by a Method of Differences. 
1 77 
and the second by 2 sine of p—^f.x by means of lemma 
No. II. and we get, a sine of px—a sine of p— ~-.x-\-b sine 
0 {pJ r q\x~b sine ofp—dH. — q.x-{-c sine of p-\-2q.x—cs\ne 
o fp — ~ — 2q.xScc.~2A cos. of p — ■ . x, and a sine of px 
- \-a sine of p— — . x-\-b sine of p -\-q.x-\-b sine of p— ~~-q.x 
-j-csineof/-}- 2q .x-{-c sine of p — — 2 q.x &c.=eB sine of 
p — q dL x ; consequently, adding these two together, we have 
by dividing by 2, a sine of px-\-b sine of p-\-q.x-\-c sine of 
p-\-%q • x &c.==A cos. of p — ~ . x -f- B sine of p— . x, ex- 
pressed generally and distinctly in terms of />, q, and x, the 
equation will therefore remain if we put z in the place of x 
throughout, and therefore the sum of the series, a sine of 
pz-\-b sine of p-\-q.z See. is given expressed generally in terms 
p, q , and of %, which is the first part of the theorem. 
Again, by multiplying the series, a sine of tH - 6 sine of 
~-\-q.x &c.=A,by 2 sine of p — x, by means of lemma 
No. I. and the series, a cos. of-x-\- cos. of — -j-q-x Sc c.=B, 
by 2 cos. of p— ^xhy means of lemma No. II. we shall have 
a cos. of p— ~^.x — a cos. of px-\~b cos. o fp— 211 — q , x- — b 
cos. of p-\-q . .r-f- cos. of p — 211 — 2 q .x—c cos. of p-{-zq . x -f. 
&c.= 2A cos. of p— q -l x, and a cos. of p— 21 . x-\ -a cos. of px 
+& cos, of q— 2 !Z ^q^Jf-b cos. of p-\-q.x-\-c cos. of q— q 
mbcccvi, A a ' 
