Green G.M. - On Isothermally Conjugate Nets of Space Curves(1915)(6).pdf
7
MATHEMATICS: G . M . GREEN this type. KHSO4 affords a notable example where three curves o f this type meet i n a triple point. F or a liquid, A s always decreases with increasing temperature on either a rising or a falling curve. O n th e rising transition curves there are 3 7 cases o f normal variation o f A v a n d 5 o f abnormal variation; o n t h e falling curves 8 normal a n d 8 abnormal cases. Th e relative compressibility, thermal expansion, a n d specific heat o f neighboring phases i s significant. I t i s natural to expect that t e phase o f smaller volume will have the smaller compressibility a n d thermal expansion, an d that the phase stable a t t h e higher temperature will have t h e higher specific heat. I f w e call this behavior normal, then o n rising curves w e find 9 cases o f normal a n d 1 1 o f abnormal compressi- bility, a n d o n falling curves 1 normal a n d 7 abnormal. T h e expansion shows 5 normal a n d 7 abnormal cases o n rising curves a n d 2 normal a n d 4 abnormal on falling curves. C p i s normal i n 5 cases a n d abnormal in 7 cases o n rising curves, an d normal in 6 cases an d abnormal in 1 on falling curves. T h e fact o f abnormal C . i s o f considerable significance from t h e point o f view o f h e quaatum hypothesis. I t means i f w e m a y apply t h e same consideratios to C p a s to C , , which i s usually done) that th e specific heat curves o f th e t w o modifications cannot b e o f t h e same characte, b u t that somewhere between t h e transition point an d absolute zero t h e o n e which i s lower a t t h e transition point must cross a n d l i e above t h e other. I n addition t o th e substances enumerated above, about 1 0 0 others have been examined without finding other forms. 1 . W . Bridgman, Proc. Amer. Acad., 47,439-558 1912); Physic. Rev., 3, 126-203 1914). O N ISOTHERMALLY CONJUGATE NETS O F SPACE CURVES B y Gabriel M . Green DEPARTMENT O F MATHEMATICS. HARVARD UNIVERSITY Premted t o t b e Academy. Auust 1 0 , 1915 Biandchi has. called a parametric n e t of curves o n a surface isother- mally conjugate if , when t he surface i s referred o these curves, the second fundamental form, D d u 2 + 2 D ' d d v + D dvi, m a y b y a trans- formation u = U u ) , v = V v ) b e made t o take on t he same shape a s does t h e first fundamental form when t h e parametric n e t i s iso- thermal; i . e . , t h e parametric n e t i s isothermally conjugate i f D' = 0 , D = D . These nets have lately attained increased importance, s o that Wilczynski s recent geometric interpretation2 o f Bianchi s condi- 5 1 6
Transcript of Green G.M. - On Isothermally Conjugate Nets of Space Curves(1915)(6).pdf
7/25/2019 Green G.M. - On Isothermally Conjugate Nets of Space Curves(1915)(6).pdf
http://slidepdf.com/reader/full/green-gm-on-isothermally-conjugate-nets-of-space-curves19156pdf 1/6
MATHEMATICS:
G .
M.
GREEN
t h i s
t y p e .
KHSO4
a f f o r d s
a
n o t a b l e
e x a m p l e
w h e r e
t h r e e c u r v e s
o f
t h i s
t y p e
m e e t
i n
a
t r i p l e
p o i n t .
F o r
a
l i q u i d ,
A s
a l w a y s
d e c r e a s e s
w i t h
i n c r e a s i n g
t e m p e r a t u r e
o n
e i t h e r a
r i s i n g
o r
a
f a l l i n g
c u r v e .
On
t h e
r i s i n g
t r a n s i t i o n
c u r v e s t h e r e a r e
3 7
c a s e s o f n o r m a l v a r i a t i o n
o f
Av a n d
5
o f
a b n o r m a l
v a r i a t i o n ;
o n t h e
f a l l i n g
c u r v e s 8
n o r m a l
a n d
8
a b n o r m a l
c a s e s .
