390 
MESSES. C. AND E. CHAMBERS — PLANETARY INFLUENCE 
each be affected by only one of the Earth’s coefficients, viz. p 2 , q 2 , p 3 , q 3 , &c. 
respectively. Again, the least positive integral value of s for which (sf+tg) or 
(13s^F8tf) is a multiple of r or 96 is 8; and therefore, if we may disregard as small 
those terms in the expression for the period of Venus which repeat themselves eight or 
more times in that period, the quantities a u b„ a 2 , Z> 2 , a 3 , b 3 &c., being unaffected by the 
disturbance due to the planet Venus, will sensibly represent the true coefficients of the 
expression for the Earth’s disturbance variation. In a similar manner it may be shown 
that A„ Bj, A 2 , B 2 , A 3 , B 3 &c. of equations (7) are sensibly equal to the true coefficients 
of the expression for the period of Venus ; for the least integral value of s for which 
( s^ft)f or 13 (s=p£) i s a multiple of 96 is s=96if£, so that only very high terms, in 
the expression for Venus, would affect the values of the coefficients of the earlier terms ; 
and further, since the least positive integral values of s and t which make (sg^tf) or 
8sipl3£ a multiple of 96 are eleven and eight respectively, and the corresponding 
terms repeating themselves eleven and eight times respectively in the periods of the 
Earth and Venus, they may, as before, be neglected. 
23. But we have adopted for thirteen periods of Venus the approximate time 8 years 
or 2922-05 days, instead of the true time 2921-11 days, which is less by 0'94 of a day. 
Having worked out the question in Section III. for three pairs of coefficients only, we 
will confine the examination to that number and to the Easterly disturbance variation 
for the sidereal period of the planet ; and it will suffice that we determine the second 
approximations to the true coefficients, rejecting terms involving i 2 , i. e. that we apply 
equation (35). 
The first approximations are 
P, = — 4-591 ; Q 1= — 1-199; P a = + 1*428; Q 2 = +1-055 ; P 3 = + 3*422 ; Q 3 = -2-786; 
the angle 
and the angle 
2 cn 3x13 
Z n 3 x 96 
2t=48° 45', 
cAx 2 err cAx 2tt 3 X 0*94 
cx ’ n x n 224‘7 
3x96 
= 0'-94; 
and the greatest value of sni is 2x(3x96)x0'-94 = 9 o 2' (cA% being the error in time 
in thirty-nine periods of Venus). Consequently sni being a small angle, the case is one 
to which the investigation in Section III. applies ; therefore 
^P.+A^ — 4-591 +-044= — 4-547," 
q— Q 1 +B 1 «=—l-199-T80 = — 1-379, 
i> 2 =P 3 +A a i= +1-428-' -088= +1-340, 
?2 =Q a +B a ^= + l-055+-115 =+l-170, > 
^ 3 =P S +A 3 i= + 3-422 + -330= + 3-752, 
?8 =Q3+B 3 «=— 2-786+-406= —2-380,. 
from which has been constructed the 
interrupted curve (Plate 53. fig. 6), 
which is seen at a glance to be almost 
identical with the thick curve con- 
structed from the first approximations 
Pi, Q„ &C. 
24. We may now examine how the sidereal disturbance period of Mercury affects the 
