奈良教育大学学術リポジトリNEAR
On a pair of surfaces mutually related, (I)
著者 MATSUMURA Soji
journal or
publication title
奈良学芸大学紀要
volume 6
number 2
page range 9‑15
year 1957‑01‑31
URL http://hdl.handle.net/10105/4934
(9)
On a pair of surfaces mutually related, ( I )
Soji Matsumura
Cosider a surface S related to any conjugate a) system of lines u=const, v=const.;
then the rectangular coordinates x,y,z of its points are particular integrals of Lapl- ace's equation of the form
where
G Ev-FGu EG-F2
E Gu-FEv EvF-GuE
ds2 =Edu2 +2Fdudv + Gdv? ,
(.2)
and ds is the linear element.
If S is the first focal sheet of a congruence, the second sheet Sx is given by
(2) *1-*+^, y^y+XJS-^-M+xJl-.
From this it follows that the coordinates x, y, z of the mean point of the line have the expressions
(3) x = x + ‑ 3X aｫ , y = y + ‑ ‑ % 2 ay Z ‑ Z + ‑ 2 32 so w e hav e 蝣 311
｣ *2 = ｣ x2 + ̲i L .s (j !L ‑¥2 + x^ x‑ I
w e n n ]>> 2 = 1 ,
2>2= 1+ A2 I
i. e .
v !* 2 : , + / E G ' F > Y l 蝣
When the parametric lines upon ]>>2=1 form an orthogonal system, F=0; so we have
2>2= l +^ EG2 Gu2 •E When 2>2 =const., l +_=r^_=co«jf.,
i. e. (3') logG= const. fEdu+ V,
Where F is a function of v only. Fron (3') follows the the theorem:
When S is a sphere and the locus of the mean points is also a sphere, then follows (3').
From this we get, by differentiation with respect to u and v, and easy reductions
by means of these equations themselves and (1), the following :
ClO) Soji Matsumura
I A\
_^f!_ -u /r_*_ 3\ r
á" ~.i._T" : I 5 ' "- )
\ I 37) / ) 3U
/ 1
f-f) -*«4 - + -,--
2 r
and similarly for y and z.
The congruences of Ribaucour may be defined as those for which the developab- les meet the mean surface of the congruence in a conjugate system.
From the above discussion it is clear that the ruled surfaces u=const., v=const.
are the delopables,and consequently, it follows from (4) that the necessary and sufficient condition that the tangents to the curves v=const, on S form a congruence of Ribaucour is that the function -^- and -\- satisfy the condition
A A
å (-f-)
JO-/JO-å + 32l°s(.i/V =0
3V 3 U3V '
i. e. (5) aul2-\uol-\-li,l-Xuv^+2Xu^v=0.
Wenn A=const., a also must be constant by (5).
In a similar manner we find that for the tangents of the curves «=const, to form a congruence of this kind, it is necessary and sufficient that
(6) »(tf/*) »(*/*) _ 32(<?/A) _Q
3U 3V 9H3V
Subtracting these two equations of condition we get a2fogQ/A2) _Q
3U&V '
so that for the tangents to the curves of both families to be congruences of Ribauc- our it is necessary that
o/V=UV,
where U is a function of u alone and V of v alone.
When the condition (5) is satisfied, the point equation of the mean surface becomes
3H3V \ ' 3D 1 3U A 3V
i.e. (7) -^-+lJL -?«-+4--*-=0,
and its invariants are
a ap aA / 1
(81 \ ;-/
-a a -1"~ ^"^
+- A 2 --t-- A2
When h and k are both zero, then
On a pair of Surfaces mutually related, ( I ) (ll)
3l
a-- 3l ,/ 1 \ . _ ^
O- <*
Hh) - 1 I
._' " L +- A 2
-
=o, -+- V -=o,
hence
3l
Equation (9J is the condition that o- ay and -^- are the partial derivatives
A /
of a function a with respect to v and « respectively.
Let
3V 3A 1_ _ _^A_.
X =3V' A 3U'
then each of equations (8) redeces to
»n _,.(3j\iaj
3V3V =0.
This may be written
3k\_ I 3k\
l og \ 3*)- (3*)
3V / \ 3U I
3U
whose integral is /=%([/+V), where U is a function of " alone and V of v alone.