T h e r e l a t i v e
c o m p r e s s i b i l i t y ,
t h e r m a l
e x p a n s i o n ,
a n d
s p e c i f i c
h e a t
o f
n e i g h b o r i n g
p h a s e s
i s
s i g n i f i c a n t .
I t
i s
n a t u r a l t o
e x p e c t
t h a t
t h e
p h a s e
o f
s m a l l e r
v o l u m e
w i l l
h a v e
t h e s m a l l e r
c o m p r e s s i b i l i t y
a n d t h e r m a l
e x p a n s i o n ,
a n d
t h a t t h e
p h a s e
s t a b l e a t t h e
h i g h e r
t e m p e r a t u r e
w i l l
h a v e
t h e
h i g h e r
s p e c i f i c
h e a t . I f
w e
c a l l t h i s
b e h a v i o r
n o r m a l ,
t h e n
o n
r i s i n g
c u r v e s
w e
f i n d
9
c a s e s o f
n o r m a l a n d
1 1
o f
a b n o r m a l
c o m p r e s s i -
b i l i t y ,
a n d
o n
f a l l i n g
c u r v e s
1
n o r m a l
a n d 7
a b n o r m a l . T h e
e x p a n s i o n
s h o w s
5
n o r m a l
a n d
7
a b n o r m a l c a s e s
o n
r i s i n g
c u r v e s
a n d
2
n o r m a l
a n d
4
a b n o r m a l
o n
f a l l i n g
c u r v e s .
C p
i s
n o r m a l
i n
5
c a s e s
a n d
a b n o r m a l
i n
7
c a s e s
o n
r i s i n g
c u r v e s ,
a n d n o r m a l
i n 6
c a s e s
a n d
a b n o r m a l
i n 1
o n
f a l l i n g
c u r v e s .
T h e
f a c t
o f
a b n o r m a l
C .
i s
o f
c o n s i d e r a b l e
s i g n i f i c a n c e
f r o m t h e
p o i n t
o f v i e w
o f
t h e
q u a a t u m
h y p o t h e s i s .
I t m e a n s
i f
w e
m a y
a p p l y
t h e
s a m e
c o n s i d e r a t i o s
t o
C p
a s
t o
C , ,
w h i c h
i s
u s u a l l y
d o n e )
t h a t
t h e
s p e c i f i c
h e a t
c u r v e s
o f
t h e
t w o
m o d i f i c a t i o n s
c a n n o t
b e
o f t h e
s a m e
c h a r a c t e ,
b u t
t h a t
s o m e w h e r e b e t w e e n
t h e
t r a n s i t i o n
p o i n t
a n d
a b s o l u t e
z e r o
t h e
o n e
w h i c h i s
l o w e r a t
t h e
t r a n s i t i o n
p o i n t
m u s t
c r o s s
a n d l i e
a b o v e
t h e
o t h e r .
I n a d d i t i o n
t o
t h e
s u b s t a n c e s e n u m e r a t e d
a b o v e ,
a b o u t
1 0 0
o t h e r s
h a v e b e e n e x a m i n e d
w i t h o u t
f i n d i n g
o t h e r f o r m s .
1
.
W .
B r i d g m a n ,
P r o c .
A m e r .
A c a d . , 4 7 , 4 3 9 - 5 5 8
1 9 1 2 ) ;
P h y s i c .
R e v . ,
3 ,
1 2 6 - 2 0 3
1 9 1 4 ) .
O N
ISOTHERMALLY
CONJUGATE
NETS
O F
SPACE
CURVES
B y
G a b r i e l
M .
G r e e n
DEPARTMENT
O F
MATHEMATICS.
HARVARD
UNIVERSITY
P r e m t e d
t o
t b e
A c a d e m y .
