376 MESSES. 0. AND P. CHAMBEES ON THE MATHEMATICAL EXPEESSION 
13. From the general expressions (23) and (28), for the coefficients^, q t we may now 
write down the particular values p 15 2L fc, §3 for the particular case in which, 
whilst neither s nor t is taken above 3, neither (s — t)z nor (s-\-t)z is ever a multiple of 
2 tt ; and at the same operation we may substitute for the general terms in which a t , a„ 
b„ b, are expressed, the series of terms obtained by giving s the values, 1, 2, 3 
successively, observing also that when s = 0 these terms vanish. We have then, 
'i = Pi — — 2) +^>, cot 2 — 4 j 2 + (cot|« + cot |) — 6 j,+ 3/i,(cot 2« + cot z) j 
+l{^(l ,i2 - 2)> +5+5i)-2' recotz + 4 *(- 2,! + _ b“ + ^) 
\ / \ sin 2 — z sin 2 - 2 
> (29) 
^^2 (' 
ncot^z+ncot ^ ) +9y 8 ( — ) — 9^ cot 2 s cot, 
( 
4 
— Qj — cot 2! — ppz + 2# 2 ^cot z 2 — cot ^s J + 3£ 3 (cots— cot 9 
-j) x n cot 2+^ (§« 2 +| — + 4p a (» cot|— wcot|«^ 
+%( T ^- T ^3-)+9 i ? 3 (ncot2-wcot22) + %( iI ^- slr ^2i)}- 
'sm 2 - sin 2 -* 2 ' 22 
> (30) 
jp* = p 2 — — 2^, +ih (cot | s — cot + 2 ^ 2 (w — s) + 2p 2 (cot 2s) — 6#, 
+ 3p 3 (cot \z + cot|) | +\Ui ( - 2n+—^Y +- 7 ^ 7 ) ~ ft ( w cot l 2 w cot 
+ fjh (3W 2 - 2% + ^ + ^2-*) - 4# a ra cot 2s + 9p. ( — 2 n + —7 + — ) 
— 9^ 3 (w cot-s+wcot^ |. 
# 2 — Q 2 — l ( — cot | — cot |s^ — 2^ cot 2s — 2p 2 w+ 2>q z (cot 3— cot |s^ j 
-f \\pi(~ n cot^— w cotes') — 4p 2 w cot 2s 
L ' 2 'sin 2 ^ sin 2 -z' 
(§*’+§— +9p 3 (w cot|— ncot\z\ +9^ 3 (— 
' 2 ' 2 'sin 2 - sin 2 -2 2 J 
> (31) 
( 32 ) 
