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On the mathematical theory of the geometric chuck
Mar., t9o3.]
Geometric C]tuck.
PHYSICAL
I95
SECTION.
Stated 3Ieeling, held February 27, T90L
On the Mathematical Theory of the Geometric Chuck. BY ]~. A. PARTRIDGE, Ph.D.
( Concludedfrom p. z46. )
Calling the i n s t a n t at which Pl and P2 are s i m u l t a n e o u s l y equal to zero, the zero point of time, we h a v e the following expression for the r e c t a n g u l a r co-ordinates of the point E at a n y time t. x = el cos al w t + e2 cos a2 w t + e~ cos (p~ + a3 wt) + e4cos (P4 +a4 w t ) + e5 cos (P2 + as w t ) . • y -~- el sin c11 w t ~- e 2 sin a2 w t q- e, sin (p3 @ a, w t ) + e4 sin (P4 -b a4 w t ) q- e5 sin (t05 + a~ w t ) .
T h e curves traced by the geometric c h u c k are all algebraic. T h e v's are all i n t e g e r s or rational f r a c t i o n s ; in the l a t t e r case the value of a, is necessarily chosen so as to m a k e all the a's integers. Hence, we have the co-ordinates x and y expressed as the sum of sines and cosines of multiples of w t . T h e sines and cosines of multiple angles can be expressed as the sum of powers of t h e - s i n e of the the simple angle affected w i t h n u m e r i c a l coefficients. If the m u l t i p l e is an even m u l t i p l e of the simple angle, its sine can be expressed in powers of the sines w i t h the factor cos w t . If the m u l t i p l e be odd, it can be expressed in powers of the sine w i t h o u t the cosine factor. T h e cosine of an even m u l t i p l e angle can be expressed as a sine of powers of sines s i m p l y affected by n u m e r i c a l coefficients, b u t if the m u l t i p l e be odd in a d d i t i o n to numerical coefficients, we h a v e the factor cos w t . Then, in a n y case, by t r a n s p o s i n g all the t e r m s affected by t h e factor cos w t to one side and s q u a r i n g we can express x 2 and y~ in t e r m s of powers of sin w t and n u m e r i c a l coefficients.
F I G . 19 .
F I G . 2o.
FIG. 2I.
FIG. 22.
FIG. 23.
FxG. 24.
Mar., 19o3.]
Geometric
197
Chuck.
T h e n e l i m i n a t i n g s i n w t b y the dialytic m e t h o d we h a v e an e q u a t i o n b e t w e e n x and y. Hence, the curves are all algebraic. T h e curves traced b y the g e o m e t r i c chuck are all of even order. T h i s follows from t h e fact t h a t from the m e t h o d of their generation t h e y are all closed curves. T h e deficiency of the g e o m e t r i c c h u c k curves is always zero. It will, of course, be inferred, from t h e m o d e of generation, that t h e curves are unicursal, b u t the analytical demonstration is v e r y simple. H a v i n g d e v e l o p e d the p a r a m e t r i c expressions for x and y in powers of s i n w t with t h e factor cos w t m a k e the following s u b s t i t u t i o n s : Let then
wt
~- 2B
cos w t - ~ cos 2~ ~
sin wt =
sin 2~--
and t a n S = d I ~ tan2~ I + tan*~ 2 tan ~
I + tan2~
i +a ~ __
2d
I + d2
Since we h a v e d e v e l o p e d x and y in powers of s i n w t and the factor cos w t , b y s u b s t i t u t i n g the a b o v e values for s i n , a t and cos w t , we have expressed x and y rationally in t e r m s of a single parameter. H e n c e , the curves are unicursal. (Jorda~l: " C o u r s d'Analyse," Vol. I, §598, 599.) T h e degree of the curves is 2n where n equals the greatest of the a ' s . In the r e s u l t a n t of two e q u a t i o n s the coefficients of each e q u a t i o n ente~r in the degree of the variable in the other. Hence, t h a t which d e t e r m i n e s the degree of the r e s u l t a n t in x and y is the degree of the h i g h e s t p o w e r of s i n w t or the degree of the h i g h e s t p o w e r of " d " after h a v i n g m a d e t h e s u b s t i t u t i o n considered in the last paragraph. Now, the h i g h e s t p o w e r of s i n w t results from the h i g h e s t m u l t i p l e of w t ; t h a t is, t h e l a r g e s t "a." S u p p o s e this l a r g e s t a = n., If n is even, then t h e hig, hest p o w e r of s i n w t r e s u l t i n g from s i n n w t will be the (n - - I)
L FIG. 25 .
FIG. 26.
FIG. 27.
FIG. 28.
FIG. ~9.
FIG, 3o.
2 FIG. 3 t.
F I C . 3 2.
Fro. 33.
FIO. 34.
\ F I G . 35.
FIO. 36.
2o0
Partridge :
[ J. F. I.,
th, b u t it will be multiplied by the factor cos w t , so that subs t i t u t i n g the a b o v e values we g e t 2d ~n-a i__d 2 ~ 1 X I + d ~' in which expression the h i g h e s t p o w e r of d is d ~". T h e h i g h e s t p o w e r of sin w t resulting from cos n w t is the n t k , which gives on s u b s t i t u t i o n (
2d
-~n '
in which d ~" is the h i g h e s t p o w e r of d.
