•'**>
Mr.
THE ELEMENTARY PEOPERTIES OF THE ELLIPTIC FUNCTIONS
MATH,
THE ELEMENTART^OPEKTIES
5^ O
OF THK ~~V~C<£ •
ELLIPTIC FUNCTIONS
WITH EXAMPLES
BY
ALFRED CARDEW DIXON, M.A.
LATE FELLOW OF TRINITY COLLK(iK, 1. \MHIil D»iE ; PROFESSOR OK M ATI I KM ATI AT QUEEN'S COLLEGE, GALWAY
il' 0 U 0 11
MACMILLAN AND CO.
AND NEW YORK 1894
All rit/ltfi rrxi mil
Math. Stat.
Add'l
GIFT
634343
MATH*. STAT.
PREFACE.
THE object of this work is to supply the wants of those students who, for reasons connected with ex aminations or otherwise, wish to have a knowledge of " the elements of Elliptic Functions, not includ ing the Theory of Transformations and the Theta Functions." It is right that I should acknowledge my obligations to the treatise of Professor Cayley and to the lectures of Dr. Glaisher, as well as to the authorities referred to from time to time. I am also greatly indebted to my brother, Mr. A. L. Dixon, Fellow of Merton College, Oxford, for his kind help in reading all the proofs and working through the examples, as also for his valuable
suggestions.
A. C. DIXON.
DUBLIN, October, 1894.
907
CONTENTS.
CHAPTELI I.
PAGE
INTRODUCTION. DEFINITION OF ELLIPTIC FUNCTIONS, 1
CHAPTER II.
FIKST DEDUCTIONS FROM THE DEFINITIONS. THE PERIODS.
THE KELATED MODULI, 8
CHAPTER III.
ADDITION OF ARGUMENTS, 25
CHAPTER IV.
MULTIPLICATION AND DIVISION OF THE ARGUMENT, - 38
CHAPTER V.
INTEGRATION, - 46
CHAPTER VI.
ADDITION OF ARGUMENTS FOR THE FUNCTIONS E, II, 53
CHAPTER VII. WEIERSTRASS' NOTATION, 63
viii CONTENTS.
CHAPTER VIII.
PAOK
DEGENERATION or THE ELLIPTIC FUNCTIONS, - 69
CHAPTER IX.
DIFFERENTIATION WITH RESPECT TO THE MODULUS, - - 73
CHAPTER X.
APPLICATIONS, - - 82
APPENDIX A.
V THE GRAPHICAL REPRESENTATION OF ELLIPTIC FUNCTIONS, 129
APPENDIX B.
HISTORY OF THE NOTATION OF THE SUBJECT, - - 136
ELLIPTIC FUNCTIONS.
CHAPTER I.
INTRODUCTION. DEFINITION OF ELLIPTIC FUNCTIONS.
§ 1. In the earlier branches of mathematics func tions are defined in various ways. Some are the results of the fundamental operations of algebra. o; + l, 2a5, x2 are such functions of x. Others are in troduced by the inversion of those operations ; such are x — 1, 1/x, ^x; and others by conventional ex tensions of them, as of*, ex. It is not easy to draw the line of distinction between the two last-named classes. Sometimes, again, geometrical constructions are used in the definition, as in the case of the trigonometrical functions.
§ 2. The elliptic functions cannot readily be defined in any of the foregoing ways; their fundamental property is that their differential coefficients can be expressed in a certain form, and as this is a somewhat new way of defining a function, we shall take one or
D- E. F. A
2 ELLIPTIC FUNCTIONS.
two examples to show that it is as effective as any of those above mentioned.
§ 3. Let us define the exponential function by the equation
d
-T- exp u = exp u.
This equation tells us what addition is to be made to the value of exp u when a small change is made in that of u, and would therefore enable us gradually to find the value of the function for every value of the argument u, provided we knew one particular value to start with. Suppose then that when u has the value 0, exp u has the value 1, that is, exp 0 = 1.
This equation combined with the former supplies a definition of the function exp u*
§ 4. From the foregoing definition we can deduce the properties of the function exp u. First of all we can find an expression for ex.p(u-\-v).
Let u + v = w, and suppose w to be kept constant while u and v vary.
Then — exp v = — -r- exp v = — exp v. Thus exp u . -j- exp v + exp v . -j- exp u = 0,
or _ (exp u exp v) = 0.
Hence exp u exp v is a constant as long as w is a constant, and has the same value whatever we may put for u and v so long as u + v = w.
* Compare the construction of trigonometrical tables, as explained in works on Trigonometry. The sine, tangent, etc., of every angle are found by adding the proper increments to those of an angle slightly less.
INTRODUCTION. 3
Put then v = 0, n = w, and we have exp u exp v = exp w exp 0 = exp(u -f v), since exp 0 = 1.
§ 5. We can also deduce the expansion of exp u in powers of u.
For — exp u = exp u,
d2 d
so that -j—g exp u = ,— exp u = exp u,
dr and — - exp u = exp u,
which = 1 , when u = 0.
Thus Maclaurin's Theorem gives
it2 ur
the convergency of which may be established in the usual way.
§ 6. As another example, define the sine and cosine by the equation
-j- sin i& = cos u, ...................... (1)
du
where cos2i6-j-sin2it = 1, ...................... (2)
and sin 0 = 0, cos 0 = 1.
§ 7. Differentiating (2), we have
cos u-j— cos w + sin u cos u = 0, du
whence -=- cos u= — sin u, ................... (3)
du
as cos u is not zero in general.
4 ELLIPTIC FUNCTIONS.
§8. To find sm(u + v) and cos(u + v) put a constant, as before.
Consider a symmetrical function of u and v, such as sin u + sin v.
d / • \
T— (sin u + sin v) = cos u — cos v.
In the same way
-7- (cos u -f cos v) = — sin u + sin v.
But cos2u + sin2w = cos2t> 4- sin2/v,
so that (cos u — cos v)(cos u + cos v)
= ( — sin u + sin v)(sm u -j- sin v) (4)
Hence (cos u + cos v) 7 (sin u + sin v) a/th
= (sin u + sin v) -, (cos u + cos v) ,
,, sin u + sin v sin(u+v) /KX
so that — = a const. = — ~- — r-f-z-, (5)
cos it + cos v cos(u + v) + l
putting w for u and 0 for v. Then from (4) and (5)
— sin u + sin v ,
— = a const, also, cos u — cos v
sm(u + -y) ~~ 1 — cos(it + v)' And we find by solving
sin2u — sin2i>
"~ sin u cos v — sin v cos u = sin w cos v + sin v cos u by help of (2).
Here again the functions may be expanded by Maclaurin's Theorem.
INTRODUCTION. 5
§ 9. The equations of definition are satisfied also if we change the signs of u and of sin u. Thus
sin( — u)=— sin u, cos( — u)= cos u.
The equations (1) and (2) are also satisfied if cosu is put for sin u and — sinu for cosu. The initial values however are now different and a constant must be added to u. Call this constant ST.
Then sin(u + £r) = cos u,
cos(u -f CT) = — sin u, if ST is such that sin or = 1 , cos sr — 0. Hence
sin( n -f 2cr) = cos(u 4- CT) = — sin u, cos(t£ + 2uj) = — sm(u + CT) = — cos u, sin(u + 4sr) = — sin(u + 2<£) = sin u, cos(u + 4nr) = — cos(u + 2cr) = cos u.
Hence the functions are unchanged when the argu ment u is increased by to, that is to say, they are periodic.
§ 10. Again, writing i for v — 1,
-r-(cos u + 1 sin u) = ((cos u + 1 sin it),
or -=— (cos 16 + f sin u) = cos tt -f t sin u,
v
and cos 0 + 1 sin 0=1,
so that cos u + 1 si n u = exp n< .
This equation includes De Moivre's Theorem, and shows that exp u is also periodic, the period being 4(£F.
These examples may be enough to show that func tions which we know already can be defined in the way that was mentioned in § 2.
G ELLIPTIC FUNCTIONS.
§11. Now the three elliptic functions sn u, en u, dn u * are defined by the equations
d
-V— sn u = en u dn u. du
• sn2u = 1 rf*
snO = 0, cnO = dnO = l. From these it follows at once that
d
7 - en u = — sn u dn it, dtt
cZ , 79
7— dn it = — Arsn. 16 en u. du
The quantity k is a constant, called the modulus ; u is called the argument.
§ 12. For different values of the modulus k (or, per haps, rather of k2, as the first power of k does not appear in the definition) there will be different values of the elliptic functions of any particular argument, in fact, sn u, en 16, dn u are really functions of two independent variables, and when it is desirable to call this fact to mind we shall write them
sn(u, k), en(i6, A1), dn(u, k).
We shall also use the following convenient and suggestive notation, invented by Dr. Glaisher : —
en 16/dn 16 = cd u, sn 16/cn u = sc 16,
dn u/cn 16 = dc it, 1 /sn u = ns u,
l/cni6 = ncu, etc.
It is usual to write k' for (1 — /»;2) , and k' is called the complementary modulus.
* Read s, n, u — c, n, u — d, n, u.
t Here and elsewhere sn'-'M, etc., stand for (sn «)-, etc., as in Trigonometry.
INTRODUCTION. 7
The reader will not fail to notice the analogy between the two functions sn u and sin u, as also that between cosu and either cnu or dnu. (Compare §§ 74-75 below.)
EXAMPLES ON CHAPTER I.
1. Find the value of tan(u-hi>) in terms of tan u and tan v from the equations
-y- tan u = 1 + tan2^, tan 0 = 0. du
2. Prove also that tan u is a periodic function of u, the period being twice that value of u for which tan u is infinite.
3. Find the value of sech(u + t>), given that
-y- sech u = — sech u tanh u, du
where sech2w, + tanh2u = 1 ,
and that sech 0 = 1, tanh 0 = 0.
4. Find, the differential coefficients with respect to u of ns u, nc^t, nctV, wK&, sd u, cs u, cdu, ds u, dc u.
Ans. — cs u ds u, sc u dc u, 7c2sd w. cd «,, nc 16 dc it, nd u edit, — nsudst*/, — fc^sdundu, — CSIMISU, &/2sc u nc u.
5. Differentiate with respect to u
(1) snu/(l+cnu). J.TIS. dn^/(l + cnit).
(2) sn u/(l + dn u). Ans. en u,/(l + dn u).
(3) cni6/(l+snu). Ans. — dnit/(l+sn u).
(4) dnu/(l + A;snu). Ans. —k cn.u/(l+ksn.u).
(5) arcsin sn u. Ans. dn u.
(6) snu/(dnu— cnu).
CHAPTER II.
FIRST DEDUCTIONS FROM THE DEFINITIONS. THE PERIODS. THE RELATED MODULI.
