187 
by a Method of Differences . 
s*ss± s'x o, or + 5x0, and consequently 5^ ought likewise to 
come out equal to o, and therefore 5, would be = § ; and 
consequently when s v in that case does not come out = o, it 
is certain that there must have been something neglected : 
and to shew how this may happen, we observe that since 
Theorem I. and II. require the differences of the coefficients of 
every term and the next succeeding term to be taken, it is 
evident that the last term will have nothing to be taken from, 
and will consequently remain through every new series ; in 
consequence of which there will be terms of the form A . sine 
or cosine of nr-\-qr.z, (in which ris a whole number and infi- 
nite, the number of terms of the series being infinite,) whose 
coefficient A will never be = o unless the series a , b, c, Sec. be 
converging : these terms are unavoidably omitted, by reason 
of their place being at an infinite distance, and can conse- 
quently never be arrived at ; but still unless it be equal to o, 
it should not be omitted ; which it cannot be unless, either in 
the above mentioned circumstance of the series a, b, c, See. 
being converging, or when the terms of the series of sines or 
cosines, are continually changing their signs, for different 
values of r; which it will always do when qz is not equal 
to o or some multiple of 360° ; provided the coefficients a, b y 
c y See. are all affirmative: and consequently the said terms 
may be omitted in every such case, there being no reason for 
taking one sign rather than the other : but if qz were equal 
to o or some multiple of 360°, since A . sine or cosine of 
Tsr^qr.z will then be simply A . sine or cos. of pz , and there- 
fore if the same sign whatever r may be, when a , b, c, d , Sc c. 
to A, have all the same signs ; and consequently cannot 
B b 2 