FIG. 37-
FIG, 3 8 .
FxG. 39.
FIG. 40.
Mar., 19o3. ]
Geometric
Chuck.
2o~
If n is odd, the h i g h e s t p o w e r of s i n w t resulting from will be the n th, which gives on s u b s t i t u t i o n
sin nwt
2d
n
T h e h i g h e s t p o w e r of d is d ~n. T h e h i g h e s t p o w e r of s i n w t r e s u l t i n g from cos n w t is the (n - - I) th with the factor c o s wt. This term s i n n - 1 w t cos w t gives the term \ i--~-d"]
\ ~ + d~J
in which the h i g h e s t p o w e r of d is d °". F r o m the a n a l y s i s we g e t the result that d zn is the h i g h e s t p o w e r of d u n d e r all circumstances. T h e p's do n o t affect this at all, since on replacing c o s (23 + n w t ) b y its e q u i v a l e n t cos P3 cos n w t ~ s i n p~ s i n n w t , t h e a b o v e reasoning a p p l i e s w i t h o u t modification. A s the d e n o m i n a t o r s are all powers of (I + d z) m u l t i p l y the expression for w and y b y (x + a~)2n and eliminate d b y the dialytic m e t h o d , and we g e t an e q u a t i o n in x and y of d e g r e e 2 n in b o t h x a n d y. Hence, the d e g r e e of the c u r v e s is 2n u n d e r all c i r c u m s t a n c e s . F o r t h e p u r p o s e of p e r f o r m i n g this d e v e l o p m e n t and elimination in a c t u a l eases I h a v e calculated the numerical coefficients of t h e p o w e r s of s i n n x and cos n x for all v a l u e s of n up to 2o. T h e y a c c o m p a n y this paper in t a b u l a r form. Another. m e t h o d of p e r f o r m i n g the elimination is to form tho e q u a t i o n of t h e t a n g e n t to the curve, w h i c h is particularly easy b y finding dx
b y simple differentiatiozl and inserting in t h e general e q u a tion of the t a n g e n t , and then finding the envelop of t h e tangent. If for any p u r p o s e polar co-ordinates are required, t h e y can b e easily o b t a i n e d b y the following transformation : ( I ) -1; = ex cos a w t + etc. (2) y = e x s i n a w t + etc. p cos O = x . p sin O ~ y.
Par[ridge
202 Multiply
,o =
(I)
[ / . F . I.,
."
by cos c~ a n d (2) by sin 0 and add ; we get
e~ cos a l w t cos 0 + e l s i n a l w t s i n 0 -~ etc. Multiply (I) by s i n 0 a n d (2) by cos 0 and
subtract; w e g e t 0 = el cos az w t s i n ~ ~ el s i n al w t cos ~ + etc. E l i m i n a t i n g w t by a n y of the m e t h o d s already described, we get an equation b e t w e e n p and ~. In order to illustrate t h e b e a u t y and variety of the curves .drawn by the geometric chuck, and to render the preceding t e x t clearer, I have d r a w n a number of curves which are a p p e n d e d to the paper. T h e following is a description of the figures. T h e first eighteen c u r v e s are produced by the combination of two circular m o t i o n s . T h e equations comprising all ,of them are : X ~
El COS Cl1 7Ut -+- e 2 c o s 0l 2 W ~ .
y =- e~ s i n a~ w t
Fig,
t.
~- e2 s i n a 2 w t ;
V1 =
I e 1 -~-
Fig.
2.
v 1 --~
I
I
~72 ~
I
e2 --'~
I
a 2 ~
2
'5 2
"5
Fig.
3.
Vl ~
@ 2
a I = e2
Fig. .Fig.
4. 5.
vz~4-2
I
:Fig. .Fig.
6.
7. 8.
"5
'5
a~ ~ e 2 ---~
3 "8
e 2 ~-
"8
a l ~-
i
v 1=-2
i
¢/2~--
I
"5
e2 ~
"5
a I
r
a2 ~
4
e1
"5
C2 ~
"125
z
a2 ~
4
"5
C2 ~
.2
v~=--2
vl~+3
H1
v ~ + 3
g l "~-
:Fig.
9.
Fig. m. F i g . II.
Fig.
12.
vl~-~+
3
v~ -~- + 3
a 1
I
eI
'5
-~ 3
a2 ~-~
"3
4 "4
i
e1
"5
a2 ~--e~ ~---
I
a2 --~
e1
"5 I
'5
4
F2 ~
a 1
V,~+3
Vl ~
3
f2 ~
"5
Fig.
a~ -~-
"5
e2 ~
4 "5
a~ ~---
4
e2 ~
0
Mar.,
Geometric Chuck.
t9o3. ]
F i g . 13.
F i g . 14.
F i g . 15.
vl =
-- 3
vt =
-- 3
va =
-- 3
a, =
I
alt =
el :
"5
e~ :
at =
I
~'l =
"5
¢~ =
I
a2=
"5
¢:t ~
"a I - ~ ¢t =
F i g . 16.
vx=--3
at = e~
F i g . 17.
