? 
298 NEW SERIES for the 
which are like funétions 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 
formule which fhall exprefs the reciprocal of the arch and its 
powers by like functions of arches deduced from one another 
by trifection, or any other fection whatever. The moft fimple 
formula, which exprefles an arch by funétions deduced from 
it by trifection, may be inveftigated as follows : 
iF aves 3 
30. From the known expreffion tan A = SL geet ne Oe 
I—3tantA * 
we readily get 
Blige be I gn stggy tan, 4 As 
tanA” 3tantA 3 3—tan'+A’ 
and hence, by fubftituting a, ta, $a, &c. fucceffively for A, 
and multiplying by the terms of the feries 1, 5, $, &c. we de- 
rive the following equations, 
aii op 3 I 8+ —fap-la 
pnd >. .3 tans a 3 3—tar ta 
I bi I _ 8 -tan5a 
3tanta gtanga 3° 3—tan’5@’ 
T niet I 3 8" tan 5@ 
gtanga™” 27tan3>a@, 33 3—tan’5'@ 
&e. 
Now conceive this feries of equations to be continued, till 
the number of equations be 7, and their fum to be taken, and 
the quantities common to each fide of the refult rejected, as in 
the inveftigations of the other formule ; then we fhall have 
tan @ 
