PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
203 
all 23 equations between the 42 quantities 
n,a,l,x; b , Jc , t , q, q,j ; c, A, r, tr, 0, %; 
b\ h\ t', q', 
P,Y,f; B,C, 
P WW; B'C' 
Developments. Article Nos. 6 to 12 
6 . We have 
(a-b-c ){n- 2) =(*-B-0)-6/3-4y-8#, 
(a-2b-3c)(n-2)(n-3)=2(l>-C) 
— 8&— 18/i— 6(5c— 3/3 — 2y— ; 
and substituting these values of tc in the formula 
v!=a(a— 1)— 2cS — 3«, 
and for a its value, =n(n— 1) — 2b— Sc, we find 
n'=n(n— l) 2 — ot{7b-\- 12c) + 45 2 + 85 + 9c 2 -f 15c 
- 8Je- 187?, + 18/3 + 12y + 12i- 9t 
— 2C— 3B— 30, 
where the foregoing equations for a—b — c and a— 2b — 3c show clearly the origin of 
the new terms — 2C— 3B— 30; these express that there is in the value of n 1 a reduction 
= 2 for each cnicnode, =3 for each binode, and =3 for each off-point. 
7. We have (n— 2){n— 8)—n 2 — in-\-8)—a-\-2b-\-8c-\-{ — 4^+6); and making 
this substitution in the equations which contain (n— 2)(n— 3), these become 
«(— 4%+6)=2(£— C)— a 2 — 4^> — 9o- — 2j— 3%, 
5( — 4 w-j-6)= 4 k — 2b 2 — 9/3 — 6y — 3i— f— 2g> — j, 
c( — 4w-f-6)= 6/i — 3c 2 — 6/3 — 4y— 2«‘ — 3<r — 
(Salmon’s equations (C)) ; and adding to each equation 4 times the corresponding equa- 
tion with the factor (n — 2), these become 
a 2 -2a=2(l~C) + 4(*-B)-<r-2;-3& 
2b 2 -2b = M- p+6y+12t-Si+2g-j, 
3c 2 —2c = 6A + 10j3+4^-2^+5 ff -^. 
Writing in the first of these a 2 — 2a—a{a— 1) — a, =w' + 2S+3« — a, and reducing the 
other two by means of the values of r, the equations become 
n'— a= — 2C— 4B+/S— a— 2j— 3%, 
2« Z +/3 + 3;+ e /=2 § , 
3r+c4-2z+x=5<r+/3+40, 
(Salmon’s equations (D)). 
2 f 2 