F i g . 18.
vx ~ - - - 3
vt =
-- 3
203
I
a2 =
"5
¢2 =
at =
I
g'2 =
et =
"5 I
¢t ~
"5
"I
--
2 .2
--
2
"25
a2 = - 2
=
ax=
2
--
c2
=
2 "4
=
a~ = ~ cz
"3 --
--
2 "5
T h e c u r v e s shown in F i g s . 19 to 3 z are f o r m e d b y t h e comp o s itio n of t h r e e circular motions. T h e g e n e r a l e q u a t i o n s i n c l u d i n g t h e m are: x = e, cos al w t + e2 cos a2 w t + e3 cos (Pn + a3 w t ) . y -= e l s i n al w t + e2 sin a2 w t + ea sin (P3 + an wt). Fig.
19.
vl
-[- 3
=
W=
F i g . 20.
F i g . 21.
+
v~ -~- +
3
a2 =
4
e2 ~
"5
e 3 ~-
13
g3 ~
"2 "5
3
al =
I
el =
v2 ~ - - - - - 3
a2 =
4
e2 =
'5
a3 =
-- 5
e~ - -
"3
vl =
+
v2
3
a I =
I
el ~"
'5
3
a2=
4
e2=
"5
ez ~---
"3
a3 = F i g . 22.
vl =
-- 3
•
v2=+
3
Fig. 23 .
ct I = a z~-2
I
el = g2=
"5 "5
a s ~--- - -
II
e3 =
.2
vt=--3
a l --~.
!
et :
"5
% = + 3
a2=--2
e2:
"5
¢'3 :
.'2
a 3 :
F i g . 2 4.
Vl =
-- 3
aI
v2=
+
a2:--2
3
"5.
P i g . ~,6.
--
II
I
:
a~ 3 =
Fig.
-- 5
--
II
e l ~---
"5
e2-~
"5
es =
"2
vl =
-- 3
al_=
I
el :
"5
v~=
+
a~ ~
-- 2
e2 :
"5
a a :
--
II
e3 :
"2
Vl=--
v2 = - - 3
3
3
al :
eI :
"~
a2 : - 2
I
e~:
"5
as ~
e, =
"2
-{- 7
#~ ~
o
,#3 =
o
Pn=t8o°
jbs = o
Ps=I5
°
Ps--9o
20~ ~
I8°°
1Ot=o
.
Partridge :
204 Fig. a 7 .
F i g . 28.
I
e1 =
2
e ~ ~-~
"5 "5
as~---+ 7
es~
"3
a] ---~ - - I
e1
"5
at----- + 7
es=
"3
al=
el=
"75
vl =
--
3
aI ~
7J2 ~
~
3
a I ~
v 1 m_ - - 3
[j. F.I..
--
Ps = o
"5
F i g . 29 .
vr=--3
3
10
"4
3 F'.'g. 3 o.
r;1 ~
-- 3
V~ ~
--
IO --
J~$
as ---~ + 24
es =
ax ~
3
el =
"75
a2=
-- 6
e2 ----
"4
a l ~--- q- 24
#l ~---
"I 7
a l ~---
3
ex =
"75
24
eI =
I~OO
=
1#. " ~
o
Ps
I80°
"I7
=
3 F i g . 31 .
vl ~-~ ~
3 IO
"4
p. = o
3 aI ~
Figures
33 to 4o are
four
circular
x =
e~ cos a~ w t
(a 4 wt
+
motions.
+
/~4); Y =
curves The
-Jr
formed general
e~ cos a~ w t
+
by the
"4 combination
equations
of
are :
es cos (a2 w t + Ps) +
e4 co~
e1 s i n al w t + ¢2 sin a2 w t + es sin (a 2 w t
p,) + e4 sin (a, wt + P4). F i g . 32~
Vl
"~- 3
at =
i
t 1 -~-
"2
-~- 3
at= a4 =
x3 40
e,= et =
"5 "3
Vl ~--- "~ 3 Vt-----+3
at = a2=
I 4
e 1 ~--is-----
"2 "5
Pi=o
V| =
at =
13
es =
"5
Jbt ~--- o
a , ~--~
I4
e, -----
"3
vt=+3 vj~-~--3
at~-~ at=
I 4
e l~-G~--"
"a "5
#t=°
at =
at=--5
el=
"5
. ~ o
a t = - - 32
e, =
"3
6fi =
"2
~---
v2-----+3 Vt =
F i g . 33.
F i g . 34.
P i g . 35.
l t i g . 36.
gtl =
--
3
"{- 3
--
3
p,-----o
I
t1~
v~=+3
a l -=-- - - 2
G =
"5
/s =o
VIL =
as=--II
et=
"S
Pt =O
at=--38
e~=
"3
aI ~ I at= 4 at----'-- 5
el "~-tt~ e,-----
"2 "S "5
a t ~---
~,t ~
"3
"~" 3
~l ~--- " ~ 3 Vs~---~3 V,=--3
22
Pl=° p , ---~ o
-L
I I
~o
i I
~
i i I
o~ ~ ~e¢~o
o
o
o
÷÷÷~÷
I l
I I I I I I I I I ~
o
~oo
o
o
~D
~oo
÷÷÷÷
'"T?,,,,, I
I! II II li 11 II N II fl II II II II It U II It II II II
I
÷ ÷ ÷÷~_~
I
~_
I I I I I
It II I1 II II II II II li II K II II II II tl II II ~ ~ ~ ~ ~ ~ ~ ~ ~ ~,~ ~'~ ~
206 Pig,
Notes 37.
