114 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
d O' 11 a! 1 d R 
dt ^ V 1 — e' 2 ' p' *d<p' 
dp' ^ n' a ' l d R 
dt ~ fj.' \/T—Y 2 ' p' ' d O' 
or, neglecting e' 2 , 
d O' 11 ' a ' 
1 
d R 
dt fjJ ’ 
7 1 
' d p' 
d p' ( n' a' 
~dTt~~ ' ~vT ’ 
1 
d R 
7 
• T¥ 
These expressions are true only when <p’ is so small that its square may be re- 
jected. This restriction, however, is convenient as well as necessary. For in 
the expansion of R we shall have to proceed only to the first power of <p’, and 
make <p' — 0 when we have arrived at our ultimate result : consequently the 
same values of 6 and <p must be employed as in the first Part. 
62. The only term of R, which by expansion will produce terms of the 
form cos (13 — 8 ) with coefficients of the fifth order, is the fraction 
— m 
V { 7 ~ + (y - yf + <7 - z) 2 } * 
For substitution in the denominator we have 
x = r cos v' (neglecting <p 2 ) 
yf = r sin v 
z = r . <p . sin (v — 0) 
x — r {cos (y — 8) . cos 6 — cos <p . sin (y — 6) . sin 0} 
y —r {cos (v — 0) . sin & -f- cos <p . sin (y — 6) . cos &} 
z — r . sin <p . sin (y — 0) 
whence the fraction is changed to 
—in 
^ /^-2 dr . cos (t/ — w ) + r 2 + 2 r'r .f 2 . cos (d — v)—‘2,r'r.f~. cos(t/ + u— 20) — 4r f r.p' < f.sin (d — 0').sin(t;— 0 )} 
where f is put for sin -77 and 2 f for sin <p, on the principle of (13). The part 
of this depending on the first power of <p' is 
— m . 2 r' r . p r f. sin ( v ' — O') . sin (v — 0) 
{> J2 — 2 dr. cos (d — v) + r~ -+• 2 d r ,f 2 . cos ( v ' — v) — 2 dr . f 2 . cos (i/ -f v — 2 0}‘ z 