A u u s t
1 0 ,
1 9 1 5
B i a n d c h i
h a s .
c a l l e d
a
p a r a m e t r i c
n e t
o f
c u r v e s
o n
a
s u r f a c e i s o t h e r -
m a l l y
c o n j u g a t e
i f ,
w h e n
t h e
s u r f a c e
i s
r e f e r r e d
t o
t h e s e
c u r v e s ,
t h e
s e c o n d f u n d a m e n t a l
f o r m ,
D d u 2
+
2 D '
d
d v
+
D
d v i ,
m a y
b y
a
t r a n s -
f o r m a t i o n u
=
U
u ) ,
v
=
V
v )
b e
m a d e
t o t a k e
o n
t h e s a m e
s h a p e
a s d o e s
t h e
f i r s t
f u n d a m e n t a l
f o r m
w h e n
t h e
p a r a m e t r i c
n e t i s i s o -
t h e r m a l ;
i . e . ,
t h e
p a r a m e t r i c
n e t
i s
i s o t h e r m a l l y
c o n j u g a t e
i f D '
=
0 ,
D
=
D .
T h e s e
n e t s
h a v e
l a t e l y
a t t a i n e d i n c r e a s e d
i m p o r t a n c e ,
s o
t h a t
W i l c z y n s k i s
r e c e n t
g e o m e t r i c
i n t e r p r e t a t i o n 2
o f
B i a n c h i s
c o n d i -
5 1 6
7/25/2019 Green G.M. - On Isothermally Conjugate Nets of Space Curves(1915)(6).pdf
http://slidepdf.com/reader/full/green-gm-on-isothermally-conjugate-nets-of-space-curves19156pdf 2/6
MATHEMATICS: G . M .
GREEN 5 1 7
t i o n
i s
o f
g r e a t
i n t e r e s t .
I n
t h e
p r e s e n t
n o t e ,
w e
p r o p o s e
t o
g i v e
a
n e w a n d
s i m p l e
g e o m e t r i c
c h a r a c t e r i z a t i o n
o f
i s o t h e r m a l l y
c o n j u g a t e
n e t s w h i c h i s
e n t i r e l y
d i f f e r e n t
f r o m
W i l c z y n s k i s .
L e t
y
1 ,
y y 2
y 3 )
y 4 ,
b e
t h e
h o m o g e n e o u s
c o o r d i n a t e s
o f
a
p o i n t
i n
s p a c e ,
a n d
l e t t h e
f o u r f u n c t i o n s
y k )
= f k
u ,
)
k = 1 ,
2 ,
3 ,
4 )
1 )
d e f i n e
a
s u r f a c e
S y
o n w h i c h
t h e
c u r v e s u =
c o n s t . ,
v
=
c o n s t . f o r m
a
c o n j u g a t e
n e t .
T h e n t h e
y
s
s a t i s f y
a
c o m p l e t e l y
i n t e g r a b l e
s y s t e m
o f
t w o
p a r t i a l
d i f f e r e n t i a l
e q u a t i o n s
o f
t h e
f o r m s
y , ,
=
a y , ,
b y ,
c y ,
d y ,
Y u ,
=
b y ,
+
c y .
+
d y .
T h e
s e c o n d
o f
t h e s e
i s o f
t h e
f a m i l i a r
L a p l a c e
t y p e ,
c h a r a c t e r i s t i c
o f
c o n j u g a t e
n e t s ;
t h e f i r s t s h o w s t h a t
t h e
c o n j u g a t e
n e t
d e f i n e d
b y
e q u a -
t i o n s
1 )
i s
i s o t h e r m a l l y c o n j u g a t e
i f
a n d
o n l y
i f
l o g
a = O .
3 )
b u b v
T h e
c o e f f i c i e n t s
i n
e q u a t i o n s
2 )
a r e
n o t
a r b i t r a r y ,
b u t a r e
s u b j e c t e d
t o c e r t a i n
i n t e g r a b i l i t y
c o n d i t i o n s .
O n e
o f
t h e
r e l a t i o n s
y i e l d e d
b y
t h e s e
c o n d i t i o n s
i s
t h a t 4
-
( b +
2 c )
c_
l o g
a ,
d v
u
au
) v
o r
b , + 2 c = 2 b - )
l o g
a .
4 )
T h e
m i n u s
f i r s t
a n d f i r s t
L a p l a c e
t r a n s f o r m s
o f
t h e
p o i n t
y
a r e r e -
s p e c t i v e l y
p
= y , - c y ,
= y , - b y ,
w h i c h
r e p r e s e n t
c o v a r i a n t
p o i n t s
o n t h e
t a n g e n t s
a t
y
t o
t h e
c u r v e s
o f
t h e
n e t
p a s s i n g
t h r o u g h
y .
