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ON SKEW SURFACES, OTHERWISE. SCROLLS. 
199 
and then 
HABC + 2a/3y = (ut supra) 
/3 3 ( i) + /3 2 (-ia 2 + -V-a~ 4 3 1 ) + /3( l« 4 -t« 3 + Wa 2 -*pa + pp) 
_a4 + ^a 3 -162a 2 + -i4pa 1210, 
(2A.B — 2a/3) 2a = 
/3 3 (-!) + /3 2 ( fa 2 -2f-a + -6p + /3(-ia^ + ^a 3 -ifia 2 + M9 a _531) 
+ 2a 4 - 36a 3 + 297a 2 - 1215a + 2268, 
((2a) 2 — 2a/3) (2 A + 2a) = 
/3 3 ( |) + /S 2 (-|a 2 +4a-^) + / 3( ^-^tf + ipa 2 -iff£a+241) 
- a 4 + Jyi a 3 - 135 a 2 + if 2 a 1064 ; 
whence, adding these three expressions, 
Order = /3 3 Q) + /3 2 (- §a + 4 ) + /3 Q a 2 -\ a + p) - 6 ; 
and by means of the foregoing expression for the weight, we then have 
Weight — Order = /3 s (±) + /3 2 (— a ) + a 2 + 3a — p) — 11a + 24; 
and therefore 
G (m 4 ) = |/3 x (Weight — Order), 
+ UM_3 a _ 1210, 
= ^ /3 {2/3 3 + /3 2 (— 6a) + /3 (3a 2 + 18a — 26) — 66a + 144}, 
which is right. 
4a 2 — 36a+144, 
Annex No. 4.—Order of Torse (m, n) (referred to, Art. 44). 
We have to find the order of the developable or Torse generated by a line 
meeting two curves of the orders m, n respectively; viz. representing by /¿, v the 
+ 18, 
classes of the two curves respectively, it is to be shown that the expression for the 
Order is 
Torse (m, n) = mv + n/x. 
— 54a+144, 
I remark, in the first place, that, given two surfaces of the orders p and q respectively, 
+72a —224, 
the curve of intersection is of the order pq and class pq(p + q — 2), or as this may 
be written, class =qp (p — 1) + pq (q— 1). Keciprocally for two surfaces of the classes 
-29a + 98; 
p and q respectively, the Torse enveloped by their common tangent planes is of the 
class pq and order qp (p — 1) + pq (q — 1). Now, in the same way that a surface of the 
order p may degenerate into a Torse of the order p, so a surface of the class p 
may degenerate into a curve of the class p; and the class of a curve being p, then 
a +18; 
(disregarding singularities) its order is =p(p — 1). Hence replacing p and p(p— 1) by 
and m respectively, and in like manner q and q(q — 1) by v and n respectively, 
we have mv + n/x as the order of the Torse generated by the tangent planes of the 
a+ 133; 
curves of the orders m and n respectively; where by tangent plane of a curve is to