726 
MR, G. H. DARWIN ON THE SECULAR CHANGES IN 
M^—Mo 3 , 2M 1 M 3 , 2M 3 M 3 , 2M 1 M 3 , |-M 3 3 
Then by adding the squares of the first and second of (23), we have 
2(M 1 3 -M 3 2 ) = + 2raW* +K 4 e 2(x+e) 
+^ 4 e-^- 9) -h2 m° K *e- 2 * + K*e- Z( x +9) .(24) 
From (20) we know that has the form 2A cos (y+B), and —M 3 the form 
2A sin (y + B) ; therefore (M 1 +M 2 )2~* has the form 2A cos (y+^+B), and 
(M x —M 3 )2 -i the form 2A .sin (y-b^7r+B). Hence if we write y—for y in M x 3 —M 3 2 , 
we obtain —2M 1 M 3 . Therefore from (24) we obtain 
-4M 1 M 3V / -l= oTe} lx ~ B) +2<xr 3 /c 3 e 3x +^e~ ix+6) 
-^ 4 e- % ~ 9) -2 rt 2 r 3 *-A- %+() .(25) 
The f — l appears on the left hand side because e? = — (— l) -i , e“; = ( — 1)“\ 
It is also easy to show that, 
2M 3 M 3 = — TS ?J K.e X ~~° — KK)e X -pzn-/c 3 e x + 39 
- -\-7&k(7Z7& — KK)e~ XJ r <7TK 3 e _(x + 29) . . . . (26) 
2M 1 M 3V / —1= — ra- 3 /ce x ~- H +utk(utct — kk)c x d-u7K 3 e* +29 
-\-TZ S Ke~ (x ~ 2e) — 7Zk(tZ7Z — KK)e ~ X — uTK 3 C _(x+39) .... (27) 
-g-— M 3 e =^ — 2txtskk-\- olt 3 k 3 c 28 + zs"K~e~^ .(28) 
It may be here noted that sts7 + kk=1, so that 
-2 uTWKK = -J {zs'-zs-' — AtSTSKK -fi K Z K~) 
These five formulas (24) to (28) are clearly equivalent to the expansion of the 
harmonic functions as a series of sines and cosines of angles of the form ay+jSZ+yiV. 
It remains to explain the uses to be made of these expressions. 
§ 4. The disturbing function. 
In the theory of the disturbing function the differentiation with respect to the 
elements of the orbit of the disturbed body is an artifice to avoid the determination 
of the three component disturbing forces, by means of differentiation with regard to 