and
Comments.
al ---= x a 2 ---~~ 2
e l --~
"2
v2 = -4- 3
e~ =
"5
Ps = o
U8 =
a 2 =
II
~'8 =
"5
JOt =
16
e4 - =
"3
ax= I a,---~--2 a s~-lt
el ----e~= es~
"2 "5 "5
a4 ~
16
g4 ~
"2
a~ ---~
1
e~ =
"2
a8 =
7
es =
"5
a4 =
34
e, =
"3
"2
vl =
--
--
3
3
--
a4 = t
F i g , 38.
Fig.
39.
v~ = - - 3 z,~=-4-3 vs~--3 v~ ----- - -
3
vs = + 3 Fig. 40.
[J. F. I . , '
v~ = - -
3
a] .-~
*'2 - - - - -
3
a2 =
vs = - - 3
as= a 4 ~
WIRELESS
--
--
0
Ps=o /~o.
t', = o,
t
e~ =
2
e2 =
"5
ps =
7
e~ =
"5
p~ -~- o
20
e'a ~
"3
O~
M E S S A G / ~ S TO A M O V I N G T R A I N .
O n t h e o c c a s i o n of t h e r e c e n t F o r t y - s e v e n t h A n n u a l C o n v e n t i o n o f t h e American Association of General Passenger and Ticket Agents, the Grand T r u n k R a i l w a y g a v e a d e m o n s t r a t i o n of w i r e l e s s teleg~'aphy on a m o v i n g train. T h e e x p e r i m e n t w a s e n t i r e l y successful. T h e d e m o n s t r a t i o n w a s m a d e b y Dr. E. R u t h e r f o r d , F . R . S . C . , a n d Dr. H o w a r d T. B a r n e s , F . R S.C., b o t h o f t h e M a c d o n a l d P h y s i c a l L a b o r a t o r y o f t h e McGi]l U n i v e r s i t y , M o n t r e a l . S i g n a l s were e x c h a n g e d b e t w e e n a s t a t i o ~ a n d a t r a i n ( w h i c h w a s r u n n i n g a t t h e rate of 5o m i l e s a n h o u r ) . N o a t t e m p t : was m a d e to c o v e r d i s t a n c e s c o m p a r a b l e in l e n g t h w i t h t h o s e a t t a i n e d b y M a r c o n i a n d o t h e r s , b u t w i t h c o m p a r a t i v e l y s i m p l e l a b o r a t o r y a p p a r a t u s it w a s p o s s i b l e to k e e p t h e t r a i n in t o u c h with t h e s t a t i o n for f r o m 8 to io m i l e s . St. D o m i n i q u e was s e l e c t e d as t h e t r a n s m i t t i n g station, w h e r e t w o l a r g e metal' p l a t e v i b r a t o r s io x 12 feet, c o n n e c t e d with an i n d u c t i o n coil of t h e u s u a l p a t t e r n , w e r e s i t u a t e d . O n t h e t r a i n itself t h e w a v e s were r e c e i v e d b y c o l l e c t i n g w i r e s c o n n e c t e d to a c o h e r e r of nickel a n d silver p o w d e r . T h e r e l a y o p e r a t e d e l e c t r i c b e l l s in t h r e e cars. T h e c o l l e c t i n g wires w e r e r u n t h r o u g h t h e g u i d e s for t h e t r a i n s i g n a l - c o r d , a n d e x t e n d e d o n b o t h sides o f t h e c o h e r e r for a b o u t o n e c a r l e n g t h . T o o b t a i n t h e m a x i m u m effect it w o u l d h a v e beer~ b e t t e r to h a v e h a d a l o n g v e r t i c a l wire, but s i n c e t h i s w a s i m p o s s i b l e , t h e h o r i z o n t a l wire w a s u s e d . A l t h o u g h t h e s e were p l a c e d i n s i d e t h e steel frame, cars, s t r o n g a n d d e f i n i t e s i g n a l s were obtained o v e r t h e d i s t a n c e n a m e d . A n o t h e r difficulty m i l i t a t e d a g a i n s t o b t a i n i n g t h e m a x i m u m s e n s i t i v e n e s s , a s o w i n g to t h e n a t u r a l v i b r a t i o n o f t h e t r a i n r e s u l t i n g f r o m its g r e a t s p e e d , it w a s i m p o s s i b l e to h a v e t h e r e l a y a d j u s t e d to its m o s t s e n s i t i v e point. I n spite. o f t h e s e difficulties t h e d i s t a n c e to w h i c h s i g n a l s c o u l d be s e n t to t h e t r a i n w a s e m i n e n t l y s a t i s f a c t o r y , a n d w i t h m o r e refined a p p a r a t u s g r e a t e r d i s t a n c e s could, w i t h o u t d o u b t , be covered. T h e success of t h i s f o r m o f w i r e l e s s telegraphy, of which this was but a pioneer experiment, opens up yet another m e t h o d of p r o v i d i n g for t h e s a f e t y of t h e t r a v e l i n g p u b l i c . - - S c i e n l i f l c American.