T h e s u r f a c e
S ,
i s
t h e
s e c o n d f o c a l
s h e e t
o f t h e
c o n g r u e n c e
o f
t a n g e n t s
t o
t h e
c u r v e s
v
=
c o n s t .
o n
S y ,
a n d
S ,
i s t h e
s e c o n d
f o c a l ,
s h e e t
o f
t h e
c o n g r u e n c e
o f
t a n g e n t s
t o t h e c u r v e s
u
=
c o n s t . o n
S . .
L e t
u s ,
w i t h
W i l c z y n s k i , 2
c a l l t h e l i n e
p a
c o r r e s p o n d -
i n g
t o
t h e
p o i n t
y
t h e
r a y
o f t h e
p o i n t
y ,
a n d
t h e
t o t a l i t y
o f
r a y s ,
w h i c h
f o r m
a
c o n g r u e n c e ,
t h e
r a y
c o n g r u e n c e .
T h e
o s c u l a t i n g p l a n e s
o f t h e t w o
c u r v e s
u
=
c o n s t .
a n d
v
=
c o n s t .
a t
a
p o i n t
y
m e e t
i n
a
l i n e w h i c h
p a s s e s
t h r o u g h y
a n d i w h i c h
W i l c z y n s k i
7/25/2019 Green G.M. - On Isothermally Conjugate Nets of Space Curves(1915)(6).pdf
http://slidepdf.com/reader/full/green-gm-on-isothermally-conjugate-nets-of-space-curves19156pdf 3/6
MATHEMATICS:
G . M .
GREEN
c a l l s
t h e
a x i s o f t h e
p o i n t
y .
T h e
t o t a l i t y
o f
a x e s ,
w h i c h
c o r r e s p o n d
t o
a l l t h e
p o i n t s
y
o f t h e
s u r f a c e
S y ,
c o n s t i t u t e
a
c o n g r u e n c e ,
t h e
a x i s
c o n g r u e n c e .
We
m a y
w r i t e
t h e
f i r s t
o f
e q u a t i o n s
2 )
i n t h e
f o r m
y,
-
b y u
- d y =
a y ,
+
c y , .
T h e l e f t - h a n d
m e m b e r
r e p r e s e n t s
a
p o i n t
i n
t h e
o s c u l a t i n g p l a n e
t o t h e
c u r v e
v
=
c o n s t . ,
a n d
t h e
r i g h t - h a n d
m e m b e r
a
p o i n t
i n
t h e
o s c u l a t i n g
p l a n e
t o
t h e c u r v e
u
=
c o n s t . ,
a t
y .
T h e r e f o r e ,
s i n c e t h e
c o o r d i n a t e s
a r e
h o m o g e n e o u s ,
t h e
p o i n t
z=y+
- y ,
a
l i e s
o n t h e l i n e o f i n t e r s e c t i o n
o f t h e t w o
o s c u l a t i n g
p l a n e s ,
a n d
t h e
l i n e
y z
i s
t h e
a x i s
o f
t h e
p o i n t
y .
We
m a y
d e t e r m i n e
t h e
d e v e l o p a b l e s
o f
t h e
a x i s
c o n g r u e n c e
a s
f o l -
l o w s .
I f
t h e
p o i n t
y
m o v e s t o t h e
p o i n t
y
+
d y ,
t h e
p o i n t
z
m o v e s
t o
z
+
d z ,
w h e r e
d y
=
y ,
d u
+
y ,
d v a n d
d z
=
z ,
d u
+
z
d v .
We
w i s h
t h e
l i n e
y z
t o
g e n e r a t e
a
d e v e l o p a b l e .
T h i s w i l l
h a p p e n
i f a n d
o n l y
i f
t h e
f o u r
p o i n t s
y ,
z ,
y , +
d y ,
z
+
d z
l i e
i n a
p l a n e ,
o r w h a t
i s
t h e
same
t h i n g ,
i f
t h e
p o i n t s
y , z ,
y ,
d u
+
y ,
d v ,
z ,
d u
+
z ,
d v
a r e
c o p l a n a r .
