ ^-^'<^ = ^' cos (< viy + ^) 



e + —^ i> = J" cos (< yi^" + B"), . 



Z Z 2 



