THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
731 
Before making these multiplications, it must be considered what are the terms 
which are required for finding secular changes in the elements, since all others are 
superfluous for the problem in hand. 
Such terms are clearly those in which 6 and 6' are wanting, and also those where 
6—6' occurs, for these will be wanting in 6 when Diana is made identical with the 
moon. It follows therefore that we need only multiply together terms of the like 
speeds. In the following developments all superfluous terms are omitted. 
Semi-diurnal terms. 
Y'2_V'2 ¥ 3 _392 - 
These are 2X'Y' £§£+2 2 
If we multiply (24) (with accented symbols) by (32), and (25) (with accented 
symbols) by (33), and subtract the latter from the former, we see that y disappears 
from the expression, and that, 
8X'Y')£§^ + 2(X ,2 — Y'~)(%' 2 —§9 2 ) = First line of (24) X second of (32) 
fl-Second of (25) X first of (33) 
Then as far as we are concerned 
2XT?P+ 
n X' 2 -Y ' 3 £ 2 -if 
2 T ^ 
=i[F jvfmW ~ 6) ~^ + 4Fsr 3 k V V 3 e- 3f +F 
+i[F 1 ^W%- 2M+2f > + 4F +F^W (9 '- 0)+2f -] . (37) 
If y had been accented in the X' functions, we should have had 2(y—y') in all the 
indices of exponentials of the first line, and —2(y—y') in all the indices of the second 
line. These three pairs of terms will be called W I? W ir , W m . 
Diurnal terms. 
These are 2Y' Z'^Z+2X'ZyZ. 
If the multiplications be performed as in the previous case, it will be found that y 
disappears in the sum of the two products, and, as far as concerns terms in &—6 
and those independent of 6 and 6', we have 
2Y'Z'gZ+2X'Z'$Z 
— ? 'k e} {e -j- GtrrK(t3 , S’ — kk)ts’k (trr ct —k'k) fi- G 2 STK 3 73-V 3 e“ 2(9 -0) "§ s 
fi- G 1 CT 3 K7rr ,3 /c , e“ 2(0 ~ e)+ £ 1 fi-GCTK(syrar— kk)z^'k'Izj'tst'~~ k K')e s fi-Go 5rfcW'' i e 2<8 "' fl)+K ’ . (38) 
If y had been accented in the X' functions we should have had y—y' in all the 
