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and the system is finally equivalent to only three independent conditions on account of the identity
(11) (L’ – L)(M — M)(N? – N) = E" A'B'C'. It may be noticed in passing that two of the conditions (9) may be thrown into the form
| M+M NEVN
[ F+ F
N-VNL -VL G - VG
T-rl: M-VM = H-VH" to which others might be added.
The conditions (9) may likewise be put in a form better known, as follows. Developping the first pair of equations, and subtracting the results, we have
(13) GH/F+HF/G + FG /H+v(FGH) = 0; adding the results, we have
(14) Fy(GH)+Gy(HF)+ H/(FG)+FGH — A'B'C' = 0; transponing the last termes of (13) and quadring: FG? H+GH? F+HF?G? +2FGA(Fy(GH)+G/(HF)+H/(FG))=FGH; substituting from (14), and dividing throughout by A'B'C', there finally results:
(15) A'B'C' — A'F – B'G? — C'H+2FGH = 0.
By means of this reduction the system (9) admits of being expressed in a brief manner, as follows: Let
112 = A'H GL
L M N E', then (9) take the form
(17) = 0, y = 0, d=0, = 0. Writing, for convenience,
-(A+B+C) = (A,B,C), (18) 1-(BC+CA+ AB)+F+G? +H2 = (A, B, C)2
| AF+BG? +CH-ABC—2FGH = (ABC)3,
the four conditions (17) may be written:
(03+(ABC),0? +(ABC)20+(ABC)3 = 0,
03 +(BCE), 0+(BCE20+(BCE); = 0, (19)
10+(CEA),0 +(CEA)20+ (CEA)3 = 0,
( 03 +(EAB), 02 +(EAB),0 +(EAB); = 0, and the two conditions resulting from the elimination of a from any three of these equations will be those to which the quantities A, B, C, F, G, H, L, M, N, must be subject, in order that the transformation from (1) to (2) may be possible. But it does not seem worth, while to proceed to the development • of the results, which would be a work of great labour.
It remains to determine the geometrical results, ensuing upon the transformation in question; the principal peculiarities of which depend on the value of the quantity w.
1. Let w=0; then the problem reduces itself to that considered in the latter part of your letter.
2. Let w be a real constant; then the surface will be cut circularly by either of the planes
lix + my +nz + kw = @
l'xt-m'y + n'z + k'w= ' (or, as they may be termed, the planes ā, @'). It may also be noticed that W, W' and ū vanish together.
3. Let w be an imaginary constant; then the surface is cut circularly by either of the imaginary planes ā=0, or @=0, and also by the real byperboloid
(lx+my+ne) (l'x+m'y +n'z) = k'w?.
w = Ag+y+v%; then if the surface be cut by either of the planes
u+kw = 0,
o+kw = 0, i. e. by a plane making angles with the planes u and w, or v and w, the ratio of whose series is = -k; it will be cut also in its intersection with the surface
+y+zo+ c+up+v%) = 0,
and if moreover
W.=w1-, 2, =)-, M, = u V-, v=vy-, the two surfaces
uvt. kui = 0
z? + y2 +7? — (9,x+uy tv,z)= 0 will both intersect (1) in the same curve. This latter surface has a double contact with the sphere
x2 + y +z = const.. and is cut circularly by the plane
w = 0, a plane which cuts the former surface in the straight line
u=0, v=0. It may further be remarked that the plane 1,= 0 cuts the original surface in the same curve as does the cone
Or? + uv = 0, whose cyclic planes are u=0, v= 0. London, March 26. 1853.
13. Lösung der Aufgaben C. und D. in Nr. 21. Band 45
dieses Journals S. 284. (Von dem Herrn Dr. phil. Lottner, Lehrer der Mathematik und Physik an der höhern
· Bürgerschule zu Lippstadt.)