80 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
18. The value of r, contracted according- to the system of (13), is 
1 — e. cos (0+1) — e 2 . cos (0 + 2) — e 3 . cos (0 + 3) — je 4 .cos (0 + 4) 
- ^ e 5 . cos (0 + 5) j 
whence q — 
— ecos (0 + 1)— \ e 1 . cos (0 + 2) — e 3 . cos (0 + 3) — -y e i . cos (0 + 4) 
125 
“384^' C0S (° + 5 ) 
and a similar expression holds for q'. Substituting these in the expression 
above, and following strictly the precept of (8), we find for the development 
of-T 
.(*) 
Principal term, 
(*) 
-C 
Terms of the first order, 
m d) 
-j - ( 1 ,0) Ci . e' cos (1 + 0) + (0,1) Cx . ecos(0 + l) 
Terms of the second order*, 
{1( 1 ,°)-| (2+) | cf. e' 2 cos (2 + 0) - ~ (1,1) C f. e' e cos ( 1 + 1) 
4- 1 4" (°,l) ~ T (°> 2 ) } cos (° + 2 ) 
Terms of the third order, 
{4 (1,°) - 4 ( 2 >°) + k ( 3 >°) } c f • 6-3 cos (3 + °) 
+ { - +0.1) + + «.!)} €^.^.00. 01+ Ij 
* In this and the succeeding expressions, when a cosine is multiplied by the sum of several diffe- 
rential coefficients of C^‘\ the symbols of differentiation are bracketed together, and is put at the 
end of the bracket. 
