PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
205 
8g=(16M— 24)3+( — 15 m+18)c — 83 2 +18 c 2 
+ 2(8*- 1 8 h) - 9(4/3 + y) - 4 ; - 93 - 6 & 
8* = 8m(m - 1 )(m - 2) + 1{ - 32m+ 5 6) + c( - 1 7n + 46) + 83 2 - 1 8c 2 
— 2(8*— 18A) +17(4/3 + y) + 4; +173+6%+ 8B, 
.2 &=w(m-1)(m-2)(m-3)+5(-4#+20w-24)+c(-6m 2 +15m-18) + 123c+184 j 
+( 8*—18A)— 9(4/3 + y) — 93+2C, 
8 m'=8m(m-1) 2 +(-32m+40)3+(-21m+30>+84 j -184 j 
— 2(8*— 18A)+ 21(4/3 +y)—12/+210—18%—16C—24B, 
4='4m(m— 1)(m— 2)+(— 16m+28)3+(— 10m+26)<?+45 2 — 94* 
— (8*— 18A)+10(4/3+y)— 4/+1O0— 6%— 6C— 8B, 
25'=— a-$-n'(n'— 1) — 3c', (n’, 4 supra), 
4+2/+3%'+2C' + 4B'= 4m(m-2)-83-11c, 
g>'— 4/— 6%'— 40— 9B' = — lln(n— 2)+<z(m'— 2)+223+30c (?4, a supra), 
24 + 4/3' + y' + 4= 4(4 — 2), (4, 4 supra) 
4*' — 3(4 + 3/3' + 2^) — 2g>' — / = (_44+6)4+24 2 , (4, 4 supra), 
67i — 2(4 + 3/3' + 2y') — 34 — %'= ( — 44 + 6)4 + 34 2 , (4, 4 supra) ; 
[or in place of either of these, 
8*' — 1 8/4 — 4§' + 94 — 2/ + 3%'= (24 — 34) { (4 — 2)(4 — 3) — a} , (4, 4, 4, a supra)], 
g' + 2/3' + 3y'+3/'=4(4— 2), (4, 4 supra), 
2£'+/3'+34+/-24=0, 
34 +24+%!+ 54— /3'— 43'= 4, (4 supra), 
(twenty-three equations, being a transformation of the original system of twenty-three 
equations). 
11. Forming the combinations 4/+6r, 24/— 8y + 18r (the last of which introduces on 
the opposite side the term +48/), we obtain 
4/+6r= c(5n— 12) — 5y— 18/3 + 34 — 2%, 
— 24/ — 8^+18r= — (8m — 16)3 + (15m — 36)c — 34/3 + 9y+ + + 9$+ 6%, 
equations which are used post, No. 53. 
12. I remark that if there be on a surface a right line which is such that the tangent 
plane is different at different points of the line, the line is said to be scrolar : the section 
of the surface by any plane through the line contains the line once. But if there is at 
each point of the line one and the same tangent plane, then the section of the surface 
by the tangent plane contains the line at least twice ; if it contain it twice only, the line 
is tor sal ; if three times the line is oscular ; and the tangent plane containing the torsal 
or oscular line may in like manner be termed a torsal or an oscular tangent plane. 
These epithets, scrolar, torsal and oscular, will be convenient in the sequel. 
