DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
where it will be sufficient to suppose r to have the values 1, 2 ; let 
X r = <pr {t), Ur = i'r ( 0 > 
be a solution of these equations. Substitute in the differential equations 
X r ~ <Pr(t) + £r> Ur ~ V'V (0 + 
and retain only first powers of the quantities £. and which are supposed to be 
small. We thence obtain a system of linear differential equations of the form 
where /3 is the skew-symmetrical matrix of constants given by 
/3 — 0 — 1 0 0 
10 0 0 
0 0 0 -1 
0 0 10 
(so that (3 1 = — ,8), and A is a symmetrical matrix whose elements are functions of t. 
We then have the theorems following :—- 
o 
(a) The roots of the determinantal equation for A, 
)8 _1 A—X| = 0, 
fall into pairs of equal roots of opposite sign ; 
( b ) The determinantal equation for p, 
Q(/3- 1 A)- P | = 0, 
is a reciprocal equation, unaltered by changing p into p~ l . 
To express the proof we require a notation for the matrix obtained from a given 
matrix u by interchanging its rows with its columns, thus placing the element u 7j in 
the [j, i) th instead of the (i, j) th place. This transposed matrix may be denoted by 
trs ( u) or by u. It is easy also to show that 
[ii (?t)] - 1 = trs [ii (— u)\. 
Then (a) is immediate from the obvious relations among determinants expressed by 
A —/3A | = | A-/3X| = | A + /3X|, 
since A = A, /3 — —f3. 
