*4 9 
by a Method of Differences, 
the same manner as in the common differential method, 
except that they here continually commence with the first 
term, a; and if p' be put — / P"=P' — ^f'—p " — ■■§?» 
p xy =p"' — ±q . Sac. s . 2 sine of ±qz—s\ — s'. 2 sine of \qz=s" , 
s ". 2 sine of \qz—s"\—s'". 2 sine of \qz=.d y &c. Then shall 
ci . cos. of p'z-\-d. cos. of p'-j-q . z-\-b'. cos. of p'-fzq .^-j-c'cos. 
of p'-f-^q .z-\- Sac.—s\ 
a. sine of p"z-\-a". sine of p''~\-q . z-\-b". sine of p" -\-zq.z &c .=s", 
a . cos. of p"'z-\-a"' . cos. ofp /,, -\-q.z-\-b"'.cos. ofp'" -\-%q.z&c.==s" 1 f 
a . sine of p iy z -j-a iv . sine of p iy -\-q .z-{-b iy . sine of p iy -{-2,q.z Sac.=s iy , 
&C, — |— &C,“j'“ &c, y J See, 
For, multiplying the series a . sine of pz-\-b . sine of p + q . z 
-f-r.sine of p-\-2q .z Szc. = s,by 2. sine of \qz by lemma No. I. 
we get, a . cos. of p— ±q .z — a . cos. of p + ±q.z + b . cos. of 
/+¥• . Cos. of p + . qz-j-c . cos. ofp-j-j;. qz—c . cos. 
of p-\-^q . z Sac. = s . 2 sine of \qz .*. putting b — >a = a' J c—b 
—b', d — c = d', Sac. p ■ — ±q=p', s . 2 sine of \qz~s' we have, 
a . cos. of p'z^a'. cos. of p'-fq . z-\-b'. cos. of p‘-\-2q. z+c'„ 
cos. of p'-\-Qq . % &cc.==s r , multiply this by 2 sine of \qz by 
help of lemma No. II. and we have — a sine of p ' — \q . z-\-a 
sine of p'-\- ^q.z— a! sine of^'-f- \q.z-\-a' sine of^'-f- q.z—b * 
sine of p'-{- j-q.z-\-b' sine of p'-\- j-q.z — c' sine ofp'-\- 
sine of p’ -\-\q-Z Sac.— s' . 2 sine of j^qz, put 6 ' — a'— a", d — b'—b ", 
d! — c'-=.d" Sac. p ' — \q—p" and — s'. 2 sine of ±qz=s", and we 
have a sine of p"z-\-a" sine of p”-\-q . z-\-b " sine of p"-\-2q . z 
Sec.—s", and because this is exactly similar to the original 
equation, ( if we put a", b", c", Sac. for b, c, d , Sac. in that, and 
