70 MR EDWARD SANG ON THE MOTION OF A HEAVY BODY 



wherefore the time of descending from F to N, and thence rising to L on the 

 other side, is between the limits 



-v'(f)«W(f) •-"«*. 



which limits approach closer to each other when the arc NF is made smaller. 



27. Resuming now the investigation as in article 24, we find that the time of 

 oscillation from K is 



time B ^(sec §B ) 2 tt J ( ^- j 



^r(sec I B ) 2 nj (^ ) . sec B v 



Continuing the same progression another step by making 



sin B 2 = (tan | Bj) 2 



we find 



time B ^ (sec i B ) 2 (sec } B,) 2 w J (^\ 



^: (sec * B ) 2 (sec h BJ 2 tt J (^j . sec B 2 ; 



these limits being now closer to each other, since B 2 is a smaller angle than B r 

 If, then, we continue the progression indefinitely, by making 



sin B 3 == (tan J B 2 ) 2 ; sin B 4 = (tan J B 3 ) 2 ; &c. 

 we shall obtain for the entire time of an oscillation in the arc 4B , 



time B = ttJ fi^\ . (sec § B ) 2 . (sec J B a ) 2 . (sec | B 2 ) 2 . &c. 



28. As an example of the calculation we may require the time of oscillation 

 over an arc of ] 80°, which is the extreme limit of a pendulum with a flexible 

 thread. In this case B = 45°, whence log tan \ B = log tan 22° 30' = 9617 2243 ; 

 wherefore log sin B 1 — 9*234 4486; B 1 = 9° 52' 45"42 ; from this again we have 

 log tan 1 B 1 = log tan 4° 56' 22 "-71 = 8-936 6506; log sin B 2 = 7873 3012, giving 

 B 2 = 0° 25' 40"-74. And once again, log tan iB 2 = log tan 0° 12' 50"-37 = 

 7-572 2861; log sin B 3 = 5-144 5722, B 3 = 0° 0' 02"-88. Here we observe that 

 the log-secant of B 3 does not differ from zero by unit in the seventh decimal 

 place ; and that, therefore, we have brought our limits so close together that the 

 difference cannot be appreciated by help of the ordinary logarithmic tables. The 



