298 NEW SERIES for the 



which are like functions of a feries of arches, formed from the 

 arch to be rectified, and from one another, by continual bifec- 

 tion. We may, however, in the very fame way inveftigate the 

 formulae which mall exprefs the reciprocal of the arch and its 

 powers by like functions of arches deduced from one another 

 by trifedion, or any other fedion whatever. The moft fimple 

 formula, which cxprefies an arch by functions deduced from 

 it by trifedion, may be investigated as follows : 



■2 tan ~ A - — tan^ — A 



20. From the known expreflion tan A — * * s i— ± ■ 



° r i — 3 tan 1 ^ A * 



we readily get 



i _ i 8 tan-^A 



tan A " 3 tan -j A 33 — tan 2 J A* 



and hence, by fubftituting a, j a, £ a, &c. fucceflively for A, 

 and multiplying by the terms of the feries 1, 7, ■£, &c. we de- 

 rive the following equations, 



I __ 



I 



8 



f-Q n ' >7 



3 

 8 



3 2 

 8 



~3 3 





tan# 



1 



3 tan f a 



1 



3 — tan 2 T «' 

 tan^ a 



3 tan y a 7 



i 



9 tan £ a 



1 



3 — tan 2 ^ «' 

 tan ~a 



9 tan ~ a 



27tan T ' 7 # 



3 __ tan 2 -j'j a 



&c. 



Now conceive this feries of equations to be continued, till 

 the number of equations be n, and their fum to be taken, and 

 the quantities common to each fide of the refult rejeded, as in 

 the inveftigations of the other formulae ', then we mail have 



1 

 tan/z 