Geometric C]tuck.
PHYSICAL
I95
SECTION.
Stated 3Ieeling, held February 27, T90L
On the Mathematical Theory of the Geometric Chuck. BY ]~. A. PARTRIDGE, Ph.D.
( Concludedfrom p. z46. )
Calling the i n s t a n t at which Pl and P2 are s i m u l t a n e o u s l y equal to zero, the zero point of time, we h a v e the following expression for the r e c t a n g u l a r co-ordinates of the point E at a n y time t. x = el cos al w t + e2 cos a2 w t + e~ cos (p~ + a3 wt) + e4cos (P4 +a4 w t ) + e5 cos (P2 + as w t ) . • y -~- el sin c11 w t ~- e 2 sin a2 w t q- e, sin (p3 @ a, w t ) + e4 sin (P4 -b a4 w t ) q- e5 sin (t05 + a~ w t ) .
T h e curves traced by the geometric c h u c k are all algebraic. T h e v's are all i n t e g e r s or rational f r a c t i o n s ; in the l a t t e r case the value of a, is necessarily chosen so as to m a k e all the a's integers. Hence, we have the co-ordinates x and y expressed as the sum of sines and cosines of multiples of w t . T h e sines and cosines of multiple angles can be expressed as the sum of powers of t h e - s i n e of the the simple angle affected w i t h n u m e r i c a l coefficients. If the m u l t i p l e is an even m u l t i p l e of the simple angle, its sine can be expressed in powers of the sines w i t h the factor cos w t . If the m u l t i p l e be odd, it can be expressed in powers of the sine w i t h o u t the cosine factor. T h e cosine of an even m u l t i p l e angle can be expressed as a sine of powers of sines s i m p l y affected by n u m e r i c a l coefficients, b u t if the m u l t i p l e be odd in a d d i t i o n to numerical coefficients, we h a v e the factor cos w t . Then, in a n y case, by t r a n s p o s i n g all the t e r m s affected by t h e factor cos w t to one side and s q u a r i n g we can express x 2 and y~ in t e r m s of powers of sin w t and n u m e r i c a l coefficients.
F I G . 19 .
F I G . 2o.
FIG. 2I.
FIG. 22.
FIG. 23.
FxG. 24.
Mar., 19o3.]
Geometric
197
Chuck.
T h e n e l i m i n a t i n g s i n w t b y the dialytic m e t h o d we h a v e an e q u a t i o n b e t w e e n x and y. Hence, the curves are all algebraic. T h e curves traced b y the g e o m e t r i c chuck are all of even order. T h i s follows from t h e fact t h a t from the m e t h o d of their generation t h e y are all closed curves. T h e deficiency of the g e o m e t r i c c h u c k curves is always zero. It will, of course, be inferred, from t h e m o d e of generation, that t h e curves are unicursal, b u t the analytical demonstration is v e r y simple. H a v i n g d e v e l o p e d the p a r a m e t r i c expressions for x and y in powers of s i n w t with t h e factor cos w t m a k e the following s u b s t i t u t i o n s : Let then
wt
~- 2B
cos w t - ~ cos 2~ ~
sin wt =
sin 2~--
and t a n S = d I ~ tan2~ I + tan*~ 2 tan ~
I + tan2~
i +a ~ __
2d
I + d2
Since we h a v e d e v e l o p e d x and y in powers of s i n w t and the factor cos w t , b y s u b s t i t u t i n g the a b o v e values for s i n , a t and cos w t , we have expressed x and y rationally in t e r m s of a single parameter. H e n c e , the curves are unicursal. (Jorda~l: " C o u r s d'Analyse," Vol. I, §598, 599.) T h e degree of the curves is 2n where n equals the greatest of the a ' s . In the r e s u l t a n t of two e q u a t i o n s the coefficients of each e q u a t i o n ente~r in the degree of the variable in the other. Hence, t h a t which d e t e r m i n e s the degree of the r e s u l t a n t in x and y is the degree of the h i g h e s t p o w e r of s i n w t or the degree of the h i g h e s t p o w e r of " d " after h a v i n g m a d e t h e s u b s t i t u t i o n considered in the last paragraph. Now, the h i g h e s t p o w e r of s i n w t results from the h i g h e s t m u l t i p l e of w t ; t h a t is, t h e l a r g e s t "a." S u p p o s e this l a r g e s t a = n., If n is even, then t h e hig, hest p o w e r of s i n w t r e s u l t i n g from s i n n w t will be the (n - - I)
L FIG. 25 .
FIG. 26.
FIG. 27.
FIG. 28.
FIG. ~9.
FIG, 3o.
2 FIG. 3 t.
F I C . 3 2.
Fro. 33.
FIO. 34.
\ F I G . 35.
FIO. 36.
2o0
Partridge :
[ J. F. I.,
th, b u t it will be multiplied by the factor cos w t , so that subs t i t u t i n g the a b o v e values we g e t 2d ~n-a i__d 2 ~ 1 X I + d ~' in which expression the h i g h e s t p o w e r of d is d ~". T h e h i g h e s t p o w e r of sin w t resulting from cos n w t is the n t k , which gives on s u b s t i t u t i o n (
2d
-~n '
in which d ~" is the h i g h e s t p o w e r of d.
