202 
PROFESSOR CATLET ON RECIPROCAL SURFACES. 
deferring for the present the explanation of these singularities. The same letters, 
accented, refer to the reciprocal singularities. Or using “ trope ” as the reciprocal term 
to node, these will be 
C', cnictropes, 
B', bitropes, 
j ' , pinch-planes, 
close-planes, 
& , off-planes ; 
but these present themselves, not in the equations above referred to, but in the reciprocal 
equations. 
3. The resulting alterations are that we must in the formulee write x— B, § — C in 
place of x, § respectively; and change the formulae for c(n — 2), [ah], \bc], into 
c(n-2)=2<r+4p+V+Q, 
[ab] =ab— 2g —j, 
[ac] =ac — 3<r — X 
respectively. 
4. Making these changes, and substituting ior[ab], [etc], \hc] their values, the formulae 
become 
a(n— 2)= x— B+ g>-}-2fr, 
6(n- 2)= g+2(3 + 3y+3t, 
c(n-2)=2<r+4{3 + y+ 6, 
a(n-2)(n-3)=2(b-C) + 3(ac-3 ( r- x ) + 2(ab-2 s -j), 
b(n—2)(n—3)= 4 {ab— 2g— j)-\-3(jbc — 3/3 — 2y— i), 
c(n-2)(n-3)= 6A+ (ac-3 ( r- x ) + 2(bc-3(3-2y-i), 
which replace the original formulae (A) and (B). 
5. For convenience I annex the remaining equations ; viz. these are 
a!= n(n— 1) — 2b— 3c, 
z!=3n(n— 2) — 3b— 8c, 
l'=\n(n-2)(n 2 -3)-{n 2 -n-3)(2b + 3c) + 2b(b-l)+3bc+%c(c-l); 
the equations 
q= ,b 2 -b-2Jc-3y—3t, 
r—c 2 —c — 2 h— 3j3, 
(q, r in place of Salmon’s R, S respectively) ; the equation 
a—d ; 
and the corresponding equations, interchanging the accented and unaccented letters, in 
