114 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
d 6' _ n a! 1 
~dt~ ~ * V 
dpi t ?i' a' 1 
dt ~ * 7 
or, neglecting e 2 , 
d S' _ n r a! 1 d R 
dt fj.' ‘ <p' ’ d <p' 
dp' ( n' a' I d R 
dt ” I " a' p' ' d b' 
d R 
* dtp' 
d R 
' db' 
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 7 — 0 when we have arrived at our ultimate result: consequently the 
same values of 0 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 - x) 2 + (y - yf + (z! - z)-} • 
For substitution in the denominator we have 
x = r cosy (neglecting <p 2 ) 
y = r sin v 
z = r . <p . sin (v — 0 ) 
x = r {cos {y — 0) . cos 0 — cos <p . sin (y — 0) . sin 0} 
y —r {cos (y — 0) . sin 0 -{- cos <p . sin (y — 0) . cos 0} 
z —r . sin <p . sin ( v — 0) 
whence the fraction is changed to 
___— m 
jy 2 — 2 dr . cos ( 7 / — u) + r 3 + 2 rV .f 2 . cos ( 7 / — v)—2r l r.J r -. cos(7/ + t> —20) —4rV.y/’. sin ( 7 / — S'). sin(u—5)} 
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 . p'f. sin ( 7/ — b') . sin (v — 0) _ 
{— 2 r 1 r . cos ( 1 / — v) + r 2 + 2 r 1 r . f 2 . cos (v r — v) — 2 /dr ,f ~. cos ( 7 / + v — 2 &} T 