§ 13. It follows from the foregoing definitions that if a function S or 8(v) of a variable v satisfies the equation
~ = CD,. ...(1)
dv
where C and D are other functions of v connected with S by the equations
1, ........................ (2)
l. ........................ (3)
then S=su(v + a,\), ..................... (4)
C=cu(v + a,\), ..................... (5)
D = du(v + a, A), ..................... (6)
where a is such a constant that
sn(a,A) = S(0), ..................... (7)
cn(a,X) = C(0), ..................... (8)
dn(a, X) = D(0), ..................... (9)
these last equations being clearly consistent.
NEGATIVE ARGUMENTS. 9
§ 14. Now, in the first place, the foregoing con ditions hold if we put
S=— snu, G — cnw, D = dnu, \ = k, v=—u, a = 0; and thus sn( — u) = — sn u, ^
cn( — u) = cnu, J- ..................... (10)
dn( — u) = dn u, J
or en and dn are even functions, and sn is an odd function.
§ 15. We have also
-r- sc u = (cn2u dn u + sn2u dn u)/cn2u
= dn u/cnzu = dc u nc u, and in the same way
d *
-,— nc u = sc u dc u, du
-j- dc u = k'*ac u nc u, du
-j- cs u = — ds u ns u, du
d
-^— ns u = — cs u, ds u, du
d
-j— ds u = — cs u ns u,
-j- sd u = cd u nd it, cZu
-j— cd u = — 7c/2sd K/ nd u, au
T— ndu=jfc*sducdu .................. (11)
Ont
By integrating these equations we shall deduce several important theorems.
10 ELLIPTIC FUNCTIONS.
§ 16. Take for instance
-j- cd u = — 7c/2sd u nd u. du
We have and dividing by dn2u,
Hence &'2sd2it-J-cd2u = l, by elimination of nd'2u, and k'2ndzu, + kzcd2u = l, by elimination of sd2u.
In the equations (1) ... (6) of this chapter we may therefore put
S = cdu, C= — k'sd u, D — fc'nd u, \ = k, v = u. The value of a is such that
sn a = 1 , en a = 0, dn a = k'. Let us write K for this value of a ; then we have
= -fc'sdu, ................ (12)
dn(u -f K ) = &rnd u. J
§ 17. From these it further follows that sn(u + 2#" ) = cd(u + Jf ) = - A/sd ^ -;- k'nd u = -snu, cn(u + 2A") - - k'sd(u+K) = -k'cdu+k'nd u = -en u,
Also sn(tfc + 3^T)= -sn(u + 7ir)= -cdu,
dn(u + 47^) = dn u.
THE PERIODS.
11
Again, sn(K— u) = cd( — u) = cdu, cn(K —u) = k's'd u, dn(K — u) = &'nd u.
Thus the function dn u is unaltered when its argument is increased by 2.K"; snu and cnu are unaltered when the argument is increased by 4>K, that is to say the functions are periodic.
§ 18. Take now the equation
d ,
-j- ns u — — cs u ds u, du
where
Here we may write $=y;nsii, C=r but sna, en a, dna are all infinite. We have, however,
Let this value of a be called L for the time being.
1
Then sn(u + L) = y ns u,
Ki
dn(u + L) = i cs u,
cn(u + 2Z/) = — en u, dn(u+2Z,)=-dnV
cn(u + 3/y) = — r ds u,
to
dn(u -f 3L) = — i cs u,
dn(u
= en M-,
= dn u.
.(13)
12 ELLIPTIC FUNCTIONS.
§ 19. Also
sn(u + K+L) = £ ns(u + K ) = £ dc u,
i ik'
k k
dn(u + K+ L) = i cs(u + K) = — ik'sc u,
= — sn u,
-...(14)
= - dn u.
§20. Hence
sn u has a period 2Z as well as 4^,
en u has a period 2J&r-|-2Z as well as 4>K, dn u has a period 4tL as well as 27iT.
We may also notice that
THE COMPLEMENTARY MODULUS.
§ 21. Now consider the first equation of the system
(11).
d
•j- sc u = dc u nc u, au
where nc2M< — sc2u = 1 ,
Hence we may put
8=i&cu, C=ncu, D = dcu, v = tu, in the equations (4), (5), (6) ; and as
S(0) = 0, (7(0)-Z)(0)=1, we have a = 0.
THE COMPLEMENTARY MODULUS. 13
Thus sn(m, k') = i sc(u, k), \
cn(iu,k') = nc(u,k), j- ............... (15)
dn(iu,k') = dc(u,k).i
These equations are of great importance. They embody what is called Jacobi's Imaginary Trans formation and enable us to express elliptic functions of purely imaginary arguments by means of those of real arguments with a different modulus.
to
§ 22. In the equations (15) put L for u.
Then sn(tL, k'} = i sc(A k) = 1,
cn(iL, //) = 0, dn(iL, kf) = k.
Thus iL stands to k' in the same relation as K to k, and we are naturally led to write
iL = K't L=-tK'.
Thus if in and n are any two whole numbers
sn(i6 + 2mK+ ZniK') = ( - l)msn u,
..... (16) dn(u + 2m^ + ZniK') = ( - 1 )ndn u.
We have then the following scheme for the values of sn, en, dn, of u+mK+mK', m and n being integers :
|
ra = 0, |
msl, |
ma 2, |
ms3 |
|
snu, |
cdu, |
— sn it, |
— cd u. |
|
cnu, |
— k'sd u, |
— en u, |
k'sd u. |
|
dnu, |
k'nd u, |
dn tt, |
k'ud u. |
fc, (dcu)/k, — n==l — t(dsu)/k, —<.k'(iicu)/k, t(dsu)/k, tk'(ncu)/k.
14 ELLIPTIC FUNCTIONS.
m = 0, 771=1, 771=2, 771 = 3.
snu, cdu, — snw, —cdu.
7i = 2 — cnu, k'sdu, cnu, —k'sdu.
— dnu, — k'udu, — dnu, — k'udu.
(nsu)/&, (dcu)/k, —(nsu)/k, —(dcu)/k. n = 3 f(ds u)//j, J/(ncu)//£, — f(ds i6)//>:, — tk\ucu)/k.
l CS It, — £/i/SC U, I CS U, — (7/SC U.
the modulus in the congruences being 4.
§ 23. These equations show that a knowledge of the values of sn u, en u, dn u does not enable us to fix the value of u, and that accordingly the value of K is not perfectly defined since we have only assigned the conditions
Writing x for sn u we have
en u = (1 - a;2)*, dn u = (1 - k2xrf,
Hence
u = f(l -
the lower limit being 0 because it and # vanish together.
Thus K=[\l- g*)-\I - k*g*)-*d£
o
This is a function of k only. The variable (• will be supposed in the integration to pass continuously from 0 to 1 through all intermediate real values and those only, and the initial value of the subject of integration will be supposed to be unity and positive. There is now no ambiguity in the value of K so long
RELATED MODULI. 15
as k2 is less than 1. Also with the same provision K is a purely real positive quantity as every element in the integration is so.
Further, k' is to be the positive value of (1 — &2) , for dnu does not change sign within the limits of integration and k' = dn K.
§ 24. Again, so long as k'2 is less than 1, K' is also a purely real positive quantity.
Thus for values of the modulus between 0 and 1 the periods 4<K and kiK' are the one real, the other purely imaginary.
We shall now show how to reduce elliptic functions in which the square of the modulus is real, but not a positive proper fraction, to others in which the modulus lies between 0 and 1.
8 25. We have -,— sn u = en u dn u, du
and we may put
( provided we have
\ = I/Ic, v = ku. Furthermore a = 0.
Thus sn(ku, I/A;) = k sn(u, k),
cn(ku,I/k) = dn(u,k), ............. (17)
, l/k)= cu(u,
The equations (17) enable us to reduce the case of a modulus numerically greater than unity to that of one less than unity.
16 ELLIPTIC FUNCTIONS.
§ 26. From the equations (15) and (17) we deduce sn(ik'u, 1/k') = k'sn(iu, k') = ik'sc(u, k), } cn(ik'u, 1//0= dn(m, k)= dc(u, k), I ..... (18) dn(ik'u, 1/k') — cu(iu,k)= n.c(u, k), J
and also, since iftf/k is the modulus complementary
to I/k,
sn(iku, ik'/k) = i sc(ku, 1/k) = tk sd(u, k),} <m(tfctMfcy]b)= nc(&tt,l/fc)= nd(w,fc),l ..... (19) dn.(iku, tk'/k) = dc(ku, I//;) = cd(u, k),)
and from (19) by help of (15)
sn(&'u, ik/k') = —ik'sd(iu, k') = k'sd(u, k),} cn(k'u, ik/k') = nd(t u, k') = cd(u, k), I. . .(20) , tk/V) = cd(( ut k') = nd(u, fc)
§ 27. The quantities corresponding to K, iKf, the quarter-periods, are given in the following table for the group of six related moduli : —
|
First |
Second |
|
|
Modulus. |
Quarter-period. |
Quarter-period. |
|
k, |
K, |
tJT, |
|
k', |
K', |
etf, |
|
1/k, |
k(K-iK'\ |
iJfeJT, |
|
Ilk', |
k'(Kf-iK), |
ik'K, |
|
ik'/k, |
kK'y |
k(K' + iK\ |
|
ik/k', |
k'K, |
k'(K+iK'\ |
the distinction being that sn = l and dn = the com plementary modulus for the first quarter-period, and that for the second sn, en, dn are infinite and proportional to t, 1 and the modulus.
§ 28. We can prove that if the modulus is a real proper fraction the elliptic functions of a real argument are real.
REALITY. 17
For as snu increases from 0 to 1, while Gnu de creases from 1 to 0, and dn u from 1 to &', the argument u increases continuously from 0 to K, so that for any value of u between 0 and K, siiu, cnu have real values between 0 and 1 , dn u has a real value between kf and 1.
Also we see from §§ 14, 17 that
su(2K—u)= snit, cuC2K—u)= —en u, dn(2K — u)= dn u, so that when u lies between K and %K.
sn u is real and between 0 and 1 ,
en u „ „ 0 and — 1 ,
dnu „ „ 1 and //.
Again, sn( — u)= — sn u,
cn( — u) = en u,
dn( — u) = dn u,
so that sn u, en u, dn u are also real for values of u between 0 and — 2 K.
Also sn.(u + 4fK) = sn.uy etc.,
so that, as any real quantity can be made up by adding a positive or negative multiple of 4>K to a quantity between ± 2K, sn u, en u, dn u are all real if u is real. They are also real if u is a complex quantity whose imaginary part is a multiple of 2lK', for
sn(u + 1iKf) = sn u, cn( u + 2iK') = — en u, = -dnu.
