PROFESSOR CAYLEY'S ADDITION TO MR. ROWE’S MEMOIR. 
755 
= ^h m i( n il x -2 J r a .il x :i J r eX-\- n ^r> + n o/ x G + n iH) 
+ n 2 m. 2 ( 
11.3P-3 + ^ + P-o/h “t" n &t J 'G H - ^ 7 /^ 7 ) 
+'>h m s( 
+ n 5 fi 5 + n 6 fi 6 +w 7 /r 7 ) 
+ ex ( 
n 5/ x 5 + 5? 6/ x 6 + W 7/ A 7) 
+ '»5 m 5 ( 
W fl/ X fi + W 7/ X 7) 
+ «6*6( 
n 7 /r 7 ), 
which is 
= E« r m r )?#s+ 6XCt'nm-\S' iifi) -\~h'nm.l < "n[ji-\-l < "n r m l n s fi Sl 
*>r *>r 
We have moreover 
^ 0 
Zlrinfi 
= EbFm/x -f 0~X~ + S"n~W[j., 
%nm 
= %'nm -\-6X -) -X'mn, 
tll/JL 
:=E??/r d - 0X -f- E P/L 
=En + 0 +E"r. 
We next calculate E 
For the singularity 
(a-l). 
to, \m,(to,—/ u,) 
( 2 
— . . . 
/ m x \mx—fj-i __ ^ ^ ^ t . . 
each branch ( 2 —gives a = m 1 —/x l5 and the value of E(«—1) for this singu¬ 
larity is ?h( w i — /M— l)+%( m 2 — 1 ^ 2 — l)-h w 3 ( m 3 —^ 3 — 1 )» which is 
= — Eb?,u — Eh?. 
For the singularity 
c s —m s I 
n s (fi s —m s ) 
( v-s # 4 .... 
z—yp *-™*) gives a=/x, 5 —m 5 , and the value of 2(a—1) for this smgu- 
larity is n b (fj.- — ra 5 — l)-|-?? 0 (|U 6 --m 6 — 1)-f-u 7 (/x 7 — ?n 7 —■ 1), which is 
= E Tljx- 
■t"n. 
For each of the 6 singularities 
/ A\ A.0 
( y—x~ K j 
we have a=X and the value of E(a— I) is = 0(X—I) : this is =0 for the value \=1, 
which is ultimately attributed to X. 
0 E 2 
