DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
157 
Repeating now the demonstration given, ‘ Proc. Lond. Math. Soc.,’ April 10, 1902, 
p. 354, let u {j a) denote the (i,j) th element of the matrix Q u, that is, 
similarly, let u~ 2) denote the (i, j) ih element of the matrix QuQu, namely, 
u. l2 u 2j 
(i) 
+ . . 
• + U in U nj l] \ dt. 
and so on. For the region chosen within the star region above explained, when the 
functions u tJ are functions of a complex variable, or for the range of values of t 
adopted when the elements u XJ are functions of a real variable, there will exist a real 
positive quantity M y not exceeded by the absolute value of u tj for the values of t 
involved. Taking a path of integration limited to such values, from the origin t ~ 0 
to t = t, this being a rectifiable curve of length s, let t x be an intermediate point of 
this path, the length of the path from the origin to t x being s v Then we have, 
considering absolute values, 
and in particular 
Similarly, 
| My (1) (t) I — Mi- I ds x < sM v , 
JO 
|^ (l) (b)|^SiMy. 
ud 2) (01^1 (MflSiMy + ... + M tV 5jM, y -) ds x ; 
Jo 
now if M denote the matrix whose element is M^, the ( i , /) th element of the 
matrix M 2 , formed by the product of M with itself, will be 
Mj'iMjj + M !2 M 2j +... + M in M n; -, 
which we may denote by (M 2 )^; thence 
and in particular 
«v la (OI5(M% 
Si ds, ^ is- (M%, 
W(M%. 
We can continue this process. The nest step will be 
