116 
PROFESSOR CAYLEY ON THE CURVES 
[a, b , c, d, e]= (a+l)(b-\-l){c-\-l)(d-\-l)(e-\-l)(ct,-\-l)(cc-\-2)(ct,-\-?>)(u-\-i) rm 
+ (a+lX^+lX«+lX^+l)(«+l)f <«+i)(«+-2X*+3)(«+4); 
I — /3 (a+2)(ct+3X«+4) 
— 7 (a+3)(a+4) 
| — 2£ (a-f-4) 
A 
[ —6s 
+ 
%b («+i)(»+i)(c+ix^+iX«+i) 
+ (<?+e+2)(a+lX^+l)(c+l) 
-22<?«fe (c+d+«+3)Ca+lX^+l) 
+6^ic^(^+c + ^+e+4)(a-il-l) 
(“+2)(a+3)(a+4) 
(a + 3)(«+4) 
(a +4) 
—2^abcde 
where a=«-|-5-f c+c£-f-<?, (3=&c., . . . g =ahcde. 
83. The complete functions ( a ), (a, b), (a, b , <?), &c. may be expressed by means of 
the linear terms [a], [a, 5], [a, c], &c. as follows, viz. we have 
(a) = [a] 
(a, b) = toJS] 
+ [«> $]■> 
{a, b, c ) = [«][i][c] 
+ [a][i, <]+[?][«> c]+ c[a, i] 
+ [«, 6, c], 
{«, d)= HP]P]M 
+t[a, b][c, d] 
+%][*, c, d] 
+ [«, 6, ^ 
and so on : this is easily verified for (a, b), and without much difficulty for (a, b , c), but 
in the succeeding cases the actual verification would be very laborious. 
84. The theoretical foundation is as follows. Writing for greater distinctness ( a) m in 
place of (a), we have (a) m to denote the number of the curves C r which have with a given 
curve U m a contact of the order a, and which besides pass through ^r(r+3 ) — a points. 
Let the curve U m be the aggregate of two curves of the orders m, m' respectively, or say 
let the curve XT'" be the two curves m, ml, then we have 
a functional equation, the solution of which is 
(d)m 
