ARCTIC SEAS.- -PART VI I. PORT KENNEDY. 
15 
From (29) we find 
cos (2m'— c)= +076 — 071 cos 2v 
+ 072 sin 2v 
sin 2(m'— c) = — 072 cos 2v— 0‘69 sin 2v 
Hence we find 
cos 2 (m— s— im)= cos. 2 (— ( m'—c ) +0) 
= 076 cos 2^6 — {071 cos 20—072 sin 20} cos 2v 
+ {072 cos 2(f)— 0‘69 sin 2(f>} sin 2v 
(50) 
(51) 
sin 2 (m—s—i m ) = sin 2 (— (m' — c)+0) 
= 076 sin 20+ (072 cos 20— 071 sin 20} cos 2v 
+ {0'69 cos 20+072 sin 2(f)} sin 2v 
If we make v—u (as an approximation) in (48) and (51), we find writing 
A=07l cos 20+072 sin 2(f) 
B=072 cos 2(f)— 0‘69 sin 2(f) 
A'=072 cos 2(f)— 0'69 sin 2 (f> 
B'=0‘69 cos 20+072 sin 20 
(52) 
M 7 cos 2 (m—s—i m )= M 
'O il cos 2(f) 
.+0-38A+0-40B 
— M{0 , 89A+072 cos 2(f ) } cos 2v 
+M {0‘89B+072 cos 2(f)] sin 2r+&c. 
7)74 sin 2<f) 
M' sin 2 (m— .s'— i m ) = M 
-0-38A'+0-40B' 
(52) 
+M{0 , 89A'— 072 sin 2(f ) } cos 2v 
+Mf0 , 89B / +072 sin 2(f)] sin2v+&c. 
Hence, from (46), since S'=Scos2o-, and cr=22°, we find 
A 2 =M{074 cos 20+O-38A+O-4OB} +0‘86S cos 2 i s 
— M{0‘89A+072 cos 2(f > } cos 2v 
+M{0‘89B+072 cos 2 0} sin 2v+&c. 
B 2 =M{ — 074 sin 20+O-38A'-O-4OB'} + 0‘86 S sin 2 i s 
— M{0‘89A'— 072 sin 2(f>} cos 2v 
— M{0'89B'+072 sin 20} sin 2v+&c. 
Comparing these expansions with Tables V. and VI. we find 
M{074 cos 20+0*38 A+0-40B} + 0*86S cos 2t,= 37 . . . (a) 
M{0*89 A+0’12 cos 20} = 13'3 ( b ') 
M{O-89B+O-12cos20} = -1O-2 (c) 
M{ -074 sin 20+0-38 A'-O-lOB'} +0-86 S sin 2i s =6‘0 . . (d') 
M{-0-89 A , -O-12sin20} = — 8-2 (e ) 
M{O-89B'+O-12sin20} = 12-2 (/') 