§ 29. Further, when the imaginary part of u is J\.'t or an odd multiple of it, sn u is real, en u and dn u are purely imaginary,
D. E. F. B
18 ELLIPTIC FUNCTIONS.
for sn(u + 1 K') = 1 /k sn u,
cn(u + iK') = — i dn u/k sn u, dn(u + t/t ') = — i en tt/sn u. Again, since sn(m, k) = isc(u, k'), cn(tu, &) = nc(u, k'), dn(m, k)= dc(u, //),
it follows that for a purely imaginary argument or a complex argument whose real part is a multiple of 27i
sn is purely imaginary,
en and dn are real.
Also, for a complex argument whose real part is an odd multiple of K
sn and dn are real,
en is purely imaginary,
for su(K+iu, k)= cd(m,k) = nd(u, k'), , k)= -&'sd(tw, />;)= , k) = 7/nd(m, k) =
§ 30. It is to be noticed that one of the periods at least is always imaginary or complex, and it may be proved that their ratio cannot be purely real.
For let wl and <o2 be two periods of a function <f>(u) so that
m and n being any integers. Also let wjo)^ be real.
Two cases arise. If e^ and o>2 have a common measure u> let
COj =_2?ft), W2 = (/O),
j> and g being two integers prime to each other.
Then integral values of m and n can be found such that
so that 0 (u -f- a/) = </>(u),
and the two periods o^, to2 reduce to one, 00.
PERIODICITY IN GENERAL. 19
§ 31. But if, on the other hand, wl and o>2 are incom mensurable we can prove that niw^ + nw.2 may be made smaller than any assignable finite quantity.
For let Ao>2 be the nearest multiple of w2 to w1 ; then
a)x - Ao>2( = o)3, say) is less than Jo>2. Let ima)3 be the nearest multiple of o>3 to o>2 ; then
o>2 - /xft>3, or w4, is less than £o>3, and so on. Then
o)2+r is less than ^o>2,
which can be made smaller than any assignable finite quantity by taking r great enough. Also each of the quantities o>3, ct)4, ..., is of the form raa>1 + 'fta>2, so that the statement is proved.
In this case then if 0(u + 7Ha>1 + 7?a>2) = 0(u), the value of the function is repeated at indefinitely short intervals, and the function must be either a constant or have an infinite number of values for each value of its argument.
§ 32. It may be proved that the same kind of con sequences will follow if a function is supposed to have three periods whose ratios are complex.
We shall represent the argument of the function on Argand's diagram, in which the point P whose coor dinates are (x, y) referred to rectangular axes OX, OF, represents the complex quantity x + iij. The statement that a straight line AB is a period will be understood to mean that if from any point P a line is drawn parallel to AB and equal to any multiple of it the value of the function is the same at the two ends of the line.
Now let OA , OB be two periods. Join AB. Through 0,A,B draw lines parallel to AB, BO, OA respectively. Through their intersections draw other lines in the
20 ELLIPTIC FUNCTIONS.
same directions and continue the process till the whole plane is covered with a network of triangles, each equal in all respects to the triangle OAB. Then any line joining two vertices of triangles of the system is a period, since each side of any triangle is one.
The triangles can be combined in pairs into paral lelograms, all exactly alike, and similarly situated, and the values of the function at points similarly situated in different parallelograms will be the same. Such a parallelogram is called the 'parallelogram of the periods.'
Suppose, however, that there is a third period OC ; then C must fall within or on the boundary of one triangle of the network. If it fall at an angular point then OC is not a new period, but is only a combination of OA and OB. If it fall on a side of a triangle, say DE, then DC and CE must be periods, and their ratio is real, since they are in the same direction ; thus this case reduces to the one already discussed.
If C fall within a triangle, say DEF, then CD, CE, CF are all periods. Let G be the point similarly situated within the triangle OAB, then OG, AG, BG are all periods being respectively equal to CD, CE, CF in some order. Any of the triangles OBG, BAG, AOG may now be taken as the foundation of another net work covering the whole plane, and since there is still a third period, we can again find a point within the fundamental triangle with which to carry on the same process. We can prove that ultimately either the point will fall on the boundary of one of the triangles, which case has been discussed above, or a period can be found shorter than any assigned finite straight line.
We shall form each triangle from the one before it as follows. Let Oab be a triangle of the series, and g the point found within it. Let Oa^Ob. Then we take Obg as the next triangle of the series.
IMPOSSIBILITY OF THREE PERIODS. 21
Let e be any finite length, then we shall prove that a period can be found shorter than e. Suppose that none such can be found among the sides of such triangles as A EG, ..., abg, ..., which have not 0 for a vertex.
The angle Oab is always acute, and can never be greater than J?r — /3 where j3 is some finite acute angle. For if there is no such limit, and Oab can be made to approach ?r/2 without limit, then since Ola ^ Oab, aOb can be diminished without limit, and therefore ab can be made less than e.
If Oh is drawn perpendicular to ab and y falls within the triangle Ohb then Oy < Ob.
Fig. l.
If not, we have
Oa — 0g = ay sin ^(Oya — Oag)-7-cos ^aOg>e sin J/3, for ag:>e, Oga > Oka > JTT, Oag < JTT — /3.
Thus Oy is less than Oa by a finite quantity, and if Oy > Ob it will be reduced by a finite quantity at the next step and so on, until after a finite number of steps we have a triangle in which Ob is the greater side. We can then replace 06 by a line which is less by at least e sin J/3, and carry on the process, reducing this line again in the same way.
Let IJL be the greatest integer in Ob-r-c sin J/3. Then after /m stages at most the shorter side Ob of the triangle Oab will be replaced by a line less than
22 ELLIPTIC FUNCTIONS.
e sin J/3, and therefore less than e. Each of these //, stages will consist of a finite number of steps by which the originally greater side of the triangle is gradually diminished till it becomes the less, followed by another step in which that which was the less originally is itself diminished.
It is proved then that if there are three periods (*>!, ft>2, o)3, either they are not independent but satisfy an identity of the form l(0l + mco2 + n(DB = 0 with in tegral coefficients, or else a period can be found whose modulus is smaller than any assignable finite quantity, so that the function has an infinite number of values for any single value of its argument. It might of course be a constant.
EXAMPLES ON CHAPTER II.
1. Prove that each of the twelve functions snu, en u, ns u, . . . , can be expressed as a multiple of the sn of an integral linear function of u with one of the six related moduli, in two ways, e.g.
dn(u, lt) = Ksn(K'-iK-iu, k).
2. What are the periods of the functions sc u, dc u,
sn u en it „
-, snucdu, sn1!*, ^
1 + en it' 1 + sn u' 3. Putting S for snitsn(u-h7(T), verify that
:(l+r)2, (2)
= (1-*02 (3)
Deduce that
EXAMPLES II. 23
and find the values of
cn{u(l+/O,^|} and dn{
4. Putting S for sn u dc u, prove that
(
5. Verify that
where s, c, rZ are sn(u, A;), cn%(w, 7u), dn(u, fc), respectively. C. If k = +/2-l, prove that
BHU(— 2)*=(— 2)*8Cttndtt,
en u( — 2)" = nc i(, nd u + A; sc u sd u, dn u( — 2)^ = nc u nd u — k sc u sd u. Hence prove that for this value of k,
7. If 7c = sin 75°, verify that
sn u( - 3)* = t sc w(4*/3 - 6 - sn2u)/(4 - 2^/3 - sn2it), en u( - 3)* = (2 -V3)(2 - V3sn2w)/cn w(4 - 2^/3 - sn% dn u( - 3)* = (2 - V3)dc u(2 - sn2u)/(4 - 2^/3 - sn2^).
Prove also that for this value of kt
24 ELLIPTIC FUNCTIONS.
8. Find the expansions of snu, cnu, dnu in ascending powers of u as far as u5.
Ana. sn u = u- en u = 1 - l
9. Trace the changes in sign and magnitude of sn, en, dn for real and purely imaginary arguments for all real or purely imaginary values of k.
CHAPTER III. ADDITION OF ARGUMENTS.
§ 33. We shall now show how to express the sn, en, and dn of the sum of two arguments in terms of the elliptic functions of those arguments themselves.
Let 1&! and u2 be the two arguments and let us write 8V cv d^ for sn uv en ulf dn uv and s2, c2, d2 for sn u2, en u2, dn u2. This notation will often be found convenient.
Suppose u^ and u2 to vary in such a way that their sum is constant, say a.
Then
Consider now some symmetric functions of u-^ and U2, as sn ux + sn u.2, sn ^cn u2 + sn U2cn uv etc. We have
- - ^(8lCl - 82C2)
26 ELLIPTIC FUNCTIONS.
Now - k\8*c* - s*e*) = - tf(8* - s22 and thus we have
(di + d2) j- (*iC2 + S2ci) = (Sic2 +
o /i I o /*
From this it follows at once that * 2 ? 1 = a const.
so long as ul-\-uz = a.
The value of this constant may be found by putting
sna
T, .
u, = 0 and tt2 = a. It is 7-7—1 — •
l+dna
Thus
Q/0 Q p 1 /-7_|_/7
§-. . * • O-i L'o *- 9^1 -*• ^^1 I ^9 34. Again, \.2_ ,2 l = — ^ 1 "
= a constant also
sna dna— 1*
cn/'ii -L 11. \
Thus
Inverting these two relations and subtracting, we have
2 _ _di + dz_ dl-d2
sn (Uj H- i^2) "~ sxc2 4- 836! 8^ — SgCj
sothat
j Ip 2 « 2,, 2
'l ^2 ~~ 2 1
Q 2_o 2
ADDITION OF ARGUMENTS. 27
By inverting and adding, we have
and
§ 35. In the same way we could prove the following relations
which we shall leave to the reader to verify.
§ 36. Any one of them is enough to give the value of cnOMj-fM^)- Adding the last two we have
/I f L __
__
and hence cn(^, + iO = M 1 72 - 2-2j1,
VI1 -^ Q /i /"/ Q n n 122 ~~~ 211
by help of the value given in § 34 for sn(ux -f it2).
§ 37. The formulae just found can be expressed in other ways. We know that
sn(u+ iK') = j ns u, cn(u + lK') = — r ds u, dn(u + iK') = — i cs u.
28 ELLIPTIC FUNCTIONS.
Put then u^ + iK' for w1 in the above formulae. We have
4- iO =
1 _ Z-2o
J. /I/ Ol
The expression on the left is — (fccn^+uAso that
These three forms, in which the denominator is 1— A^Sj2^2, are those generally quoted. It may be verified by multiplication that they are the same as the former set. Thus, in the case of dn^ + it 2),
+ k\^) - s^W + ^ V)
'
for d? + /^22c12 = 1 - A; W = c?!2 + A;Vc22-
The other verifications are left to the reader.
