22 MR. E. CUNNINGHAM ON THE NORMAL SERIES 



Now fl (w+cr) = Sl(w)a{Sl- 1 (w)<rQ(w)} (Dr. BAKER, loc. cit., p. 339), and 



~ l , where o = / e ' t 6 ' 1 



and w u is the value of w at t = C, so that 



n(|-4++* 



\^ t' t 



I ,.\-<+e l * \ / /t\ -,'+,' //X-Si'+Si 1 



f /'(f) ' '"A /' a21 (f ' a (f) ' 



/ \t n / \ / \Co/ \ c o/ 



- W O J M //\-9| 2 + , 3 4-1 //\-, 2 + , s 



?.o ,i ....rU - .*" 



I 



there being no exponentials in the last matrix since 



'>' = 0, unless 6>/ = <?/. 



(/I /3 \ /p\ 



-| + )n( -, ), r is a matrix having zero in and to the left of the 

 t* 1 1 \t a j 



diagonal, so that Q ( ) has zeros in the same places. 

 \t / 



r /r\ 



-gQljj) therefore has zeros in and to the left of the diagonal, and also in the (n 1) 



\ L ' 



places to the right of the diagonal, and also wherever F has a zero, and so on. 



r r 



Thus Q-^Q-^... vanishes after a finite number of steps. Further, none of these 



l> v 



expressions contain log t, since F contains no integral powers of t. 

 Thus 



721, 731- 



all the places which were occupied by zeros in F being also occupied by zeros in this, 

 and yy contains only a finite number of powers of t, positive or negative, and no 

 logarithms. 



We may specify a little more exactly the form of the term y {j . A typical term of 

 T/t 2 is 



and Bj0i is not an integer and c,-, is unity or zero. 



