by a Method of Differences. igi 
. r , sine of *z . sine of zz 0 2 At Qz 1 , 2; 0 
have sine of z -\ 8 1 p-i- &c.=-Qz~ + 77 
and this requires no correction whilst z is from o to 360 in- 
clusively of both ; this is evident at first sight, as we are not 
now obliged as before to avoid correcting when 2=0 or 360°, 
as the equation from which this is derived does not fail in those 
cases. 
2. If in the equation sine of nz — sine of n -{- q . z . 4- sine 
of M-3g% — &c. = failing, when qz = 180° or 
an odd multiple thereof, we put n =1, q = 2 we have sine of 
z — sine of 3%-f* sine of 5 Z &c. = o, failing when z = 90° or 
any odd multiple thereof ; if 'we multiply this by z and find 
the fluent we shall have cos. of z — C0S; - of -j- C ° J ° f — &c. 
3 1 5 
= correction, which must not be sought when z = 90°, or 
odd multiple thereof ; if it be sought when z = o, we shall 
have it = 1 — 3- + i &e. = ’-O, which will answer whilst z 
is exclusively from — go° to -}- 90°. If the correction had been 
sought when 2=180° we should have it = — 1+-5- — -f &c. 
— — 2 Q, answering whilst % is from go 0 to 270°. 
3. Again, from Cor. 1. Example 2. Theorem I. cos. of z -j- 
cos. of 32+ cos. of 52-b &c. is equal to o, failing (from above) 
when qz = o or a multiple of 360°, and therefore when z = o 
or a multiple of 180°: if we multiply by the z and take the 
fluent we have sine of z 4 - of 3Z 4 - s -. e - of See. = correc- 
3 5 
tion, which should not be sought when 2 — o or any multiple 
of 180, when z = 90° it becomes sine of 90° J 9 °° 
'+ ~ ' , e ° f j * 90 &c. that is 1 — 3- + -5- &c. or its equal — for the 
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