by a Method of Differences. 
189 
■~z y is equal . to the fluxion of sine of \z, is equal to the fluent 
. 2] == — sine of k . 
of ~ — sine of £'. 
fluxion of sine of \z cos of k 
sine of \z 
log, sine of \z— cos : ,°l£- .z+ a correction : now this correction 
must not be sought when 2 = 0 or a multiple of 360°: for in 
that case from what has been just now said, the primary equa- 
tion fails, or rather there is a supplemental value only then to 
be prefixed ; therefore the easiest method which offers, is 
when 2^=180°, we then have the sine of £+2= ~ k , sine of 
kfffz— sine of k, sine of £+32— — sine of k &c. and sine 
of £2=1, consequently putting Q for £ of the periphery of a 
circle whose radius one, the expression will become — sine 
k + sln e 2 ^l - slne 3 °— + 25 &c. or sine of k x log. of 1= 
— cos. of , £1.0+ correction .-. correction = sine of k . log. of 
2+ cos. ofTl . O, which correction being prefixed we have, 
• f i — r""” ■ sine of k 4 - 2-z , sine of & + 3s; ■ r ~T i M 
sine of £+2-1 1 &c.= sine of £ x log. of 
L_ — l o — — x cos. of £ ; which is only true whilst 2 is 
2 sine of \z. 1 ^ 2 J 
between o and 360°; for though our primary equation fails 
only when 2: is o or some multiple of 36 o°, and is true in 
every other case, whatever 2 may be, whether more or less 
than 360° ; still it cannot be so in this derivative equation : 
for suppose K to be the said supplemental value, which is 
equal to o in every other case but that mentioned above, the 
derivative expression, will in that case contain the supplement, 
the fluent of K . z producing a correction which will remain 
when it is once introduced, though K may afterwards vanish, 
namely, when 2 becomes neither o, nor any multiple of 3 6<f 
