2o6* NEW SERIES for tie 



28. This feries, however, admits of being exprefled under 

 another form, better adapted to calculation, and to effect this 



transformation, let us begin with the term * * A . In 



tan 4 A 



this quantity let — — — be fubflituted for tan* A ; it then 



becomes, after proper reduction, ZJE — £5L_ co ^ . A- 



• r 1 — 2 cof 2 A +cof 2 A 



gain, in this expreflion let — ^— — ^— ' be fubflituted for 



cof 2 2 A, we then get 



3 + 4 tan' A _ 13 — cof 4 A -f T 2 cof 2 A 

 tan 4 A 3 + cof4A — 4cof2A' 



The remaining terms of the feries, which are fimilar to one 



1 1 



another, and of the form 4 tan 2 — A+ 3 tan 4 — A, admit of a 



' 2» 2« 



T 



i — cof- A 

 like transformation -, for by fubftituting for 



1 -f- cof - A 

 n 



1 + cof- A 



tan 2 -i- A, and again — for cof. A in the remit, we 



2« 2 n [ '_ 



get 



_ 





13 — cof- A — 12 cof- A 

 4 tan 4 ~A + 3 tan4 : i-A = 



f 



I 



3 + coffA + 4<eofiA 



By 



