AND ON THE REMOTE HISTORY OF THE EARTH. 
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expanded in a series of solid harmonics of r, 9, </>, each multiplied by a simple time 
harmonic, which will involve n and fl. 
For brevity of notation nt, fit are written simply n, fl, but wherever these symbols 
occur in the argument of a trigonometrical term they must be understood to be multi¬ 
plied by t the time. 
We have 
cos PM = sin 9 cos MR+ cos 9 sin MR sin MRQ 
and 
cos MR= cos MN cos NR+ sin MN sin NR cos i 
= cos f2 sin ((f)—n) -f- sin fl cos ((f)—n) cos i 
also 
sin MR sin MRQ= sin MQ= sin 12 sin i 
Therefore 
cos PM= sin 9 sin ((f) — n) cos /2-f- sin 9 cos ((f)— n) sin fl cos i-\- cos 9 sin fl sin i 
=-^sin # {sin [<£ — (« —/2)]-j- sin [(f>— (w+/2)]} 
+ ^sin 9 cos i[sin \_(f)—(n —42)]— sin [(f) —(n +42)]} + cos 9 sin fl sin i 
Let 
Then 
i . i 
p= cos q= sm - 
cos PM=p 3 sin 9 sin [(f)—(n—fl)]-\-2pq cos 9 sin fl-\-q 3 sin 9 sin [(f)— (n -1-/2)] . (2) 
Therefore 
cos 3 PM=^-y» 4 ’ sin 3 9{ 1 — cos [2(f) — 2(n — /2)]} + 2 p~cp cos 3 9(1 — cos 2/2) 
sin 3 9[1 — cos [2(f)— 2(n-p/2)]} -\-2p s q sin 9 cos 9{ cos ( <f> — )i)— cos \ (f> — (n — 2/2)]} 
+ 2 pep sin 9 cos 9 {cos [(f> — (n + 2/2) ]— cos (c f> — n)} pp~p sin 3 9{ cos 2/2— cos ( 2(f )—2 n)} 
Then collecting terms, and noticing that 
\('P [J r < f) rin 3 0 + 2jfq* cos 3 0=-g—^(1 — GjPg' 3 )^ — cos 3 9) 
we have 
— 8 = cos 3 PM — \ 
WTT* O 
= — \ sin 3 9 {y> 4 cos [2(f) —2 (n— /2)] +2 p z q~ cos [2(f) — 2n\ + q x 'cos [2(f) — 2 (n 4- fl) ]} 
— 2 sin 9 cos 9 [p^qcos [(f) — (n — 2/2)] — pq(p 2 —q ~) cos ((f) —n) —pep cos [<£ — (» + 2/2)] } 
+ (^— cos 3 #) {3p 3 g ,3 cos2/2-|-|(l — 6^ 3 ^ 3 )}.(3) 
Now if all the cosines involving (f> be expanded, it is clear that we have V consisting 