§ 38. By putting ut 4- K for u^ we may form another set from each of the two we have. The
ADDITION OF ARGUMENTS. 29
four sets of formulae are embodied in the following scheme : —
Numerator
«1<-.jrf2 Numerator of
'W-W Numerator of
Denominator of each :
l-ltPafaf, 8lCtd.r8*cidi* c^
§ 39. The above formulae give the sn, en, dn of itj — w.2 by simply changing the sign of s2.
Thus sn(Wl_tt.2) = fi|^^, etc.
— /l/ 0^ 69
By combining different formulae we easily find the following, writing A for 1 — tfsfe/ : —
A 811(14 + uz) sn(w1 — u 2) = sx2 — s22,
A cn(u1 + u2)cn(^1 - ?/2) = 1 - s^ - «22 + & W»
A dn^ + w2)dn(-a , - uz) = 1 - L\* - k\* + A;2^ V>
A sn(u1 + ito) cn^ — it2) = s1c1eZ2 + 82czdv
A sn(i(,1 + it2)dn(u1 — K.2) = S1c2^i1 + s2ci^2>
A cn(u1 + w2)dn(?^ x — u2) = c1c2cZ1(^2 — A/2^.
A { 1 ± sn(u1 + u2) } { 1 ± 811(14 — w2) } = (c*2 ± 81rf2)2>
A { 1 ± /,: sn^! + u,) } { 1 ± /,: sn(t4 - M2) } = (c^ ± A:«1C2)2,
n(t6! + u.2)±k 011(14 + U,)}{dn(t61-tt1
^ = ±cn(i4 + w2)}{l ±cn(w1-«2)}=(c1±c2)2,
30 ELLIPTIC FUNCTIONS.
A { dn(i&! + u2) ± A/sn^ + u2) } { dii(ul- u 2) ± fc'sn^- w2) }
= (c2dl±k\)2, etc., etc.
The verification of the above results will give the reader useful practice in the algebraical handling of the elliptic functions.
§ 40. Since u = v + a is the integral of the equation du = dv, a being the constant of integration, the different addition-formulae may be considered as forms of the integral of the same differential equation. Also if we write x for sn u, y for sn vt the differential equation becomes
which therefore has an integral that is algebraical in x and y, although neither side can be integrated by means of algebraical functions. This fact was known for a long time before elliptic functions were invented. Euler succeeded in integrating the equation
where X is a quartic function of x and Y is the same function of y.
Let X = ax* + foe3 + ex2 + ex +/,
Y= ay* +by* + cy* + ey +/ Then the integration is as follows : — Write X', Y for dX/dx and dY/dy. We have v _ Y — z
EULER'S EQUATION. 31
Thus
* y
= -(x-y)2{a(x + y)
Also ~-X^ = ±X^X\ ij*
dx dy
dx dy d(x + y) d(x-
neilCe . — — - r — - r — - 7 — - j —
y^-F*\ $}C \ x-y )'
Therefore
vb __ y 2 v^— V*
and
(j being the constant of integration.
This is the integral sought.
Further information, with references, will be found in Forsyth's Di/erenticd Equations, pp. 237-247.
$ 41. Suppose in the addition-formulae that u^ is real, and u.2 purely imaginary. Then sv cv dv c2, d2 are all real, and s2 is purely imaginary. Thus the imaginary part of sn(^1 + u2) is
This cannot vanish unless s2 = 0 or oo , or c: = 0 or (^ = 0.
But c^ cannot vanish as u^ is real, and if Cj = 0 we have u^ = an odd multiple of K.
32 ELLIPTIC FUNCTIONS.
Also since U2 is purely imaginary, if s2 = 0 or ac we have u2 = & multiple of iK'.
If then a complex argument have a real sn, its real part must be an odd multiple of K, or its imaginary part a multiple of iK'.
In the same way if the sn be purely imaginary, s1 = 0 or oo , or c2 = 0 or d2 = 0. These are all im possible but the first, so that the real part must be a multiple of 2K.
§ 42. From this it follows that sn has no other period than ^K and 2iK'. For if A were such a period it must be complex, say A^+iA^ Then sn(u + ^1 + ^2) *s real or imaginary according as u is real or imaginary.
If u is real we have
A 2 = a, multiple of K't
for u + Aiia not generally an odd multiple of K. If u is imaginary we have
J.1 = a multiple of 2 A".
Hence there can be no periods other than those already found. The same holds for en and cln.
§ 43. Suppose now that there are two arguments ft.* and u.^ for which sn, en, and dn are all the same. Then it follows from the addition- formulae that
sn(ux 4- u2) = 8^14 + u3), etc.,
whatever u^ may be.
Hence u2 — u3 is a period for sn, en, dn, and must be a quantity of the form ±mK+4*mK'.
Thus all arguments having the same sn as u are included in the formula
UNIFORMITY. 33
all having the same en in the formula
±tt+4mJT+2n(J5r+dr);
. and all having the same dn in the formula
§ 44. An important property of the elliptic functions, which has been assumed once or twice in the foregoing pages (as in § 41) is that they are uniform, that is to say that each of them has one single definite value for each value of its argument. Many examples might be given of functions for which
this is not the case ; or is one.
The property may be proved as follows : — Suppose sn u = x, and let us examine the behaviour
of u and x when x is in the neighbourhood of a value a. Put x — a + £, and let a be the value of u when x = a.
Then <|= {1 _(« + ^}-*{l - (fca + W*.
The right hand side of this equation can be ex panded in a series of powers of £ which will always converge absolutely so long as | (• | (the modulus of £) does not exceed the least of the quantities
1-
-i — a
k
(See Chrystal, Algebra, ch. xxvii., § 11).
By integrating every term on the right we get another absolutely convergent series since the term in £r is multiplied by £/(r+l), a constant (complex) multiple of a quantity that decreases as r increases.
Hence the value of u is given as the sum of an absolutely convergent series.
Therefore (see Chrystal, ch. xxx., §18) ^ can be expanded in a convergent series of powers of u — a
D. E. F. C
34 ELLIPTIC FUNCTIONS.
within limits which are not infinitely narrow, and within those limits £ is defined as a continuous uniform function of u (Chrystal, ch. xxvi., §§18, 19). This applies to every finite value of a but ± 1, ± l/k.
If a has any of these values we may put x = a + £2, and deduce the same conclusion.
Lastly, in order to consider very great values of x we put os = l/(p, and find that I/a; is in that region a continuous uniform function of u.
Hence in all the plane there is no point where any branching-off of two or more values of x takes place, and therefore x is a uniform function of u.
The uniformity of en u and dn u can be proved in the same way.
EXAMPLES ON CHAPTER III. 1. Verify from the formulae of this chapter that g— sn( Wj -f u2) = cn(
2. Find the sn, en, dn of ul-\-u2-{-u3 in terms of those of uv u2t us, and show that the results are symmetrical.
3. If 14 + u2 + u3 = 0, show that
Cv-j CvnCvo "^ A> ^I^O^Q — ~ /b ,
EXAMPLES III.
4. If ux + u2 + u.j + tt4 = 0, show that
d^d^d^ — &2c1c2c3c4 + k^k'^s^s^ = k'2,
6'0 Cf
ds 1
8A CA
I 1 1
-s±c3)(d1-d.2) = 0, -c/Z3)(s1-s2) = 0,
(These relations may be put in many more forms by such substitutions as u^ + K, u2, UB — K, u4 for ult u.2,
5. If u^
= 0, then
=0.
6. If 8(u) be written for snudcu and <S/ differential coefficient then
7. Verify the formulae of § 39.
8. Prove the following : —
- — a) — cn(u + a)
sn it sn a = ^
dn( u — a) + dn(u + a) 1 dn(?,6 — a) — dn(u + a)
cd(u — a) + cd(u + a) cnucna = ,;
nd(^6 — a) + n
k'2 nd(u — a) — nd(u + a) " ^ ' ed(tt-a)- cdfa+a)'
36 ELLIPTIC FUNCTIONS.
dc(u — a) -f dc(u + a)
nc(u — a) + nc(it + a)
,,2 nc(tt — ct) — nc(tt + a) ' dc(u — a) — dc(u 4- a)'
sn( w + a) 4- sn(it — a) dn( w> + a) + dn( u — a)
1 dn(tt — a) — dn(^t4-a)
dn u dn a =
sd u en a =
sn( u + a) + sn(u — a) sc it dn a = — —
sn 16 cd a =
sn it dc a =
a) + cn(u — a)
cn(u — a) — cn(u 4- a) — sn(u — a)'
a) 4- sd(w- — a)
?T - r nd(u — a)
nd(u + a) — nd(u — a) sd(i64-a)- sd(u-a)'
(ifr — a)
nc(ii + a) + nc(?£ — a)
a) — nc(u — a) Bc(tt— a)1
ds(tt + a) 4- ds(u — a) dn u nd a = —7— — 7—
ns(u + a) 4- ns(u — a)
_ ns(u 4- a) — ns(u — a) "~ ds(u 4- a) — ds(u — a)'
ds(u — a) 4- ds( it + a) sn it ns a = --7 —
cs(?t — a)— cs(u-|-a)
_ cs(it — a) 4-
ds(te- — a) — ds( it + a )'
EXAMPLES III. 37
sd(u + a) + sd(u — a) sc u iid a = — TZ-
't6 — a)
— cd(i& — a) ~ A;'2 ' sd(u + a) — sd(^ — a)'
dc(u + a) + dc(n — a) en u ds a = —
sc(u + a) — sc(w — a)
en u nc a =
a) — ns(u — a) —
CHAPTER IV.
MULTIPLICATION AND DIVISION OF THE ARGUMENT.
§ 45. By putting u± = u.2 in the addition-formulae we easily find the values of sn2w, cn2u, dn 2u in terms of sn u, en u, dn u. Writing S, C, D, s, c, d for these quantities respectively, we have
G= (c2 - s2c/2)/( 1 - &V) = (1 - 2s2 + k
§ 46. Moreover, these equations can be solved for , c, d if $, C, D are supposed known.
We have D - (7 = 2/c/2s2/(l - £
7/2 n n
, , , , -a-*- iv vx A/ _L/ Ly
Thus «2 =
—j^, by subtraction,
1— D
= /2/1 .r , by subtracting again.
A' ^1 -f- (yj
HALVING OF THE ARGUMENT. 39
Hence we find the following formulae for |t& : —
/l-cnu\* l/l-dnwX* sn hu = \ - — I asTI .
