OF OBSERVATIONS OF COMPLEX PERIODICAL PHENOMENA. 
377 
j) 3 = P 3 — g | ■ — 2#i H-jp, (cot 2z — cot z) — 4^2 + 2p 2 ^cot -g — cot ^ J + 3# 3 (rc — 2 ) 
+ 3^3 cot 3z j + ^ ji>, ( — 2ra + ^ 2 ^ + — g_i{n cot 2 z-n cot z) 
+ 4p 2 ( — 2 n + — + — ^ ) — 4^ ( w cot §s — n cot 
' sin 2 %s sin 2 ' Z > 
W33) 
+ 9 i> 3 (f w 2 — 2w + 3 +^ 3 ^) — cot 3z j . 
§' 3 = Q 3 — ijg'iC — cot z — cot 2z) + 2q 2 ( — cot ^ — cot |z^ — 3^ 3 cot 3z — 3 p 3 w j 
+ w cot z - cot 2z) + #, (^, — ^ 2 ^) + 4p 2 ( - n cot | - - n cot \z^j 
+ 4 «> (t^- tV) “ 9p ‘ n COt 3z + 9?1 (I ^“riSs) }• 
'sin 2 - sin 2 -,? 7 ' 7 J 
> (34) 
J 
For the values of^, q x , &c. in the last terms of equations (29) to (34), we must now 
insert their first approximations, j9 l ='P 1 , q t = Q 1} p 2 =P 2 , £ 2 =Q 2 , &c. (and in the last 
terms but one, second approximations), as follows : — 
P, = Pj — 3 |q,(w— 2) + P, cot z — 4 Q 2 + 2P 2 (cot |z + cot 
— 6 Q 3 + 3P 3 (cot 2z+ cot z)j ; 
= Qi — 1|— Qj cotz— P i^+2Q 2 (cot^— cot|z^ + 3Q 3 (cotz— cot 2z)j ; 
P 2 =P 2 -|{ - 2Q, +P 1 (cot §z-cot f) + 2Q a (w— 2) + 2P 2 cot 2z- 6 Q 3 
+ 3P 3 (cot|z-cot|)j; ^ (35) 
22 =Q 2 — 2 |q, (— cot|— cot|z^ —2Q 2 cot2z—2P 2 w+3Q 3 (cot|—cot|z^j; 
i> 3 = P 3 — g | — 2Qj + Pj(cot 2z — cot z) — 4Q 2 + 2P 2 (cot ^z — cot 
+ 3Q 3 (% — 2) -{- 3P 3 cot 3zj> ; 
Q 3 — 2 “[Qi( — c °t z — cot 2z) + 2Q 2 ( — cot | — cot|z) — 3Q 3 cot 3z — 3P 3 | . 
q 3 = 
MDCCCLXXV. 
3 E 
