of Eclinhurgli, Session 1882-83. 
185 
a sin a - & sin j3 
a cos a 
I /l-;“ 
- b cos /5 ( a cos a-b cos 
a sin a - 5 sin B\^ fa sin a - 6 sin Sf ) 
-3/ + I 
\a cos a-b cos /?/ cos a-b cos ^ 
Hence, by equating the components along the initial axis 
1 { ^ a sin a - 6 sin 2 fa sin a-b sin j3f 
{ 
a cos a-b cos S i a cos a-b cos /3J \a cos a-b cos /3, 
1 b b^ 
= — cos a + -2 cos (2a - + - 3 - cos (3a - 2j3) + 
Another identity is obtained by equating the components along 
the perpendicular axis. 
By treating (1 + a)^ in a similar manner we get 
.1 1 . o . 1-3 , o 
1 + 7 T a cos a - pr — 7 a^ COS 2 a + — 77 a^ cos oa - 
2 2.4 2.4.0 
^ r 1 / a sin a \2 1.3.5 
(l + acosa)q l + -2747078 
/ a sin -a Y 1 
\1 -H cos, a) ) 
and 
a sin a 
2.4 
a 2 sin 2 a -i- 
1 . 3 
2.4.6 
a^ sin 3( 
. 1 f 1 a sin a 1 . 3 « sin a \3 ) 
— (1 -1 - (2 cos a) I 2 i_|_( 7 jcosa 2 . 4 . 6 \1 -H a cos a/ 
An expansion for log {of' -I- + 2ab cos 0]^ is derived as follows 
b 
Now 
Log (a -h b) = log a + log ^1 + . 
log a - log a -h i log a , 
. b\ b l/b\2 l/b\3 
log (l + 7 j- 7 - 7 r ( 7 ) +-3 ( 7 ) - 
= -^ cos (/3 - a) - y ( Y - a) + 
and 