\l+dnu/ /Al-fcnit/
ksuu
nu\* k'( 1-dnu V en $u = — 1 — - ) = ( - —
\ 1 + dn u J k \dn u — en u/
( 1 - dn u)\dn u + en u)* ksuu
dn u + en uy , ,/ 1 — en u V - ) = Je ( , - )
1 + en u / \dn u — en uJ
_ (dn u + en u)^(l — en u)^ snu
47. In particular
dn & =
being purely imaginary and of the same sign as its argument ;
mJeJT-Af *(!+&)*,
being a positive quantity ;
dnJdT «(!+*)*,
being also positive.
These three may also be deduced from the others by using the complementary modulus.
40 ELLIPTIC FUNCTIONS.
Als< >
dn
These three are most conveniently found from the former six by the addition-formulae.
MULTIPLICATION OF THE ARGUMENT BY ANY INTEGER.
§ 48. By repeated use of the addition-formulae we can find the elliptic functions of 3-u., 4i6, ..., in terms of those of u.
We may prove the following facts about the form ulae for sn nu, en nu, dn nu : —
Firstly, when n is odd,
sn nu = sn u X a rational fractional function of sn2tt, en nu = en u X a rational fractional function of sn%, diinu = duu x a rational fractional function of sn2^.
In each case the denominator is the same function, and is of the degree n2 — 1 in sn u ; the numerators are different, but are of the same degree, ?i2 — 1.
Secondly, when n is even,
sn nu = sn u en u dn u x a rational fractional function
of sn2w,
en nu = a rational fractional function of sn-u, dn nu — a rational fractional function of sn%.
MULTIPLICATION OF THE ARGUMENT. 4-1
In each case the denominator is the same, and its degree is n2 in snit; this is also the degree of the numerators of en nu and dn nu ; the numerator of sn nu-T-mi ucnu dn u is of the degree n2 — 4.
Clearly we may say a rational function of cn'2-it or di\2a instead of sn2u without altering the meaning or the degree to be assigned.
§ 49. These statements are evidently true when n = 1 or 2. Suppose them to be true for the values m and m + 1 of n ; one of these values will be even, and the other odd.
Write Sp, Cp, DP) Np for the three numerators and denominators of sn^m, cnpu, dnpu respectively, and s, c, d for sn u, en u, dn u. Then
= scd x a rational integral even function of s of degree 4m2 — 4,
= a rational integral even function of 6- of degree 4m2,
a rational integral even function of s of degree 4m2,
2m m
= a rational integral even function of s of
degree 4m2. Also
= a rational integral odd function of s of degree 2m*+2(*w + l)»-l, that is, (2m + 1)2;
v/ftw-f 1 =: ^7»^> m^M + 1^» m + 1 — ^rn^m^m + l^m + 1
= a similar function of c ;
42 ELLIPTIC FUNCTIONS.
= a similar function of d ;
\T _ A72 A72 _ Z-2.Cf2.Cf2
iV — IV miV m + 1 ^ °*5
= a rational integral even function of s of degree (2m + l)2-l.
Hence, if the theorems hold for the values m, 771+ 1 , they hold also for 2m and 2m + 1. Now they hold for 1 and 2, and therefore for 2 and 3, 4 and 5, and universally.
§ 50. Also these expressions will be in their lowest terms. Consider for instance Cm, a rational integral function of c of degree m2. This must vanish when ever en mu = 0, that is, whenever
p and q being any integers.
Hence the roots of Cm = 0 as an equation for c are
, _. . ,
the values of en - *-- — - - . This expression has in
m2 different values found by making
p = 0, 1 ... m— 1, and q = 0, ±1 ... ± J (m — 1 ) or ± |m,
in turn. Thus the degree of the numerator of en mu cannot possibly be lower than m2 and the expression we have found for en mu is in its lowest terms.
Also as G+ S = N
and Cm, Nm have no common factor, /<m and Dm can have no factor in common with either.
§ 51. We may notice that when Nm is expressed in terms of s, the coefficient of .s2 in it vanishes.
DIVISION OF THE AEGUMENT. 4,3
For N = tf
Now s is a factor in Sm and $TO+i, so that if the term in 8Z is wanting in Nm and ^m+i it will be wanting in N2m and N.2m+i.
Now .A^ = 1 , ^V2 = 1 — & V, from which by induction the theorem follows.
By changing u into u + iK' we find that the co efficient of s™2-2 vanishes in Sm when m is odd and in N,n when m is even.
DIVISION OF THE ARGUMENT BY ANY INTEGER
§ 52. If we know the value of sn u, the multiplica tion-formula gives us an equation to find sn u/n. When n is odd,
77
sn - is the root of an equation of the degree n2,
n whose coefficients are rational in sn u. When n is even,
f\»
sn2- is the root of a similar equation. n
We may show that the solution of these equations depends only on that of equations of the nth degree.
$ 53. Take the case when n is odd.
Since sni6 = snt£ + 4>/f+2'iA'/, it follows that
sn-(u+4?pK+2qiK') is also a root, and as this ex
pression has n2 values it includes all the roots. Call it \(p, q).
Then clearly any symmetrical function of X(£>, 0), X(p, 1), ••• > M./>, n — l) will be unchanged by adding any multiple of 2iK' to u. Such a function then will have only n values, given by putting p = 0, 1 , . . . , n — 1
44 ELLIPTIC FUNCTIONS.
in turn. It will therefore be a root of an equation of the ?ith degree only.
Thus A(_p, q) is the root of an equation of the tith degree whose coefficients are also given by equations of the Tith degree, rational in sn u.
The same form of argument holds in the case when n is even, and also in the case when en u or dn u is the function given and we have to find the sn, en, or dn of u/n.
EXAMPLES ON CHAPTER IV.
1. Find the values of the sn, en, and dn of
mK') for all integral values of m and n.
2. Prove that sn \K is a root of the equation
What are the other roots, and which is the real one? Ans. sn(i/f ±f</O> sn(3K+%iK'). The last is real.
3. With the notation of this chapter, show that A^m-fi ± C^m+i, expressed in terms of c, has 1 ± c for a factor, the other factor being a perfect square.
4. Show that Nzm — C*m has 1 — c2 for a factor, and that the other factor of it is a perfect square, as is also N2m+C2)n.
5. Prove that when expressed in terms of d, N2m+i±D2m+i has l±d for a factor, the other factor being a perfect square, that N^ — D^m has l — d2 for a factor, and that the other factor, as also iV2,u + D2,w, is a perfect square.
6. Show that N2m ± S^m can be expressed as a perfect square, as can also the quotient of N9m+\±S^m+i by I±(-l)m8.
7. Prove similar facts with regard to Nm±kSm, k'Nm±Dm,Dm±Cm,Dm±kCm.
EXAMPLES IV. 45
8. Prove that
(cNm - Cm)* + (Nm + l-Cm + ,)(Nm _ ! - Cm _ i),
are independent of the argument u.
9. If /UL, v are any two Tith roots of unity, show that the Tith power of
p=Q q=0
is a rational function of sn u and en u dn u.
Hence show that the value of sn u/n may be found by the extraction of ?ith roots, if sn 2K/n and sn ZiK'jn are supposed known.
10. Use the last example to find expressions for
sn \u, sn \u.
11. When n is odd, prove that
and that
12. When n is even, prove that
tfHtt+ '
CHAPTER V. INTEGRATION.
§ 54. We must now examine how far it is possible to integrate, with respect to u, any rational algebraic function of sn it, en u, dn ut or, as we shall write them, s, c, d.
In the first place, suppose the function to be j?/ r^Y
0 and i//> being rational integral algebraic functions.
We may make the denominator rational in s by multiplying it and the numerator by
\/s(s, -c, d)\l^8, c, -d)\ls(s, -c, -d), and by means of the relations
by means of the same relations we may reduce the numerator to the form
Xi(s) + CX2(S) + dXs(s) + cdXi(s)>
the denominator being x(s) and x> Xv X2> Xs> X4> a^ rational integral algebraic functions.
§55. Now
INTEGRATION. 47
which can be integrated by the ordinary rules for rational fractions ;
J x(s) * Jx(8)
and this can be reduced to the integral of a rational function by the substitution
i 1 — 02 which gives (1 — s2)* = z --
Also
which can be reduced by putting
7 _?£_
~l+z2'
The problem is thus reduced to the integration
§ 56. The first step will naturally be the expression of Xi(8')/x(s) as a series of partial fractions.
When this has been done the expressions to be integrated will fall under one of the two forms
a being any constant, real or imaginary. We will consider these in turn.
Let \8mdu = vm. Now
*,. J
= (m - 3)sm - W2 - sm ~ 2d2 - k?sm ~ 2c2
= (m - l)k*8m - (m - 2)(1 + /-2).sw - 2 + (m - 3),s?" - 4,
48 ELLIPTIC FUNCTIONS.
and therefore, integrating, we have
where C is a constant.
Thus when m > 3, vm can be expressed by means of vM-.2 and i>m_4; and in the case when m = 3, VB can be expressed by means of vr
Thus when m is odd the integration of vm depends only on that of vlt and when m is even on that of v2 and vQ.
§ 57. Now V-L = I sn u du
— 2 1 sn 2x dxr putting
4 sn x en x dn x -,
j*4 snx
Tims the integral of an odd power of sn u can always be expressed by means of the functions sn, en, dn, log.
§58. Again, v0=\du = u, i sn2 u du.
-1-
It is not possible to express v2 by means of known functions, and a new symbol has to be introduced.
THE FUNCTION E. 49
The letter E is generally used, and the definition of its meaning is
Eu
fM = I dn2t6 du,
so that v2 = (u — Eu)/k2.
The value of Eu when u = K is generally denoted by E simply, so that
E=( du2udu.
o
The Greek letter Z was used by Jacobi for a slightly different function, defined as follows : —
Thus ZK=0.
One advantage in the use of this notation is that there is not the same risk of confusing the product Eu with the function Eu.
§ 59. We now turn to I (s — a) ~ mdu, which we shall call wm. Put s — u = t.
~(8-a)~m+lcd du"
-a)- mo2d2 - (s - a)
1)_(_m+1)(1_h/
- m + l)W(t + a)4 - t(t + a){ 1 + & - %k\t + a)2}]
Integrating, we find that wm can be expressed by means of known functions, and wm_i, wm.-Z, WU-t, D. E. F. D
50 ELLIPTIC FUNCTIONS.
WM .. 4, provided always that (m — 1 )(1 — «'-)(! — &V2) docs not vanish.
If a'2 = l or I//;2, then w,M_i can be expressed in terms of u'w_2, wm-S, w,n-4 for 2m — 3 does not vanish.
Hence for these special values of a the integral can be reduced to w0, w.v w.2, that is to VQ, vv v2, and no new function need be introduced.
But in general the reduction can only be carried on as far as u\ , since when m = 1 the coefficient of ium in the formula of reduction vanishes. We must introduce a new function to express wlt and w.2, ws ... can be expressed by means of this and known functions.
