ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES, 215 
and the direction-cosines of the normals to the surfaces fx = const, (f) = const, will be 
— cos 6 cos 4>, — cos 0 sin (/>, sin 0, 
— sin (f), cos (p, 0, 
respectively. Hence we have 
U = — {tc cos (p V sin (p) cos 0 -{- w sin 0, 
V = V cos (p ~ It sin <p, 
W = (u cos (p V sin 6) sin 0 -}- iv cos 0, 
from which we obtain 
Again 
— U cos + W sin 0 = u cos (P -f- v sin (p. 
7 3"^ o-i/r ^ 0i/r . ,3 I . 
/o ^ cos <p cos (J — ^ sill 0 cos 0 4-^ sin t>, 
Cfx av cu oz 
and therefore from (2), we find 
, d\Ir / dtc ^ \ , /i ^ \ , /I , 3 ^^ • /I 
^ = — ( -g- — 2(ov j cos (p cos ^ ^ d" zojuj sm <p cos 0 ~r ^ 0 
djjb 
Similarly 
and 
3U . 
= gy -fi 2 c(jV cos 0. 
, C\lr C-dr C\!r . 
h:, - cos (h — ^Sin (p 
■'00 01/ dr 
= + 2coiij cos (p — — 2ojv) sin (p 
3V 
= -f- 2(0 (W sin 0 — U cos 0), 
cyp dyjr . „ ; . 3i/( . ^ , 30- ^ 
<3 g - = sm 0 cos 9 gy sin u sin 9 -r yy cos 0 
= — 2ci)V^ sin 0 cos (p -fi -}- 2(oii^ sin 0 sin 0 -f yy cos 0 
= -„ r — 2(oV sm 0. 
ct 
Allowing' for the diftereiices in the notation, the three equations just obtained agree 
with those given by Professor Lamb.'“ If we suppose U, V, W, 0, each proportional 
to they may be written 
* ‘ Hydi’odyuamics,’ p. 344. 
