by a Method of Differences. 
169 
. z -f- e cos. of p -f 3 q . z, Sec. = s . 2 sine of qz, therefore pat- 
ting a'=c — a , b'—d — b, c'—e—c, Sec. p'= p — q, s'=s . 2 sine 
of qz, we get a cos. of p’z-\-b cos. of^'-j- q . z -}- <2' cos. of/> # 4“ 2 ? 
. z + V cos. of p -j- Qq . z, See. = s', and multiplying this by 
2 sine of qz, by help of lemma No. II. we get a . sine of p*~\- q 
z — a: sine of p' — q.z -f- b sine of p'-{- zq.z — b sine of p'z-\-a' 
. sine of p'-{- 3 q . z — a' sine of p'-\- q.z -\-b' sine of p'-\- 4 q . z 
— b' . sine of p’~ j- 2 q . z See. = s' . 2 sine of qz, therefore putting 
a r — a — a ", b' — b = b", c r — ■ a' = c" Sec. p 1 — q — p" , — s' . 
2 sine of qz = s" we have, a . sine of p"z -j- b . sine of p" - f- q. z 
-j- a", sine of p"-\- 2 q.z + b”. sine of p"-\-3q-z Sec. = s", which 
being exactly similar in form to the original series, the suc- 
cessive series, which will be of a similar form to the second 
or first of the series, will be deduced by the like operations 
and substitutions. O. E. D. 
Corollary 1. The 7rth successive value of s is == +$ . 2 sine of qz\ v 
or + s'. 2 sine of qzY'* 1 , the upper sign to be taken when * 
being divided by 4 leaves o or 1, otherwise the under sign 
and the -a-th successive value of p —p — ■ tr.q. 
Corollary 11. These operations are performed by differences 
whether the signs be all positive, or alternately positive and 
negative. 
Example 1. Required the sum of the series n sine of pz-\- 
n -j- r sine of p q . z n 2 r sine of p 2 q.z Sec. 
Her ea,b,c, d, Sec. = n,n-\-r,n-\-2r,n-\-3r,?i-\-/^r,Sec."l 
a, b, a' , b', Sec. — n, ?i-\-r, 2 r, 2 r, 2 r, &c. V 
a, b, a", b", Sec. = n, n-\-r, 2r—n, r — n, o, &c.J. 
s — s . 2 sine of n sine of p"z -j- n -J- r sine of^>"-j-^ 
MDCCCVI. Z 
