114 
PROFESSOR CAYLEY ON THE CURVES 
=( a +i)(a+iX*+ix«i+i) 
and the factor in { } being =rm\rm— a— l] 3 , the coefficient of run is 
(a+l){b+l){e+l)(d+l)l-*-Y\\ 
which is 
= -(a+lXHl)(«+l)(^+l)-(«+l)(«+2X* + 3)- 
2°. To obtain the term in A, writing rm — 0, z = 0, and observing that 
[D]'=A + 1, [D]»=(A+1)A, [D]*=(A+1)A(A-1), [D]*=(A + l)A(A-l)(A-2), 
&c. give the terms A, A, — A, +2A, — 6 A, &c. respectively, the term in A is 
(a+l)(b+l)c+l)(d+l) r [-a- 1> . PA 
+ [_«_2] > j3. 1 
+[— a — 3] J y . —1 
+ i. 2 
- «(a+lX« + 2X*+3) 
+ ft (“ + 2)(a + 3) 
+ 7 ( a +3) 
+2S 
3°. For the term in z, writing rm— 0, D=l, and observing that [z]\ [z] 2 , [«] 3 , [z] 4 
give respectively the terms z, —z, 2 z, —6z, this is 
~-td («+lX5+l)(c+l){[-a-l] 3 + [-«-2]V }. 1 
+Xcd (a+lXfl—l) {[— a— 2] 2 +[— a— 3]'a" } . -1 
-tbcd{a— 1) { [_ a _3]>+ a "'}. 2 
+ abed . —6 
where the terms in { } are 
— ( a + 1 — 2)(a 3), («+2- a "X« + 3) and -(a + 3-a'") 
that is, 
— (^H-l)(«+2)(a+3), (c+6Z+2)(a-(-3) and — (6-fc-Fd+3) 
respectively : whence the whole expression is 
Xd (a + V)(b + 1)(<? + 1)(^+ 1) . (« + 2X*+3) 
- Xcd (c+d+2){a+l)(b+\). (« + 3) 
-|-25$C£? {b-{-c-\-d+%)(a-\-l) 
— 6 abed 
the expression multiplying (a-}-2)(a+3) is 
(a+iX«+iXc+iX<*+i)2fc(fl+iX«+iXc+iX^+i>; 
and we have moreover 
(a+lX^ + l)(c+l)(^+l)=(l+a+/3+7 + ^)i 
