49(3 Mr. R. H. M. Bosanquet on the Beats 



ing them into multiple arcs, and writing p£ = 0, #£ — e=0, the 

 equation becomes 



-7T2 =Ecos0 + Fcos0 



-[|(^+/ 2 )+Sy(e 4 +/ 4 +4^) 



+ 1/3 { (V + 2/ 2 > cos + (2e 2 +f 2 )f cos } 



+ ~{* + y(e 2 + 3f 2 )} cos 20 + '^ {« + 7(<V+/ 2 )J- cos 20 



+ |(e 3 cos30+/ 3 cos30) 



+ | (e 4 cos 40 +/ 4 cos 40) 



+ ^/{« + f 7 ( e 2 +/ 2 )}(cos(6/ + (/))+cos(0-0)) 

 + f/3e 2 /{cos(20 + 0) + cos (20-0)}- 

 + W 2 { cos (0 + 20) + cos (0-20) \ 

 + f7e 2 / 2 {cos 2 (0 + 0)+ cos 2(0-0)} 



+ 7^" {cos (30 + 0) + cos(30-0)} 



+ 7^- {cos (0 + 30)+ Cos (0-30)} 



68. On performing the double integration, we shall find the 

 constant term in the above multiplied by t 2 , an inadmissible 

 result. It is only necessary to look back to the result of the 

 complete process, when we find that the term in question is 



represented after integration by -% — , where w 2 is the small 



coefficient of the term we have neglected. This indicates that 

 the position of equilibrium is indeed displaced, but through a 

 finite amount ; as this does not affect our results, we omit the 

 term in question. 



69. Remembering that 6=pt and $ — qt—e y 



E - F 

 e= — a, /= -kt 

 p 2 ' J q 2 



the remainder of the equation becomes 



