80 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
18. The value of r, contracted according to the system of (13), is 
a | 1 — e . cos (0 + 1) — 4" e 1 . cos (0 + 2) — -g- e 3 . cos (0 + 3) — ~ e 4 . cos (0 + 4) 
384 ^ (0 + 3 ) 
whence q = 
— e cos (0 + l)—~e 2 . cos (0 + 2) — e 3 . cos (0 + 3) — e i . cos (0 + 4) 
125 
-^f+cos (0 + 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 
(k) 
of-r,. 
Principal term, 
- cf 
Terms of the first order, 
m ( k ) 
+ (1,0) Ci . e'cos (1 + 0) + (0,1) Ci . ecos (0 + 1) 
Terms of the second order*, 
r i 1 'i ( k ) i ( k ) 
| T (1,0)— — (2,0) |C, . e' 2 cos (2 + 0) —-^-(1,1) C, . e+cos(l + 1) 
+ j h (0,1) - | (0,2) ] Cf . c 2 cos (0 + 2) 
Terms of the third order, 
1 4 (1,0) - ~ (2,0) + L (3,0) ) cf . & cos (3 + 0) 
+ ( — -^-(l,l)+-^-(2 ,l))C 4 . e' 2 ecos (2+1) 
* In this and the succeeding expressions, when a cosine is multiplied by the sum of several diffe- 
(k) (k) 
rential coefficients of C) , the symbols of differentiation are bracketed together, and C) ' is put at the 
end of the bracket. 