We
h a v e
o n
d i f f e r e n t i a t i o n
o f
e q u a t i o n s
2 )
y o m
=
a C 2 )
i 1 2 ) y ,
y 1 2 ) y
1 2 ) y ,
y
=
a O 3 ) y ,
+
t O S ) y,
y O S ) y ,
5 O 0 ) y ,
w h e r e
i n
p a r t i c u l a r
a 1 2 )
=
C t , 1 )
= - b 2
+
b ,
y 1 2
=
b c
+
c
+
d ,
c
_ 1
a 0 8 ) = b _ -
-
l o g a ,
8 ) =
b c
+
b - b ,
+
d ) ,
5 ) .
a b v
a
Y o 8
=
[
b c + c c
-
b )
+ c - c , - - d ] ,
a
s o
t h a t
o n
u s i n g
t h e s e a n d
e q u a t i o n s
2 )
we f i n d
Z
=y
c
y .
+ y ,
a
a
_
_y
+[ 1 2)+c
+ c)Iy,+Oy,
c
y i = , ,
yw+
^ )
y
=
b i
a l o g
a )
y
[ ) y ,
[ )
y
J .
+
) Y
a= y , + v+ ~ ) ~ y .
5 1 8
7/25/2019 Green G.M. - On Isothermally Conjugate Nets of Space Curves(1915)(6).pdf
http://slidepdf.com/reader/full/green-gm-on-isothermally-conjugate-nets-of-space-curves19156pdf 4/6
MA
THEMATICS:
G . M.
GREEN
i n
w h i c h t h e
c o e f f i c i e n t s
o f
y
d o n o t c o n c e r n
u s .
C o n s e q u e n t l y ,
zdu+zdv
=
[ c ' d u +
( b '
-
l o g
a )
d v ]
y ,+
[ B 1 +
b
) d u + ( 0 d v ]
y
+
[ ( 2
+
) )
d u
0 3 )
+
d v
y
) y .
N o w ,
i f
t h e
p o i n t s
y ,
z ,
d y ,
d z
a r e
t o
b e
c o p l a n a r ,
t h e d e t e r m i n a n t o f t h e
c o e f f i c i e n t s
o f
y , ,
y , , y ,
i n t h e
e x p r e s s i o n s
f o r
z ,
d y ,
d z m u s t
v a n i s h ;
o n
e x p a n s i o n
t h i s
d e t e r m i n a n t
y i e l d s
t h e
q u a d r a t i c
i n
d u :
d v ,
a
[ y 1 2 )
+
)
]
d u 2 -
Sdudv
-
a 0 3 ) d v 2
= 0 ,
6 )
w h e r e ,
o n
u s i n g
4 ) ,
w e
f i n d
Z
=
d
+ a b 2
- c 2
b c
+
b c
a b
-
c.
7 )
T h e
q u a d r a t i c
6 )
d e t e r m i n e s
t h e
d i r e c t i o n
i n w h i c h
y
m u s t
m o v e ,
i n o r d e r
t h a t
t h e
a x i s
y z
m a y
t r a c e
o u t
a
d e v e l o p a b l e ;
t h e r e
a r e t w o
s u c h
d i r e c t i o n s
a t e a c h
p o i n t
o f
S y .
We
m a y
r e g a r d
6 )
a s
a
d i f f e r e n -
t i a l
e q u a t i o n
d e f i n i n g
a
n e t
o f
c u r v e s
o n
S y
h a v i n g
t h e
p r o p e r t y
t h a t
i f
t h e
p o i n t
y
t r a c e s o u t
a c u r v e o f t h i s
n e t ,
t h e
c o r r e s p o n d i n g
a x i s g e n -
e r a t e s a
d e v e l o p a b l e
s u r f a c e .
We
c a l l t h e t w o
c u r v e s
o f t h e n e t
w h i c h
p a s s
t h r o u g h
t h e
p o i n t
y
t h e
a x i s
c u r v e s
o f
t h e
p o i n t
y .
