284 NEW SERIES for the 
it is changed to 
f 1+cof2a, 2 
1—col2za 3 
ES —(3 1 —cofa ai 1—cof ja 1 1—coftea 
Faaan 4 1+cola 4 1+ cofta 43 4+ cofia 
I 1—cofta ; 
er 1 corte +, &e.)s 
or, putting 1 4 inftead of a, and 7 a inftead of 4 a, and fo on, 
2 4 2 
in order that the formula may contain only the cofines of the 
arch and its fub-multiples, and dividing the whole expreflion 
by 4, 
I 1+ cofa I 
4 1—cola is 6 
pe i he es r—cof4a, 1 1--collg 1 1-cofza 
rag - 4° 1+ colfta 43 1+ colta 4+ 1+ cofia 
1. 1—coly; 4 
eaters oe &c.) 
and this is the fecond feries which I propofed to. inveftigate, 
reduced to its moft convenient form. 
18. We may determine two limits to the rate of conyergen- 
cy of the feries juft now found, in the fame manner‘as we have 
found the limits of that of our firft feries ; and, indeed, the rea- 
foning employed in the one cafe is immediately applicaye to 
the other. For if the firft feries, which is 
is 
I + = tan < a+ 7 tan ; at; tan gid Py &e. 
ig ek. 
aon tania 2 
be 