~ldu
§60. Now though \(s — a}~ldu and (
cannot be found in terms of known functions, their sum can.
For by the addition-theorem
x 2 sn u en a dn a sn(u + a) + m(u - a) = -—
Now each of the terms on the left can be integrated since we have found Isnitdu. Hence if a be so chosen that kma = I/a, we have an expression for
or s-
The new function that is introduced is therefore only needed to express
f (« - a) - ldu - i(s + a) - ldu, and the one actually chosen is
&2sn a en a dna
(Vfctena
du.
THE FUNCTION II. 51
This is denoted by II(i6, a), and u is called the argu ment, a the parameter.
It has been shown then that any rational function of sn u, en u, dnu can be integrated by help of the new functions E and II. The properties of these will be considered in the next chapter.
EXAMPLES ON CHAPTER V.
CK CK
1 . Prove that k2 \ siizu du = I ns2u du — k'.
o 11 .LI. L / /9 f du en 16 dn 16 7 /9 r-,
2. Prove that k 2 - — = , - + k2u — E
J 1 — sn u 1 — sn u
3 Find f ^u f du f rZi6 f cZi
J l+/csni6' JF+chTu' J 1 +cn u' J F^t
A; en 16 dn u 1 //-it, sn u en t6
sn u dn u ^ 1 , _ - . sn u en u
— + u — Eu, jJu — Eu) — -^ — j - .
1 -fen u A;2V 1— dni6
4. Show that
an a cna dna du
r
-
, — sn-a
TT/ . ,, ,
«n(u,a)— Jlog ; (
te su\a — u)
o 5. Prove that
I ns u (Zu = log sn J 16 — log en Ji6 — log dn J u, I cs u du — log sn \u 4- log en Ju — log dn Jit, I ds udu = log sn Jit — log en J it + log dn Jt6.
52 ELLIPTIC FUNCTIONS.
6. Verify the formulae
f " sn a en a dn a dn% du „ , , l cn(u-a)
J - cn^-sn^dA~ =n^a
("k^nam
J ~dS^:
o
7. Prove that
dn(u--a)
IL(ku,ka, l/A;) = n(tt,tt,A;),
cnu— a sna-dna n(«w, «a, //) = II(«, a) + J H
the modulus on the right being k throughout.
CHAPTER VI.
ADDITION OF ARGUMENTS FOR THE FUNCTIONS E, H.
§ 61. Expressions can be found for E^ + u^) and (16! + u2, a) in terms of functions of u^ and u%. As in the former case, suppose i^-f it2 = &, a constant. Take the function Eu^ + Eu2.
Thus jE'u1 + £*?-62 — Ps1s2sn 6 is constant, and putting
= b, i£2 = 0, we find its value to be Eb.
Hence
Eu^ + Eu2 — E(u-^ + u2) It follows that Zuj + Zit2 — Z(u1 + w,2) = A:2sn u
§ 62. Putting u2 = /<T we have
E(u + K) - Eu = E- Hsn u sn(u
cd 11.
54 ELLIPTIC FUNCTIONS.
E(u + 21Q - E(u + K) = E-k
= E '+ &2sn u cd u
Hence E(u + 2m#) -Eu = 2mE.
§ 63. Let us apply Jacobi's Imaginary Transforma tion (§21) to Eu. We have
(ni, //) = J"dn2(m, tt)du = i P
XT cZ snitdnu , „ 79 „ , snw,
Now -7— — = dn2u — Arsnnt + —
du en u
n, 7 ,x sn u dn 16 ^
Hence x^ctt, A; ) = i --- \-tu-i Eu,
the modulus when not expressed being k ; no constant is added for both sides vanish with u.
Thus as en K = 0, E(iK, //) and therefore also E(<K', k) are infinite. Let us find the value of
= Eu + E— Fsn u cd u. Thus
. en u dn u
Put now iK' for it, and write
R for j&(^', //). Then tE(K+iK')-E' = iE-K',
THE FUNCTION E. 55
§64. Since
E(K + u) = Eu + E- Hsn u sn(u + K) we have
Thus E(mK) = mE if m is any whole number. Also
E(u + 2m/Q - Eu = E(2mK) = 2m E. In the same way
Em(K+ iK') = mE(K+iK') = mE+im(K' - E'). E(u + 2mK+ 2mi K') - Eu = 2mE+ 2mi(K' - E').
Thus
E(u + 2mK+ 2mK') = Eu+ 2mE+ 2m(K' - E').
This equation shows that the effect on the function Eu of adding any multiple of 2K or ZiK' to its argu ment is to add the same multiple of 2E or 2i(K' — E') to the function.
§ 65. The quantities K, K\ E, E' are connected by an important equation which we shall now prove. Clearly
JK/CK+iK' (J
0 K
Thus
rK , rK+iK' v
K'.E= \ ( dn*vau\dv, J VJ )
JK CK+IK (d
0 A'
The right-hand side may be transformed by putting sn ufm.v = x, dn uduv = y.
56 ELLIPTIC FUNCTIONS.
We have
3(aj, y) _ I en u dn u sn v , — &2sn ^ en u dn i> 3(it, v) ~~ j en v dn v sn u, — &2sn t> en v dn u
= — &2cn w en i>(sn2/y dn% — si\z
= en u en v(dn-v — dn?u).
The subject of integration is then .
en u en v
Now &2cn2u cn2^ - ?/2 = F/»/2^2 - k'z,
so that the transformed integral is
k dy dx
if,
+ /;2//%c2-A
As to the limits, snv takes all real values from 0 to 1, and sn u all real values from 1 to I//.:.
Thus, if x has an assigned value > 1 , sn u and sn i> are nearest when
and furthest apart when
sn u = 1 //<:, sn v = /££. The value of £/ will therefore range from
k'(I-kW)* to 0. For y2 = 1 + Ate2 - 2£2a - 7c2(sn u - sn ^)2,
which is least when sn u and sn v are furthest apart, and greatest when they are nearest.
Also, if x has an assigned value <1, sn?/, and snt> are nearest when
sn u = I , sn v — x, and furthest apart when
sn u = 1 /k, sn v = kx.
VALUE OF AN INTEGRAL. 57
The value of y will therefore range from
k'(\-kW)^ to 0 still. The integral is therefore
ff'V-^ kdydx LTT
— Y, that is, — ir. (y*+#*»«*-V*$
Any doubt there may be as to the sign of this result is removed by the consideration that in the original double integral
dnv > k' > dn u ,
so that the subject of integration is always negative, while du is positive and dv has the sign + < .
Hence K. E(K + iK')-(K+iK')E= - $nr.
Substituting the value that was found above for '}, we have
§ 66. The following result will be useful after wards : —
•K
I
o We may prove it thus
du=( E(K-u)du = ±{ [Eu + E(K-u)}du
0 0
= jf {E+k*fmufmKsn(K-u)}du
o
f
!
dnu
o
- J log faK=±
58 ELLIPTIC FUNCTIONS.
ADDITION OF ARGUMENTS FOR THE FUNCTION II. § 67. Again if u^ + u.2 = b,
_ &2sn a en a dn ast2 &2sn a en a dn as22 l-k*8*8D*a Y^-~'
&2sn a en a dn a(sx2 — s22)
Now we have seen that '
What we have to do is therefore to express s^2 -j- ^2 in terms of 8r92 and 6. Now
= ^(cy 1 + /^ V) - tAfftaV + ^i
So that
- 2srs2cn 6 dn 6}
which reduces to ( 1 - fc2^2)2^2 + -922).
Hence s^ + s22 = ( 1 + k\*8*)sm*b - 28rs2cn 6 dn 6, and
/;2sn a en a dn a sn b
>22)sn26 - 2s1s2cn 6 dn b } +^4s12ts\22sn4a
THE FUNCTION II. 59
The denominator
= (1 — &2sn2a sn'26) + 2&2$1s2 . sn2<x en b dn b
2%n2a(sn2a - sn26) = (1 — A;2sn2a sn26){ 1 + k\s2sii a sn(a + b)}
{ 1 + 7iAs1s2sn a sn(a — ?>)}. The numerator
= K1 - ^sn2a sn26)Psn a{sn(a + ?>) - sn(a - b)}. Hence
_ / 9 \ 3tt^ r 2
1 7c2sn a sn(a — 6) d a-6) d
a sn(a + b)
_ , 1(2
~ 2 du^ ° 1 + /^SjSgSn a sn(a — 6)'
Integrating then, we have
, a) — Ii(uv a) — H(u2, a)
_L, 1 + csn 'itjSn t62sn asnu1 + '^2 + a ~ 2 * 1 — A;2sn u1sn u2sn a sn( i^ + U2 — a)'
^ 68. There is another interesting property of the function II which we shall now prove. It connects II(i6, a) with II(a, u)t the same function with argu ment and parameter interchanged. We have
_ d T..., . 2A;2sn a en a dn a siizu 2 7-II(u, a) = - r — j-t — 9 - y — du 1 — irarfa sn2u
sn
— a)}.
60 ELLIPTIC FUNCTIONS.
Thus
(u, a)
= k2sn u{sn(i6 + a)cna dna -f sna cn(u + a)dn(u + a)} + &2sn it { sn(it - a)cn a dn a - sn a cn(u - a)dn(i& - a) } . But by the addition-theorem
sn2(w/ + a) — sn2a
. ^ /
_
sn(i6 + a)cn a dn a + sn a cn(u + «)cln(u + a) sn2(u — a) —
sn(u — a)cn a dn a — sn a cn(u — a)dn(u — a)' for u = (u + a) — a = (u — a) 4- a. Hence
n*(u - a)- = 2 dn2a - dn2(u - a) - dn2(u + a).
In the same way
32
a, u) = 2 dn% - dn2(a - u) - dn2(a + u),
so that --{II(%, a)-H(a, u)} =dn2a-dn2u,
oK/dot-
J;{n(«, a)-n(a, w)}=wdn2a-j&w,
for
Finally then II(w, a) — II(a, u) = u.Ea — a. Eu. This may also be written uZa — aZtc.
EXAMPLES ON CHAPTER VI. 1. Prove that
EXAMPLES. VI. 01
2. Prove that
E(u + K+iK')-Eu = E(K+iK') + ju
3. Prove that
4. Prove that kE(ku, l/k) = E(u, k)-k'2u.
5. Prove that
kE(iku, ik'/k) = LU — iE(u, fe)+c£fen(tt, k)cd(u, k).
6. Find the values of E\K, E\iK ', E$
Ans. ±
7. Show that
8. Prove the formula
fu+a
2H(u, a) = 2uEa - Ev dv.