I n
l i k e
m a n n e r ,
w e
m a y
d e t e r m i n e
t h e
d e v e l o p a b l e s
o f t h e
r a y
c o n -
g r u e n c e ,
i . e . ,
t h e
n e t o f
c u r v e s
o n
S y
h a v i n g
t h e
p r o p e r t y ,
t h a t
i f
t h e
p o i n t
y
t r a c e s
o u t
a
c u r v e
o f t h e
n e t ,
t h e
c o r r e s p o n d i n g
r a y
t r a c e s
o u t
a
d e v e l o p a b l e
o f
t h e
r a y
c o n g r u e n c e .
T h e
d i f f e r e n t i a l
e q u a t i o n
d e f i n -
i n g
t h i s
n e t
o f
c u r v e s ,
w h i c h
w e
c a l l
t h e
r a y
c u r v e s ,
i s w i t h o u t
d i f f i c u l t y
f o u n d
t o
b e
a
H
d u 2
- d u d v -K d v 2 =
O ,
8 )
w h e r e
Z
i s
g i v e n
b y
7 ) ,
a n d
H
=
d
b c
-
b ,
K
=
d
+
b c
c
9 )
a r e
t h e
L a p l a c e - D a r b o u x
i n v a r i a n t s
o f t h e
g i v e n
c o n j u g a t e
n e t .
I f we
us e
9 ) ,
we
f i n d
f r o m
5 )
t h a t
a
0 3 )
=
H
+
2 b
- b , ,
7 1 2
+
)
=
K
+
2 c
,
t h e
l a t t e r
o f w h i c h
b e c o m e s ,
o n
u s e o f
4 ) ,
1 2
+
C )
K
2 b ' - b
_-
l o g
a .
a
U
b
u
b v
5 1 9
7/25/2019 Green G.M. - On Isothermally Conjugate Nets of Space Curves(1915)(6).pdf
http://slidepdf.com/reader/full/green-gm-on-isothermally-conjugate-nets-of-space-curves19156pdf 5/6
MATHEMATICS:
G .
M .
GRIEN
T h e
d i f f e r e n t i a l
e q u a t i o n
6 )
o f
t h e
a x i s
c u r v e s
m a y
t h e r e f o r e
b e
w r i t t e n
aK+
2 b - b , -
l o g
a
d 2 - -
d u
d v
- H
b
b)
d v
=
0 .
1 0 )
a u
a v
T h e
d i f f e r e n t i a l
e q u a t i o n
o f
t h e
a s y m p t o t i c
c u r v e s
i s
a d u 2
+ d v 2
=
0
1 1 )
T h e
p a i r
o f
a s y m p t o t i c
t a n g e n t s
a t
y
i s o f
c o u r s e
h a r m o n i c a l l y
s e p a -
r a t e d
b y
t h e
t a n g e n t s
t o t h e
c u r v e s
o f
o u r
c o n j u g a t e
n e t .
T h e
d i f f e r -
e n t i a l
e q u a t i o n
a d u - d v 2
= O
1 2 )
d e f i n e s
a
new
n e t
o f c u r v e s .
I t
e v i d e n t l y
h a s
t h e
p r o p e r t y ,
t h a t t h e
t a n g e n t s
t o
t h e t w o
c u r v e s
o f
t h e n e t a t
t h e
p o i n t
y
s e p a r a t e
h a r m o n i -
c a l l y
b o t h
t h e
p a i r
o f
a s y m p t o t i c
t a n g e n t s
a n d t h e
t a n g e n t s
t o t h e
t w o
c u r v e s
o f
o u r
c o n j u g a t e
n e t .
I t
i s
m o r e o v e r
t h e
o n l y
n e t
w h i c h
h a s
t h i s
r o p e r t y ;
s i n c e
i t
a l s o i s
a
c o n j u g a t e
n e t ,
we
c a l l
i t
t h e
a s s o c i a t e
c o n j u g a t e
n e t .
We s h a l l
d e i n e
a n o t h e r
n e t o f
c u r v e s w h i c h
w i l l
b e
o f
i m p o r t a n c e
i n
o u r
g e o m e t r i c
i n t e r p r t a t i o n . .