FIG. 37-
FIG, 3 8 .
FxG. 39.
FIG. 40.
Mar., 19o3. ]
Geometric
Chuck.
2o~
If n is odd, the h i g h e s t p o w e r of s i n w t resulting from will be the n th, which gives on s u b s t i t u t i o n
sin nwt
2d
n
T h e h i g h e s t p o w e r of d is d ~n. T h e h i g h e s t p o w e r of s i n w t r e s u l t i n g from cos n w t is the (n - - I) th with the factor c o s wt. This term s i n n - 1 w t cos w t gives the term \ i--~-d"]
\ ~ + d~J
in which the h i g h e s t p o w e r of d is d °". F r o m the a n a l y s i s we g e t the result that d zn is the h i g h e s t p o w e r of d u n d e r all circumstances. T h e p's do n o t affect this at all, since on replacing c o s (23 + n w t ) b y its e q u i v a l e n t cos P3 cos n w t ~ s i n p~ s i n n w t , t h e a b o v e reasoning a p p l i e s w i t h o u t modification. A s the d e n o m i n a t o r s are all powers of (I + d z) m u l t i p l y the expression for w and y b y (x + a~)2n and eliminate d b y the dialytic m e t h o d , and we g e t an e q u a t i o n in x and y of d e g r e e 2 n in b o t h x a n d y. Hence, the d e g r e e of the c u r v e s is 2n u n d e r all c i r c u m s t a n c e s . F o r t h e p u r p o s e of p e r f o r m i n g this d e v e l o p m e n t and elimination in a c t u a l eases I h a v e calculated the numerical coefficients of t h e p o w e r s of s i n n x and cos n x for all v a l u e s of n up to 2o. T h e y a c c o m p a n y this paper in t a b u l a r form. Another. m e t h o d of p e r f o r m i n g the elimination is to form tho e q u a t i o n of t h e t a n g e n t to the curve, w h i c h is particularly easy b y finding dx
b y simple differentiatiozl and inserting in t h e general e q u a tion of the t a n g e n t , and then finding the envelop of t h e tangent. If for any p u r p o s e polar co-ordinates are required, t h e y can b e easily o b t a i n e d b y the following transformation : ( I ) -1; = ex cos a w t + etc. (2) y = e x s i n a w t + etc. p cos O = x . p sin O ~ y.
Par[ridge
202 Multiply
,o =
(I)
[ / . F . I.,
."
by cos c~ a n d (2) by sin 0 and add ; we get
e~ cos a l w t cos 0 + e l s i n a l w t s i n 0 -~ etc. Multiply (I) by s i n 0 a n d (2) by cos 0 and
subtract; w e g e t 0 = el cos az w t s i n ~ ~ el s i n al w t cos ~ + etc. E l i m i n a t i n g w t by a n y of the m e t h o d s already described, we get an equation b e t w e e n p and ~. In order to illustrate t h e b e a u t y and variety of the curves .drawn by the geometric chuck, and to render the preceding t e x t clearer, I have d r a w n a number of curves which are a p p e n d e d to the paper. T h e following is a description of the figures. T h e first eighteen c u r v e s are produced by the combination of two circular m o t i o n s . T h e equations comprising all ,of them are : X ~
El COS Cl1 7Ut -+- e 2 c o s 0l 2 W ~ .
y =- e~ s i n a~ w t
Fig,
t.
~- e2 s i n a 2 w t ;
V1 =
I e 1 -~-
Fig.
2.
v 1 --~
I
I
~72 ~
I
e2 --'~
I
a 2 ~
2
'5 2
"5
Fig.
3.
Vl ~
@ 2
a I = e2
Fig. .Fig.
4. 5.
vz~4-2
I
:Fig. .Fig.
6.
7. 8.
"5
'5
a~ ~ e 2 ---~
3 "8
e 2 ~-
"8
a l ~-
i
v 1=-2
i
¢/2~--
I
"5
e2 ~
"5
a I
r
a2 ~
4
e1
"5
C2 ~
"125
z
a2 ~
4
"5
C2 ~
.2
v~=--2
vl~+3
H1
v ~ + 3
g l "~-
:Fig.
9.
Fig. m. F i g . II.
Fig.
12.
vl~-~+
3
v~ -~- + 3
a 1
I
eI
'5
-~ 3
a2 ~-~
"3
4 "4
i
e1
"5
a2 ~--e~ ~---
I
a2 --~
e1
"5 I
'5
4
F2 ~
a 1
V,~+3
Vl ~
3
f2 ~
"5
Fig.
a~ -~-
"5
e2 ~
4 "5
a~ ~---
4
e2 ~
0
Mar.,
Geometric Chuck.
t9o3. ]
F i g . 13.
F i g . 14.
F i g . 15.
vl =
-- 3
vt =
-- 3
va =
-- 3
a, =
I
alt =
el :
"5
e~ :
at =
I
~'l =
"5
¢~ =
I
a2=
"5
¢:t ~
"a I - ~ ¢t =
F i g . 16.
vx=--3
at = e~
F i g . 17.