9. Verify that
= u( 1 - V) + log cln(tt + i^t) - J log /^
10. Prove that the limit when a is indefinitely diminished of II(i£, a)-h a is u — Eu.
11. Show that Enu — nEu- is equal to a rational fractional function of sn u multiplied by en u dn u.
By partial fractions or otherwise show that
nEnu — n2Eu = -^— log Nn,
where Nn denotes the common denominator in the expressions for sn nu, en nu, dn nu.
62 ELLIPTIC FUNCTIONS.
12. In the same way prove the formula (n being odd)
13. Prove the formula for addition of parameters in the function II, namely,
II(u, a + b)-U(u, a)-U(u, b)
_ j 1 + /v2sn a sn 6 sn u sn(u + a + 6) ° 1 + Fsn a sn 6 sn u sn(u — a — b) — k'2u sn a sn 6 sn(a + b).
*"\
14. Find the value of —Ii(u, a) and prove that
ru Evdv.
o
15. Prove, by putting % + t' = 2r, u — v = 2t, and integrating, that
II(u, a) + U(v, a)-IL(u + v, a)
_ L , { 1
-2 iOg r
CHAPTER VII. WEIEESTEASS' NOTATION.
§ 69. For some purposes it is convenient to use the notation of Weierstrass, which we shall now explain shortly.
We write <pu for a2ns2cm-|-/3, where a is any con stant and ft is a constant which we shall determine.
Differentiating, we have
<@'u — — 2a3ns au cs au ds au. Also cs2ai6 = ns2au — 1 ,
ds2au = ns2au — k2. Thus (&'u)* = ±(pu - pxpw -ft- a>)(yu -ft- a^/,-2).
Now choose ft so that the coefficient of £>% on the right may vanish. Then
and where
i/2 = - 4/=l(/3 + a2) - r/3= 4/8(/8 +
The equation
64 ELLIPTIC FUNCTIONS,
with the particular equation
constitutes the definition of Weierstrass' function § 70. Conversely, if $>u = x}
J
The periods of the function $m are 2K/a, They are denoted by 2o>, 2o/ respectively, and their sum by 2o>". We then have
=ev say, jpw" = /3 -f- a?k2 = ez, say, put =/3 = e3, say ;
and ev ezt es are the roots of the equation
4x3-r/2a-(/3 = 0
in descending order of magnitude. Thus #/o> = ^ V = $>'w>" = 0.
§ 71. We may write <p(u, y2, y3) for ftu when we wish to specify the quantities (/2, gs.
Thus if we put /ma for a in the original definitions <@u is changed into /x2#>Mu> and </2, (/3 are changed into H*<j.2 and fi«gv
Hence p(«, ^2, gs)
In particular
Also by a second differentiation we have
WEIERSTRASS' NOTATION. Go
§ 72. The addition-formula for $>u is easily found from the formula
snOv For
'1 °2
_ 2
~\T3 B"» • * < 2 4 ' ' 4
This, translated into Weierstrass' notation, as ex plained in §69, gives, if we take v^au, V2 = av, and remember that 1 +k2= - 3/3 /a2,
the formula sought. Again,
Now p/2
, 3 Q'U — Q'V n/0 , !/#>% — p'-yV2
so that ^— - =2(2»it + 0t;)-- , ( ),
du <pu — <pv '2\^u — ^v/
' — ' and
D. E. F.
66 ELLIPTIC FUNCTIONS.
§ 73. Instead of the function E or Z, Weierstrass uses £u, defined by the equation
Differentiating, we find
£'u=-pu.
The term L is put outside the sign of integration
because $u is infinite at the lower limit, but <@u -- 2 is finite.
The value of g(u + v) is found as follows : —
Hence
2 pu —
where G is a quantity independent of u.
Also £u -- = 0, when u = 0 ; and for the same value
2 1
of u, ®u-\ — o = 0. and &u -- ^ is finite. u3 u2
rm 1 ®'u — <a'v 1 . T
1 hus 3 - — h — is zero when u = 0 and
2 <u — <>v u
Hence f(u + v)-tu-tv = -- -
2 pu —pv
The definition of f shows that since ^ is an even function, f is an odd function. Thus
and if
EXAMPLES VII. 67
1 <0'u — <&'v we have ++= -
2 v — ?16> ~ 2 —
The theory of these functions will be found de veloped in Halphen's Traitt des Fonctions Elliptiques et de leurs Applications (Gauthier-Villars).
EXAMPLES ON CHAPTER VII. 1. Prove that
/2. If u + v + w = Q, show that quantities a and b
may be found such that
"
/3. In the last question prove that
a = — 2(£u 4- fi> + gw).
4. If the equation 4*x3—g2x—g3 = Q has only one real root, prove that one corresponding value of k is a complex quantity whose modulus is unity, and that
in this case k^snuk~* is real if u is real. / 5. Show that c^^
6. Prove the formulae
(1) (IP
68 ELLIPTIC FUNCTIONS.
7. Writing rj, »/, »/' for f«, fo>', fo>", prove the formulae
(1) >i + yj' = ti',
(2) f (u + 2mo> -f 2m V) = fu + 2m^ + 2m^', if m and m' are integers ;
(3) jyo/ — ^'ft) = J£TT,
{w+w fw" g i—
& dv = («-
(5) 2f2u= 8. Show that
= (V J V>x-
0 0
9. If a and 6 have the same meaning as in Ex. 2, show that
d l y'x-a&x- b=_ &'u p'v _tfw
dx °
CHAPTER VIII. DEGENERATION OF THE ELLIPTIC FUNCTIONS.
§ 74. For certain values of the modulus the elliptic functions degenerate into trigonometrical or expo nential functions.
Thus let k = 0, then dn u = 1 always, and
d
-^-snu — cnu. du
where cnzu -f sn% = 1 ,
and snO = 0, en 0 = 1.
Therefore sn u is sin u and en u is cos u (§6),
§ 75. The six related moduli in this case are equal in pairs, the three values being 0, 1, GO . If & = 1, then dnu = cn u, and we have
-j— sn u — cn?u = 1 - sn%, sn 0 = 0. du
Put sn u = tanh 0 and we have
3(\
sech2^- = 1 - tanh20 = sech20. du
Thus 9 = u, as they vanish together.
70 ELLIPTIC FUNCTIONS.
Hence sn(^, l) = tanhu,
cn(u, l) = dn(it, l) = sech u,
E(u, 1) = I dn2(u, l)du = sn('&, 1 ) = tanh u.
o
K is the least positive value of u for which sech u = 0, that is K=cc ,
Z(u, l)
§ 76. For the case when k = oo we have sn(u, k) = -^ sn( /^, TJ = T sin ku,
du(u, k) = en ( ku, -,- ) = cos ku.
\ K//
These formulae show the behaviour of snu, cnu, dn u when u is a quantity comparable with 1/&.
The table of periods for the related moduli (§27) shows that in this case both the periods are infinite, their ratio being —1.
§ 77. When & = 0, the real quarter-period is finite, its value being JTT ; the imaginary period is infinite.
When 7c=l, the imaginary quarter-period is finite and equal to JTT* ; the real period is infinite.
It may be shown that in this case the limit of ' is finite, and in fact = — 1.
*The notation sgu, cgu for sn(w, 1), cn(w, 1) is sometimes used, in honour of Gudermann. As however the functions have names already, being the hyperbolic tangent and secant, we have not used the others.
The function arcsin tanh u is generally called the Gudermannian of u and written gd u. (See Chrystal's Algebra, chap, xxix., §31, note.)
DEGENERATION. 71
For we proved that
•K Eu du.
-log/0 = J
o
Thus $(EK-2K-logV) = r(Eu-l)du.
Also E(u, 1 ) — 1 = tanh ^ — 1
= _2e- *•/(!+ e-2«),
so that |{^(t*, l)-l}du = log(l + e-2tt) = -log 2, between the limits 0 and oo . Hence
= Lim — ^-~ — — i-log k'= — 1 ,
-~ as E=l in the limit.
EXAMPLES ON CHAPTER VIII.
1. When k vanishes, prove that
sina —
2. Show that
Ii(u, a, 1) = J log cosh(it — a)sech(t^ + a) + ?«- tanh a.
3. Prove that the degeneration of $u takes place when #23 = 27(/32.
4. Show that gd(t gd u) = iu.
5. By the substitution
b cot 9 — a tan $ = (a + 6)cot 0, prove that
rf , rl
I (a2sin20+&2cos20)'M0=l (ol%i
where 2^ = ^ + 6, 61 = o6, and a, 6, av ^ are all positive.
72 ELLIPTIC FUNCTIONS.
6. If in the last question a2, 62 are formed from al5 6j as these from a, b, and if this process is carried on, show that in the limit, when n is increased indefinitely,
an
7
= &» = | / f
(This quantity is Gauss' Arithmetico-Geometric Mean between a and b.)
CHAPTER IX.
DIFFERENTIATION WITH RESPECT TO THE MODULUS.
§ 78. The elliptic functions depend on two variables, the argument and the modulus. We must now show how to differentiate them with respect to the modulus.
Write s, c, d for smt, criu, duu, and let <r, y, S denote
|sn«, !«!«, J^clnu.
Since then -j- = cd,
du
we have -j- = yd + cS.
Since we have
Eliminating y and S,
CCA + sa-(d2 + JW) + ksW = 0. ci/u
Now ^cd^-s^ + kW),
d ( a- \ , A's2 A
so that -j—l - 7 ) + T» = °-
du\cdJ d2
74 ELLIPTIC FUNCTIONS.
rl
It/
cZ/o- foc\ &c2 fc'2-d2
j- ( — j — TTTj ) = — TTo = 77/2 '
v\cd k2d/ k2 KK* (T _ Jcsc u Eu
ancl ^Wd+jrw*
each side vanishing when it = 0. Hence
en tt dn u - 'cn u dn u,
^sn^it en u - rsn u dn « + an u dn «, /c A? A^/c
it - ku sn it en u + ~#u sn u en u.
§ 79. From the last we may further find ^jEu, as follows : —
Now
u . s2) =
u < cs,. 7 Hence ^ — =- — r— j^Eu-k
DIFFERENTIATION OF THE PERIODS. 75
Integrating,
x 7 Eu = — v>2 cn2uEu — ku sn2^ + p sn u en u dn u, since again both sides vanish with u.
§ 80. These equations enable us also to find
dK dE -jj-, -j--, etc. dk dk
We have cn(K, k) = 0, and therefore
dK K E
--, sn K dn K= 0
by dinerentiatmg. Ihus •^77 =
Again, when u = K, r^u — ~ ^-
Thus
0*
= -kK+
/i v1 Also
dk~ * dk
E-k'*K=E-K
' E'-WK' dE' E'-K'
., dK' k2K'-E'
S°that =-->
§ 81. Again
=-. = kK.