T h e
q u a d r a t i c
a
H
d u 2
+
d u d v
-K
d 2
=
O
1 3 )
h a s
f o r
i t s
r o o t s t h e
n e g a t i v e s
o f
t h e
r o o t s
o f
8 ) .
I t
t h e r e f o r e
d e f i n e s
a
n e t
s u c h
t h a t
t h e
t a n g e n t s
t o
t h e
t w o
c u r v e s
t h e r e o f
a t
t h e
p o i n t
y
a r
t h e
h a r m o n i c
c o n j u g a t e s
o f t h e
t w o
r a y
t a n g e n t s
t h e
t a n g e n t s
t o
t h e
r a y
c u r v e s )
w i t h
r e s p e c t
t o
t h e
o r i g i n a l
c o n j u g a t e
t a n g e n t s
t h e
t a n g e n t s
t o
t h e
c u r v e s
o f
t h e
o r i g i n a l
c o n j u g a t e
n e t ) .
F o r
c o n v e n i e n c e ,
l e t
u s
c a l l
t h e
c u r v e s d e f i n e d
b y
1 3 )
t h e
a n t i - r a y
c u r v e s ,
a n d
t h e
t w o
t a n g e n t s
t o
t h e
a n t i - r a y
c u r v e s
a t t h e
p o i n t
y
t h e
a n t i - r a y
t a n g e n t s
o f
t h e
p o i n t
y .
L e t u s
now f i x
o u r
a t t e n t i o n
u p o n
a
p o i n t
y
o f
t h e
s u r f a c e
S y ,
a n d
l e t
u s
r e g a r d
e q u a t i o n s
{ 1 0 ) ,
1 2 ) ,
a n d
1 3 )
a s
b i n a r y q u a d r a t i c s
w h o s e
r o o t s
g i v e
r e s p e c t i v e l y
t h e
p a i r s
o f a x i s
t a n g e n t s ,
a s s o c i a t e d
c o n j u g a t e
t a n g e n t s ,
a n d
a n t i - r a y
t a n g e n t s
o f t h e
p o i n t
y .
T h e
J a c o b i a n
o f
t h e
f o r m s
1 0 )
a n d
1 2 )
i s
a
d u 2 + 2 a
o H- K
+
b
a v l o g
a
d u d v
+
d v
0 ,
1 4 )
a n d i t s
t s
g i v e
t h e
p a i r
o f l i n e s
t h r o u g h
y
w h i c h
s e p a r a t e
h a r m o n i c a l l yb o t h
t h e
p a i r
o f a x i s
t a n g e n t s
a n d t h e
p a i r
o f
a s s o c i a t e d
c o n j u g a t e
t a n g e n t s
o f
y .
T h e
J a c o b i a n
o f t h e
f o r m s
1 2 )
a n d
1 3 )
i s
5 2 0
7/25/2019 Green G.M. - On Isothermally Conjugate Nets of Space Curves(1915)(6).pdf
http://slidepdf.com/reader/full/green-gm-on-isothermally-conjugate-nets-of-space-curves19156pdf 6/6
P H Y S I O L O G Y :
P . D .
LAMSON
a
Z d u 2
+
2 a
( H l - K )
d u d v +
Z
d , =
0
1 5 )
a n d
d e f i n e s
t h e
p a i r
o f
l i n e s
t h r o u g h
y
w h i c h
s e p a r a t e
h a r m o n i c a l l y
b o t h
t h e
p a i r
o f
a n t i - r a y t a n g e n t s
a n d
t h e
p a i r
o f a s s o c i a t e
c o n j u g a t e
t a n g e n t s
o f t h e
p o i n t
y .
T h e
t w o
J a c o b i a n s
1 4 )
a n d
1 5 )
c o i n c i d e i f
a n d
o n l y
i f
2
l o g
a =
0 ,
3 b i s )
bub
i . e . ,
i f
a n d
o n l y
i f
t h e
o r i g i n a l
c o n j u g a t e
n e t
i s
i s o t h e r m a l l y
c o n j u g a t e .
