[484 
484] 
THE ORBIT IN THE PLANETARY THEORY. 
545 
C. VII. 
69 
espectively ; 
ar distance 
' + y\ 
viz., this is a linear function of x, x', y, that is of p, q, p', q\ with coefficients which of 
course involve the other variable elements and the time; but it will be remembered 
that ©, ©', 4> are not variable elements, but are absolute constants. The variations 
dd dfL 
of p depend upon and those of q on , and the quantities p, q, p, q',... 
disappear from these differential coefficients ^ ; that is, disregarding periodic 
CLQ OjJ) 
terms, and the variations of the elements, we obtain ^^ as absolute constants, or 
dt dt 
reckoning the time from the epoch belonging to the fixed orbit of m, we have p, q 
as mere multiples of the time (p = At, q = Bt, where A and B are constants); agreeing 
with the statement preceding the investigation. 
Observe that the p, q, as used above, have reference not only to the fixed orbit of 
m, but also to the node thereon of the fixed orbit of m : we may, if we please, write 
p = tan cf) sin (© + 0), q = tan </> cos (0 + 6), that is, p = q sin © + p cos ©, Q = q cos © — p sin © 
(or p = p cos © — q sin ©, q = P sin © + q cos ©), and in place of p, q introduce into the 
formulae p and q, which have reference only to the fixed orbit of m, and similarly 
writing p' — tan cf)' sin (©' + 6'), q' = tan <// cos (© + 6'), instead of p', q' introduce p', q' 
which have reference only to the fixed orbit of m'. 
I remark that a table for the relative positions of the orbits of the eight Planets 
for the Epoch 1st January, 1850, is given in Leverrier’s Annales de VObserv. de Paris, 
t. ii. (1856), pp. 64—66. 
the term