F i g . 18.
vx ~ - - - 3
vt =
-- 3
203
I
a2 =
"5
¢2 =
at =
I
g'2 =
et =
"5 I
¢t ~
"5
"I
--
2 .2
--
2
"25
a2 = - 2
=
ax=
2
--
c2
=
2 "4
=
a~ = ~ cz
"3 --
--
2 "5
T h e c u r v e s shown in F i g s . 19 to 3 z are f o r m e d b y t h e comp o s itio n of t h r e e circular motions. T h e g e n e r a l e q u a t i o n s i n c l u d i n g t h e m are: x = e, cos al w t + e2 cos a2 w t + e3 cos (Pn + a3 w t ) . y -= e l s i n al w t + e2 sin a2 w t + ea sin (P3 + an wt). Fig.
19.
vl
-[- 3
=
W=
F i g . 20.
F i g . 21.
+
v~ -~- +
3
a2 =
4
e2 ~
"5
e 3 ~-
13
g3 ~
"2 "5
3
al =
I
el =
v2 ~ - - - - - 3
a2 =
4
e2 =
'5
a3 =
-- 5
e~ - -
"3
vl =
+
v2
3
a I =
I
el ~"
'5
3
a2=
4
e2=
"5
ez ~---
"3
a3 = F i g . 22.
vl =
-- 3
•
v2=+
3
Fig. 23 .
ct I = a z~-2
I
el = g2=
"5 "5
a s ~--- - -
II
e3 =
.2
vt=--3
a l --~.
!
et :
"5
% = + 3
a2=--2
e2:
"5
¢'3 :
.'2
a 3 :
F i g . 2 4.
Vl =
-- 3
aI
v2=
+
a2:--2
3
"5.
P i g . ~,6.
--
II
I
:
a~ 3 =
Fig.
-- 5
--
II
e l ~---
"5
e2-~
"5
es =
"2
vl =
-- 3
al_=
I
el :
"5
v~=
+
a~ ~
-- 2
e2 :
"5
a a :
--
II
e3 :
"2
Vl=--
v2 = - - 3
3
3
al :
eI :
"~
a2 : - 2
I
e~:
"5
as ~
e, =
"2
-{- 7
#~ ~
o
,#3 =
o
Pn=t8o°
jbs = o
Ps=I5
°
Ps--9o
20~ ~
I8°°
1Ot=o
.
Partridge :
204 Fig. a 7 .
F i g . 28.
I
e1 =
2
e ~ ~-~
"5 "5
as~---+ 7
es~
"3
a] ---~ - - I
e1
"5
at----- + 7
es=
"3
al=
el=
"75
vl =
--
3
aI ~
7J2 ~
~
3
a I ~
v 1 m_ - - 3
[j. F.I..
--
Ps = o
"5
F i g . 29 .
vr=--3
3
10
"4
3 F'.'g. 3 o.
r;1 ~
-- 3
V~ ~
--
IO --
J~$
as ---~ + 24
es =
ax ~
3
el =
"75
a2=
-- 6
e2 ----
"4
a l ~--- q- 24
#l ~---
"I 7
a l ~---
3
ex =
"75
24
eI =
I~OO
=
1#. " ~
o
Ps
I80°
"I7
=
3 F i g . 31 .
vl ~-~ ~
3 IO
"4
p. = o
3 aI ~
Figures
33 to 4o are
four
circular
x =
e~ cos a~ w t
(a 4 wt
+
motions.
+
/~4); Y =
curves The
-Jr
formed general
e~ cos a~ w t
+
by the
"4 combination
equations
of
are :
es cos (a2 w t + Ps) +
e4 co~
e1 s i n al w t + ¢2 sin a2 w t + es sin (a 2 w t
p,) + e4 sin (a, wt + P4). F i g . 32~
Vl
"~- 3
at =
i
t 1 -~-
"2
-~- 3
at= a4 =
x3 40
e,= et =
"5 "3
Vl ~--- "~ 3 Vt-----+3
at = a2=
I 4
e 1 ~--is-----
"2 "5
Pi=o
V| =
at =
13
es =
"5
Jbt ~--- o
a , ~--~
I4
e, -----
"3
vt=+3 vj~-~--3
at~-~ at=
I 4
e l~-G~--"
"a "5
#t=°
at =
at=--5
el=
"5
. ~ o
a t = - - 32
e, =
"3
6fi =
"2
~---
v2-----+3 Vt =
F i g . 33.
F i g . 34.
P i g . 35.
l t i g . 36.
gtl =
--
3
"{- 3
--
3
p,-----o
I
t1~
v~=+3
a l -=-- - - 2
G =
"5
/s =o
VIL =
as=--II
et=
"S
Pt =O
at=--38
e~=
"3
aI ~ I at= 4 at----'-- 5
el "~-tt~ e,-----
"2 "S "5
a t ~---
~,t ~
"3
"~" 3
~l ~--- " ~ 3 Vs~---~3 V,=--3
22
Pl=° p , ---~ o
-L
I I
~o
i I
~
i i I
o~ ~ ~e¢~o
o
o
o
÷÷÷~÷
I l
I I I I I I I I I ~
o
~oo
o
o
~D
~oo
÷÷÷÷
'"T?,,,,, I
I! II II li 11 II N II fl II II II II It U II It II II II
I
÷ ÷ ÷÷~_~
I
~_
I I I I I
It II I1 II II II II II li II K II II II II tl II II ~ ~ ~ ~ ~ ~ ~ ~ ~ ~,~ ~'~ ~
206 Pig,
Notes 37.
and
Comments.
al ---= x a 2 ---~~ 2
e l --~
"2
v2 = -4- 3
e~ =
"5
Ps = o
U8 =
a 2 =
II
~'8 =
"5
JOt =
16
e4 - =
"3
ax= I a,---~--2 a s~-lt
el ----e~= es~
"2 "5 "5
a4 ~
16
g4 ~
"2
a~ ---~
1
e~ =
"2
a8 =
7
es =
"5
a4 =
34
e, =
"3
"2
vl =
--
--
3
3
--
a4 = t
F i g , 38.
