118 
PROFESSOR CAYLEY ON THE CURVES 
and comparing with the equation for (a, b), we see that [a, b~\ is not to be considered as 
acquiring any divisor when b is put —a, but that the divisor is introduced as a divisor 
of the whole right-hand side of the equation in virtue of the remark as to the divisor 
of the functions (a, b), (a, b, c) ... in the case of any equalities between the numbers 
(a, b, c . . .). This is generally the case, and the foregoing expressions for [a, b\ [«, b, c], 
&c. are thus to be regarded as true without modification even in the case of any equa- 
lities among the numbers a, b, c . . . . 
86. To complete according to the foregoing method the determination of the expres- 
sions for (a), (a, b), . . , we have to determine the linear functions [a], \_a, 5], See., which 
are each of them of the form fm-\-gA -f- hz, where (/' g, h) are functions of r and of 
a, b, See . ; and I observe that the determination can be effected if we know the values of 
(a), (a, b), Sec. in the cases of a unicursal curve without cusps and with a single cusp 
respectively. Thus assume that in these two cases respectively we have 
(a)=(a-\-l)(rm—a), 
(a) = (a -J- \)(rm — a) — a. 
Writing first A=— 1, z=0, and secondly A = — l,*=l,we have 
(a-\-l)(rm— a) =fm—g, 
(a^-l)(rm—a)—a=fm—g-\-h, 
whence 
f=(a-\-V)r, g~[a-\-l)a, h=—a, 
giving the foregoing value 
[a] = (a-f l)r?n + (a-j- l)aA — uz. 
Similarly, for two contacts assume that we have in the two cases respectively 
(a, b)=(a-\-l)(b-\-Vj\rm— a— $] 2 , 
(a, b)={a-\-l){b-\-l)\rm—a—bf—{a(b-\-l)-{-b{a-\-l)}\rm—a—b — Y\ l . 
Starting here from the formula [a, b~] = (a, b) — [«] \b~\ —fm + gA -f- hz, and writing suc- 
cessively A = — 1, z— 0, and A=— 1, z— 1, we have 
(«+l)(5-|-l)[m— a— bf— {(a-\-l)(rm— -a)} {(b-\-V)(rm— b)} —fm—g, 
(<z-f l){b-\-l)\rm— a— bf— {a(b -{-!) + b(a-\- l)}[m— a— b— l] 1 
— {{a-\-l){rm—a)—a} {(b-\-V)(rm—b)—b} -=fm—g-\-li ; 
the first of which, putting therein a-\-b=a, ab=fl, is at once reduced to 
(a-\-V)(b a — l)-|-a(es-|-l)— /3} = fm—g , 
whence f= — {a-{-l)(b-\-l)((x,-\-l)r, g= — (a+l)(5+l)(a(a+l) — j3). And taking the 
difference of the two equations, we have 
— {a(b-\-l)-\-b(<x,-\-l)}(rm—a—b—l) 
+ a{b -f- Y)(rm — b) ■+• b{a -J- 1 )(rm—a) —ah— h, 
that is h=(a+l)(b-\-V)(a-\- b)— ab ; whence [a, b ] has the value above assigned to it. 
87. The actual calculation of [a, b, c] would be laborious, and that of the subsequent 
