I’D') 
t’KOPESSOlJ C. j. .JOLY OX OXATEENIoKS 
AND EKOJECTITE OEOMETEY. 
T^-hich has a united plane through a given line. If the line lies in one or more of the 
planes of the developable (442), the quartic degrades in the manner explained. 
1 14. (Jtlierwise we may say tliat (446) and (447) determine a .system of functions 
J wliicli destroys tlie line q + xq' point by point. Or counting unitv as 
one function, it may be said that a tive-.system is inquired to destroy a line point by 
point. Hownver, when the line intersects the critical curve once, twice, or thrice, it 
can be destroyed seriatim liy a fbur-, three-, or two-system of functions. For example, 
in tlie case ot triple intersection we may wnate 
-f y, = t^x -f ty ; 
d +Fn '1-^ + y) — (^u‘^ + b) + 7 ) = 0 (450); 
and, going one step further, in the case of a quadruple chord 
/(Fo, + 7 ) = b ( 7 '^' + 7 ).(451). 
Ihus a quadrujile chord of the critical curve is a line locus of united points of 
a determinate function. And because the number of <piadriiple chords of a curve is 
(‘ Three Itimensions,’ Art. 274) 
d" — 71in' -j- 78/n- — 48ni/i -j- I'S'Ih -f- 12/i') , . (452), 
or 20 for m == 10, ]i = 25, we learn that twenty functions of the four-system have line 
loci of united points—(quadruple chords of the critical curve. 
The formula (314) gives 80 as the order of the surface of triple chords. 
115. IVie locus of a point which determiius a function having a united point in a 
given plane is a sextic surface. 
The functions fy, and A), lieiiig Hamilton’s auxiliary functions for/),(f/) = t\q^q), 
the relations 
IIj, {(q) = t'q ; (‘4 {q) = fq ; Fp ( 7 ) = t"’q .(453) 
are satisfied, provided (q is a united jioint of / {pq), t', t" and t'” being suitable scalars. 
If 7 lies in a given plane, these eipiations, with that of the given plane, afford the 
relations 
sy = 0, S7i/;(/) = o, = S7f;(0 = o . . (454), 
linear in (j and of orders 0 , L, 2 and 3 \\\q>. Fxpressing that <q is a common point, 
we have the etpiation of the sextic surface 
(/, ii;{if g;{1), zv(0) = o.( 455 ). 
116 . Ihe sextic surface has a double curve oj the seventh ondcr answering to qiairs 
oj'united qioints in the qjlane. 
If the first, second and third of equations (454) regarded as planes in q intersect m 
a common line, the fourth plane wall also pass through that line. The condition for 
a common line is 
ul -f vllp (I) d" ivGjf ij) = U 
( 456 ), 