Fig.
39.
v~ = - - 3 z,~=-4-3 vs~--3 v~ ----- - -
3
vs = + 3 Fig. 40.
[J. F. I . , '
v~ = - -
3
a] .-~
*'2 - - - - -
3
a2 =
vs = - - 3
as= a 4 ~
WIRELESS
--
--
0
Ps=o /~o.
t', = o,
t
e~ =
2
e2 =
"5
ps =
7
e~ =
"5
p~ -~- o
20
e'a ~
"3
O~
M E S S A G / ~ S TO A M O V I N G T R A I N .
O n t h e o c c a s i o n of t h e r e c e n t F o r t y - s e v e n t h A n n u a l C o n v e n t i o n o f t h e American Association of General Passenger and Ticket Agents, the Grand T r u n k R a i l w a y g a v e a d e m o n s t r a t i o n of w i r e l e s s teleg~'aphy on a m o v i n g train. T h e e x p e r i m e n t w a s e n t i r e l y successful. T h e d e m o n s t r a t i o n w a s m a d e b y Dr. E. R u t h e r f o r d , F . R . S . C . , a n d Dr. H o w a r d T. B a r n e s , F . R S.C., b o t h o f t h e M a c d o n a l d P h y s i c a l L a b o r a t o r y o f t h e McGi]l U n i v e r s i t y , M o n t r e a l . S i g n a l s were e x c h a n g e d b e t w e e n a s t a t i o ~ a n d a t r a i n ( w h i c h w a s r u n n i n g a t t h e rate of 5o m i l e s a n h o u r ) . N o a t t e m p t : was m a d e to c o v e r d i s t a n c e s c o m p a r a b l e in l e n g t h w i t h t h o s e a t t a i n e d b y M a r c o n i a n d o t h e r s , b u t w i t h c o m p a r a t i v e l y s i m p l e l a b o r a t o r y a p p a r a t u s it w a s p o s s i b l e to k e e p t h e t r a i n in t o u c h with t h e s t a t i o n for f r o m 8 to io m i l e s . St. D o m i n i q u e was s e l e c t e d as t h e t r a n s m i t t i n g station, w h e r e t w o l a r g e metal' p l a t e v i b r a t o r s io x 12 feet, c o n n e c t e d with an i n d u c t i o n coil of t h e u s u a l p a t t e r n , w e r e s i t u a t e d . O n t h e t r a i n itself t h e w a v e s were r e c e i v e d b y c o l l e c t i n g w i r e s c o n n e c t e d to a c o h e r e r of nickel a n d silver p o w d e r . T h e r e l a y o p e r a t e d e l e c t r i c b e l l s in t h r e e cars. T h e c o l l e c t i n g wires w e r e r u n t h r o u g h t h e g u i d e s for t h e t r a i n s i g n a l - c o r d , a n d e x t e n d e d o n b o t h sides o f t h e c o h e r e r for a b o u t o n e c a r l e n g t h . T o o b t a i n t h e m a x i m u m effect it w o u l d h a v e beer~ b e t t e r to h a v e h a d a l o n g v e r t i c a l wire, but s i n c e t h i s w a s i m p o s s i b l e , t h e h o r i z o n t a l wire w a s u s e d . A l t h o u g h t h e s e were p l a c e d i n s i d e t h e steel frame, cars, s t r o n g a n d d e f i n i t e s i g n a l s were obtained o v e r t h e d i s t a n c e n a m e d . A n o t h e r difficulty m i l i t a t e d a g a i n s t o b t a i n i n g t h e m a x i m u m s e n s i t i v e n e s s , a s o w i n g to t h e n a t u r a l v i b r a t i o n o f t h e t r a i n r e s u l t i n g f r o m its g r e a t s p e e d , it w a s i m p o s s i b l e to h a v e t h e r e l a y a d j u s t e d to its m o s t s e n s i t i v e point. I n spite. o f t h e s e difficulties t h e d i s t a n c e to w h i c h s i g n a l s c o u l d be s e n t to t h e t r a i n w a s e m i n e n t l y s a t i s f a c t o r y , a n d w i t h m o r e refined a p p a r a t u s g r e a t e r d i s t a n c e s could, w i t h o u t d o u b t , be covered. T h e success of t h i s f o r m o f w i r e l e s s telegraphy, of which this was but a pioneer experiment, opens up yet another m e t h o d of p r o v i d i n g for t h e s a f e t y of t h e t r a v e l i n g p u b l i c . - - S c i e n l i f l c American.