We
m a y
s t a t e
o u r
r e s u l t
a s
f o l l o w s :
A
n e c e s s a r y
a n d
s u f f i c i e n t
c o n d i t i o n
t h a t
a
c o n j u g a t e
n e t
o f
c u r v e s o n
a
s u r f a c e
b e
i s o t h e r m a l l y
c o n j u g a t e
i s
t h a t
a t e a c h
p o i n t
o f
t k e
s u r f a c e
t h e
p a i r
o f
a x i s
t a n g e n t s ,
t h e
p a i r
o f
a s s o c i a t e
c o n j u g a t e t a n g e n t s ,
a n d
t h e
p a i r
o f
a n t i - r a y
t a n g e n t s
b e
p a i r s
o f
t h e s a m e
i n v o l u t i o n .
By
m e a n s
o f t h e
v a r i o u s n e t s
o f
c u r v e s
d e f i n e d
i n
t h e
c o u r s e
o f t h e
a b o v e
i n t e r p r e t a t i o n ,
w e
h a v e
b e e n
e n a b l e d
t o d e d u c e
a
n u m b e r
o f
p r o p e r t i e s
o f
i s o t h e r m a l l y
c o n j u g a t e
n e t s .
We
h a v e
i n c l u d e d
t h i s
m o r e
e x t e n d e d
d i s c u s s i o n
i n
a
l o n g e r p a p e r ,
w h i c h
i s
a
s e q u e l
t o
t h e
o n e o n
c o n j u g a t e
n e t s t o
w h i c h
r e f e r e n c e
h a s
a l r e a d y
b e e n
m a d e .
L .
B i a n c h i ,
V o r l e s u n g e n
i l b e r
D i . f e r e o n a l g o m e t r i e ,
t r . M.
L u k a t ,
2 t e
A u f ,
p p .
1 3 5
e t
s e q .
2
J .
W i l c z y n s k i ,
T r a n s .
A m e r . M a t k .
S o c . ,
1 6 ,
3 1 1 - 3 2 7
1 9 1 5 ) .
G .
M.
G r e e n ,
A m e r .
J . M a t I . ,
3 7 ,
2 1 5 - 2 4 6
1 9 1 5 ) .
C f .
§ L
4
I b i d . ,
e n d
o f
§ 3 .
THE RO LE O F THE LIVER I N
ACUTE
POLYCYTHAEMIA:
THE
MECHANISM
CONTROLLING
THE RED
CORPUSCLE
CONTENT O F
THE
B L O O D
B y
P a u l
D .
L a m s o n
PHARMACOLOGICAL
L A B O R A T O R Y . J O H N S
H O P K I N S U N I V E R S I T Y
P i e n d
t o
te
A c a d e m y ,
A u u o t 1 8 .
1 9 1 5
I t
i s
v e r y g e n e r a l l y
c o n s i d e r e d
b y
a l l
e x c e p t
t h o s e w h o h a v e
p a i d
s p e c i a l
a t t e n t i o n
t o t h e
s u b j e c t ,
t h a t t h e n u m b e r o f
r e d
c o r p u s c l e s
p e r
u n i t v o l u m e
o f b l o o d
i s ,
i n
t h e
n o r m a l
i n d i v i d u a l ,
a
f a i r l y
f i x e d
q u a n -
t i t y
s u b j e c t
t o
g r a d u a l
c h a n g e
o n l y .
A
m o r e
c a r e f u l
s t u d y
s h o w s
h o w -
e v e r
t h a t
t h i s
n u m b e r
i s
s u b j e c t
t o
v e r y
r a p i d
a n d
g r e a t
c h a n g e s ,
a n d
i n s t e a d
o f
b e i n g
c o n s t a n t ,
t h a t
i t
i s
c o n t i n u a l l y
c h a n g i n g
u n d e r
p h y s i o -
l o g i c a l
c o n d i t i o n s .
Q u e s t i o n s n a t u r a l l y
a r i s e a s t o
w h a t
f a c t o r s
w i l l
c a u s e
a
c h a n g e
i n
5 2 1