Hence
a- av _ V 1 _ V ,_U+V 2V
A U+V X U-tV U' ' U
and (1) becomes ««,+^-pr«M+^-y8,=o, giving
Ul + V, U*+ Vn U.-i-V.,
.V,;= -*- ' ' å *å .\fz=z £. ' ~ & rj-- J ' ' d
u +v u +v u+v (4)
so the parametric curves the surface S form a conjugate system, where dash means differentiation. Consider a surface F referred to any conjugate system of lines, u=const., v=const.; then the rectangular coordinates *, y, z of its points are particular integrals of Laplace's equation of the form (7). If F is the first focal sheet of a cong- ruence, the second sheet -F'i is given by
,
ax , , ay , 3Z
Xl=x+l^r> yi=~~y+x~^T' Zl =Z rX-^r<
<fU a ll a lt
From this it follows that the coordinates x, y, z of the mean point of the line have the expressions
x=x+-^ -2L, y=y+-L. Jy-t z=z+J^- J?L.
2 3U ' 2 3U 2 3U
This condition that the point equation of 5 may have equal invariants is
(12) Soji Matsumura
rt \ I \
3U 9V
i.e. when I is constant, then a must be a function of v only. Hence, if the tangents to the curve v=const, are to form a congruence of Ribaucour, we must have in cong- ruence of (5)
X=UV,
where U and V are arbitrary functions of u and v respectively.
§2
1. We consider here
3U3V A 911 I BV
(5)
again, when (2) (ss)=l, we get
(3) (si sO = i/a, (4) (sis') =i/ff,
(5) 1=1, when (eie')=1.
where
(6) Si= £+-E., (7) £'=-§-,
7 _ E _ aS
«"-:-•E{" av
so we have the
«« ,-.*.,-./ *y,
iiieureiii: 1ne cumncs uj irw angles ueiween £j, g unu ^\, £å u/e vquut vu X '"*" a ' and I= 1 is the condition that jj, g' touches one another, in which case the surface r, is always a sphere.
2. When the inverse surface of the surface g' with respect to the sphere S is equal to y/ then we have
(9) _?_=2 (-E-, $) f--|-, i.e. flO) A=<r. (10) « the condition sought.
3. We consider the surface (IB x>it=exp.(-±-+-?rv^s.
Then we have
(12) (E"0«. = 0, so j(/' is a surface of Translation.
4. We can easily show that
(E'"Si)=«^. (-J-+-j- w). JCss) +-?-(ss,)}.
When (ee)= 1 , (e'Ei)= 1 > then, from (13) we have
l=exp.( * +_*_»), 25) E
On a pair of Surfaces mutually related, ( I ) (13) so we have the
Theorem : When 5 is a sphere, and g''', g touch each other, then j coincides with j;/'.
5. We consider two surfaces
(16) iu=a%u, (17) lv =llv ,
then from (16), (17) we have
(18) O--0 Uv+Ovlu-Xulv=0.
From (1), (18) we have
(19) e,/(.a-X) =a/X, --*2-=i/X, 0-A
i.e. (20) -JLa-=<t or (21) a = * ^+co«rt.
Theorem : When fl), (16), (17) ex^i, then we have (21).
6. When tj is the inverse surface of i" with respect to the sphere $, then (22) q = -2(.exp. {u/l + o/l.v} z,$')$ +exp.(u/X + a/fcv')£, so we have
(23) (CT) = -2(e*/>. {u/X+a/i»v} j, f((fE) +e.v/>. (w/A+tf/A.^Css) -2(e^>. {«/A+ff/A.«} R f) y(fEO +e*/>. («/A+ff/^««) y (IE).
If we put (ei)= 1, (fl)= 1 in (23) then (24) OjEO=«#.{-^-+-f-«},
so we have the
Theorem : When f, j ore spheres and touch each other, then the cosine of the angle between
%I,«(24).
(6)
7. The equation of the affine moulding surface may be expressed in the form
I x=v, y=exp(_~j 0dvXfi20exp<i,av( f tvedv) + £72), z=exp{- Jydv)(/ tfexpQf ^»edv)+ LT3),
The surface obtained by reciprocal polar with respect to the unit sphere from the surface (.x,y,z) is given by the surface {X, Y, Z) where X, Y, Z are the soluti- ons of the equation:
,, 32w ,/Hv V\ 9W Hu 3W
liiDI " å 1 \ TT -å å å å å å --1 M^ TT- å å "\J•E
9U9V \ H Vj 3U H 3V
When the surface with coordinates (X, Y, Z) is the central one of another surf- ace, in other words, when (1) and (26) coinside, then
( 97) Hv_ vr-_?_ J^i= 1
[l" H V I ' H I '
where
GEv-FGu 1 ,_ EG-F2
°"~ E vn.. T?rr~,'> n A
Gu- FEv EvF-GuE
i^i \ 9
E =V F= s (- G =S( a *: )2.
(14) Soji Matsumura
H=
蝣 / .
x , y , z
