84 
PROFESSOR CAYLEY ON THE THEORY OF RECIPROCAL SURFACES. 
as standing for c 2 —c—2h — 3(3 ; only then q and r are not in all cases the classes of the 
two curves respectively. 
3. In the formulae No. 6 et seq., introducing the new singularity we have as 
follows : — 
(a — b — c)(n— 2) =(z— B — 0+2<y) — 6/3— 4y — 3 1, 
(a-2b-3c)(n-2)(n-3)=2(l-C-3a,)-8k-18h-12(bc-3(3-2y-i); 
and substituting these in n'=a(a— 1) — 2b— 3c, and writing for n’ its value 
=a(a— l) — 2b — 3 k, we have, as in the memoir, 
n'=.n(n — 1 ) 2 — 7i{ lb + 1 2c) + 4 b 2 + 85 + 9c 2 +15 c 
-8jfc-8A+18j3+12y+12i--9tf 
— 2C-3B-30; 
viz. there is no term in a. 
Writing (n— 2){n— 3) = «+2&+3c+( — 4w + 6) in the equations which contain 
(n — 2){n~3), these become 
a{ — 4w+6) = 2(S— C)— a 2 — 4% — 9<r—2J— 3%— -15<y, 
b( — 4m+6) = 4 k — 2b 2 — 9/3 — 6y — 3*— 2g — j, 
c(— 4w+6) = Qh — 3c 2 — 6/3 — 4y— 2«— 3<r— 3<y, 
(Salmon’s equations (C)) ; and adding to each equation four times the corresponding 
equation with the factor (n— 2), these become 
a 2 -2a=2(h-C)+4:(z-B)- ( r-2j-3x-3*, 
2£ 2 — 2J=4#— /3+6y+12£— 3«+2g— y, 
3c 2 — 2c= 6/2+ 10/3+4$ — 2«'+5<r — % + a. 
Writing in the first of these a 2 — 2«=?f + 2^ + 3* — a, and reducing the other two by 
means of the values of q, r, the equations become 
n'— a= — 2C— 4B+* — a— 2j— 3^— 3a, 
2$'+/3 + 3*+^=2|, 
3r + c + 2i + 5<r +/3+4 $+&>. 
The reciprocal of the first of these is 
a’=a— n-\-x! — 2/ — 3%' — 20 — 4B' — 3d ; 
viz. writing a=n(n — ]) — 2b — 3c, and z=3n(n— 2) — 6b — 8c, this is 
<r'=4n(n-2)-8b-llc-2j'-3x!-2C'-4:B'-3d; 
and it thus appears that the order d of the spinode curve is reduced by 3 for each off- 
plane d. 
