112 
PROFESSOR CAYLEY ON THE CURVES 
(a, b, c, d, e)—{a-\-V)(b-\-\)(c-\-l)(d-\-V){e-\-\) 
[m— a ] 5 
+ [m— a— l] 4 a [D] 1 
+ [m- a— 2] 3 /3 [D] 2 
+[m- a — 3] 2 y [D] 3 
+[m— a— 4] ! c$ [D] 4 
.+ * PI 1 
-[2«(a+l)(S+l)(«+l)(^+l) 
[m- a — l ] 4 
+[m— a— 2] 3 a' [D]‘ 
+[rm— a— 3] 2 j3' [D] 2 
+[m— a— 4]V [D] 3 
.+ ^ [D]‘ 
3W 
+[S«fe(«+l)(J+l)(e+l) ' \rm-a- 2] s 
I +[m-a-3]V [D]’ 
| -f [m— a — 4]’/3" [D] 2 
! + / P>] 
W 
-[S«fe(a+l)(5+l)r [m-a-3] 1 1][*]* 
i +[m— a — 4]y" [D] 1 1 
1+ • /3"'[D] 2 J 
+[2bcde(a+ l)f [m-a-4] 1 V][>] 4 
1+ ‘TM 1 / 
— abode [*] 5 . 
78. In all these formulae there is, as before, a numerical divisor in the case of any 
equalities among the numbers «, b, c, &c. And D denotes, as before, the deficiency, viz. 
its value now is -l)(w— 2)— z; or observing that the class n is = 
m 2 — m— 2eS— 3*, we have or say D=l— =1 + A if 
A= — Tfi -|- -|- tt k . 
79. It is to be observed with reference to the applicability of these formulae within 
certain limits only, that the formulae are the only formulae which are generally true ; 
thus taking the simplest case, that of a single contact a , the only algebraical expression 
for the number of the curves C r which have with a given curve U a contact of the order 
«, and besides pass through the requisite number ^r(r+3)— a of arbitrary points, is that 
given by the formula, viz. 
(a) = {a + 1) (r m — a + AD) — ax. 
Considering the curve U m and the order r of the curve C r as given, if a has successively 
