s 
360 PROFESSOR A. R. FORSYTH ON 
Substituting we have 
-pj / \ LsJ- n/ , n fW] 2p "W f W 1 3 F>») 
PW=-p^ p W+l^[ ——n>v;,A ~w 
2! lP'(o,„)i 3! + 
l„Sm~ I 5 
1 L p/ K)J 2! 
/ s 3 f/ON O'Ta 'l 
s-\ / \ 7 s~\// \ « I 0 m o m \y v mj 
Q{x m )— ■ (®/«) -j 
and therefore 
A 
/ pq*a) 
7 QW _S * V i + 
^ P"(^») + 
/ 
{P'KOI 2 2! 
P'K) Q,(am) ‘ ‘ • 
-£»«(! di. m S m ~ B M S M ) SBy, • 
correct to the fifth order. Moreover 
<ZU=iT 
m=l V QW 
sjds m ( 1 — A m sJ—B m sJ) 
m=p 7 
^ v i)l _ <> 
or 
> 1=1 P (®Wi) 
)/i = p 7 
U=-2 “ 
>1— ! P'( a *) 
correct to the seventh order. Further 
A„ 
5 -l-'m rj 
~o r- t-r S>n 
o o 7 
X-ni) dx n 
2du /d aij— a m ~^~ + A/ Q(* m ) a^— a„ 
+ 
and 
dx r 
x r —a n 
■ 2a r> r/l s r ds r (1 “ha^^s,- -b a;- m s r ) 
which with the help of (26) gives 
(26) 
(27) 
du m — ds m { 1 A. m s m ) S & r ,nSr ds r (^ 1 -(~a ; . )OT s r | a ^ m s r )(1 A- r s r ~ B/^v ) 
r=l 
where %' denotes that r may receive all values between 1 and p except to. Thus 
'M'm - 
B r =p 
r 2 B r m 
5 r=l 
(28) 
correct to the fifth order ; and as s nl is of the order u this expansion will be sufficient 
for the expansion of U in (27) accurately to the seventh order. The equation (28) 
holds for m= 1, 2, . . . , p. 