76 ELLIPTIC FUNCTIONS.
Putting k* = c, k"2 = c in this we have it in the form d
which is unchanged if c and c' are interchanged. It must therefore also hold when K is put for K, as can easily be verified.
The most general solution of the equation
is accordingly y = AK+ BK',
where A and B are any constants.
§ 82. In the same way d i]dE\_dE dK _E-K E-k'2K_ JcE
rllf\ d If I ill* ill* I* Ir1*/m% 7/2"
CC/t/ \ Cv/v / CC/A/ Ct/A-' A/ fvfv /\j
This equation is not satisfied by E' also, but we saw (§ 63) that
so that K' — E' is suggested as a second solution. Now ^JT-AT)— f-
Thus
Hence the most general solution of the equation
is z = CE+D(K'-E'),
C and D being any constants.
EXPANSION OF THE PERIODS. 77
§ 83. The differential equations just found for K and E may be solved in series, and thus the expansions of K, Kf, E, Ef in powers of k may be found.
Take
and put ?/
for the exponents of k in successive terms must clearly differ by 2. Then
The coefficients are therefore given successively by
/S-\- *2T l\?l
the relation //r=f — ^ , ^ ) /x,._i, and the values of s
by the equation s2 = 0. This equation has equal roots, so that we find the second solution by differentiating the first, namely
with respect to s before putting in the value of s. Hence, if
l 3 ... (2r — 1) and '
the complete primitive is
78 ELLIPTIC FUNCTIONS.
§ 84. We may therefore choose A and B so that this expression shall be the value of K or Kf.
Now we have seen that when k = 0,K=^7r. But yl = 1 , 2/9 = x for this value of k. Thus
Suppose that K' = A yl + By2.
§ 85. In the same way, from the equation for E we may find series for E and K' — E', or we may use the formulae
K' - Ef = W Putting 0i = (i
; ^-a
we find E=\T
K'-E' = A where
l)
Hence E =
and as when 7^ = 0,
E' = \, y\ — z\ = {\ an(l 2/2 ~~ 02 = ~~ we have iy = — 1 .
EXAMPLES IX. 79
§ 86. A , as well as B, may be found as follows : — We found (§ 66) that the limit of \(EK- 2K- log //), when k = 1 , was — log 2.
Thus in the limit, when k = 0,
, + %2) - log fc+ 2 log 2 = 0.
The coefficient of log A; on the left is -B*-2B-1. This must vanish, so that, as we found before,
B=-l.
The absolute term is — AB — 2 A + 2 log 2. This must vanish, so that
A=2 log 2.
Hence K' = ^log 2 — y2,
It is noticeable that the series yv 0X are hyper- geometric. Thus, in the notation of hypergeometric series,
EXAMPLES ON CHAPTER IX.
1. Prove that K increases with k so long as the latter is a positive proper fraction, while E decreases as k increases.
2. Show that
u te'2 k Sn2t6
and hence find — 7- II(u, a).
80 ELLIPTIC FUNCTIONS.
3. Prove that if Nn is the common denominator of sn nu, en nu, dn nu, and is equal to unity when u = 0, then
4. Writing # for sni6, transform this differential equation into the following, in which x and k are the independent variables : —
+a{(2™2-l)&2(l-^
C%C
(For Examples 3 and 4 use the result of Ex. 11, Chap. VI.) 5. Show that
6. Prove also that
and that
- 4020s™.
EXAMPLES IX. 81
7. Show, by differentiating the equation
or otherwise, that
8. Prove also that
9. Verify by differentiating that EK' + E'K-KK' and rjw —rj'co are constants.
10. Interpret the following differential equation, satisfied by <pu : —
11. Verify the values of --jj- and 1T when one of
dk dk
the related moduli &', I/A:, I/A:7, tA'/A:7, t/;'/^ is sub stituted for k.
12. Deduce the expansions of K and .Z? in powers of k by means of the equations
jr
K " 2
= P(l -Psi
0
13. From the equations of Ex. 12 find the values
, dK , dE
or -«- and -jj-.
dk dk
D. E. F.
CHAPTER X. APPLICATIONS.
§ 87. The usefulness of the Elliptic Functions con sists chiefly in this, that by means of them two surds
of the form (a + 2/3x + yx-y can be rationalized at once. One such surd could be made rational by
an algebraical substitution: thus (1— x2)* becomes
(l-y2)/(l + y2) if 2y/(l+f) is put for a:, and (1+z2)^ becomes (l + y2)/(l — y2) if 2y/(l—y2) is put for x ; but generally speaking no rational algebraical or trigonometrical substitution will rationalize two such surds.
§ 88. Let the two surds be s* and v~ where
We shall suppose the coefficients in s and cr to be real.
Also let S = A + 2Bx + Cx2
where A , B, C are found from the equations
so that in fact S =
1 — x x
c b a
y ft «
APPLICATIONS. 83
Let £ r] be the two roots of the equation S=0.
Then it is known that s and <r can both be expressed as sums of multiples of squares, of x — g,x — rj, and in fact it is easily verified, since
that s(f - 1;) = ( c£+ b) (x - ^ - ( cn + and o-(^->;) = (rf+^)(^-^)2
Also by tracing* the rectangular hyperbolas
each of which has the line £=rj for an axis, it is at once seen that the values of f and rj which they furnish are real except when the line g=t] is the transverse axis in each, and each hyperbola has one vertex lying between those of the other. This is the case in which 8 = 0 and er = 0 have both real roots, arranged so that one root of each falls between those of the other. We see also that in the identity
the product of the coefficients of the squares on the right is - {c2& + bc(g+r)} + b2}, that is ac-b2.
Hence s is expressed as the sum of two squares if s = 0 has imaginary roots, as their difference if s = 0 has real roots. The same holds for &.
If then £ and tj are real we may by the real rational
substitution y = (x — rj}/(x — g), express s^ and or in terms of ?/ and two surds ( ± 1 ±K2y2) , ( ± 1 ±/x2//2)L'
§ 89. Such a surd as ( — 1 — ipy*) will be imaginary for all real values of y. The other cases we shall take in turn.
84 ELLIPTIC FUNCTIONS.
I. To rationalize (1 -*2?M (1 -/*V )*• (Take K > p.] Put
then (1 - K2ifT = en tt, (1 - /x2/)1 - tin w.
II. (1-KV)4, Put Ky =
then (1 - Ky )4 = sn u, (1 + /x2*/2 f = V2 + /c2)Mn it.
f f (/c2-M2)-)
Put cy^ac-ju, — - — Ji
_ „!
then
IV. (/C2y2_1)i> (l_M«y«)* Here /c must > /x, or both surds cannot be real.
Put
then
/x V. (/c2?/2-!)^, ("
Put lC?/=-
then (/c2?/2 - 1 )4 = *c w, ( 1 + M2//2 )' = (* 2 + P
RATIONALIZATION OF SURJJS. 85
VI. (*y- 1)*, (MY2- I)1- (Take K > M.)
Put
then (Vy2 - 1)* =-dfl u, (My2 - 1 ) - = cs u.
In each case the value of x is given in terms of u by substituting for y in
It hardly need be. said that if £ were infinite, we should put y = x — t], and then we could go on as before.
§90. If £=»;, the process fails. But in that case is and cr have a common factor x —
Let s = (x — Put Tims s(c — y <y2)2 = (Sy* — d — £c
so tliat 8a and o-^ can be expressed by means of a single surd of the form ( A -f By2)**. This surd can again be rationalized by putting
Hence if £=rj, the surds can be rationalized by an algebraical substitution.
§ 91. The above does not apply to the case when
s = c(x — d)(x — e), cr = y(x — 3)(x — e), t/, 3, c, e being real quantities in order of magnitude.
86 ELLIPTIC FUNCTIONS.
In this case put
x — d 8z-d
Then «= cS-
Thus sa and cr are expressed by means of two surds
only, and those of the form (Ayz -\- B)'2 , which we have already shown how to rationalize.
§ 92. It is easy to verify, and important to notice,
that in each case -r— is a constant multiple of 8*<r". du
§ 93. An expression of the form
ax* + /3xs + yxZ + to + € ( = X> say )
can always be expressed as the product of two real quadratic factors by the solution of a cubic equation. Hence any expression which is rational in x and X2 can be rationalized by a substitution such as we have just discussed.
The exceptional case of § 91 need not arise. It will not be possible unless the roots of X = 0 are all real. In that case there will be three ways of resolving X into real quadratic factors, and only one of the three will lead to the exceptional case.
If a = 0, X becomes a cubic instead of a quartic ; but by a linear substitution for x of the form
the expression is made rational in y and Y* where
r=x(M2/+>)4,
so that Y is a quartic in y having ^y + v for one of
GEOMETRICAL APPLICATIONS. S7
its linear factors. Thus there is no real distinction between the cases of the cubic and the quartic.
§ 94. It must not be supposed that the rationaliz ing of these surds can only be accomplished by the particular substitutions which we have used. The number of substitutions that might be used is un limited. We have tried to choose the simplest. The comparison of the different substitutions that would rationalize the same surd or pair of surds belongs to the theory of Transformations, which is beyond our limits.
APPLICATION IN THE INTEGRAL CALCULUS.
§ 95. When an expression has to be integrated which contains two surds, each the square root of a quadratic, or one surd which is the square root of a quartic, linear functions being counted as quadratic and cubic functions as quartic, then it follows from what we have proved that the integral can be expressed by means of the functions sn, en, dn, E, II.
For the subject of integration can be made a rational function of sn u, en u, dn u by a properly chosen sub stitution, and such a function can be integrated as explained in Chapter IV.
GEOMETRICAL APPLICATIONS.
$ 96. The elliptic functions have an important use in the theory of curves, plane and twisted. This depends on the following theorem : —
The coordinates of any point 011 a curve whose deficiency is 1 can be expressed rationally by means of elliptic functions of a single parameter. (Compare Salmon, Hiyher Plane Curves, §§44, 3(i(l.)
Suppose the equation to the curve to be £7=0, and
88 ELLIPTIC FUNCTIONS.
that it has multiple points of orders, kv /K7..., its degree being m. Then the deficiency is
Km - l)(m - 2) - 2 p(£ - 1 ), and we have 2 ^k(k — 1 ) = J w(?n — 3).
Take a system of curves of the degree m — 2, each having a point of order k — 1 , where U = 0 has one of order k, and passing also through m — 2 other fixed points on the curve.
The number of arbitrary coefficients in the equation to such a curve is i(m-flXm""2), and the number of conditions assigned is 2 ^k(k— l) + m — 2, that is \ ,(??i + 1 )(m — 2) — 1 . Hence there will be one arbitrary coefficient left, and as all the equations to be satisfied by the coefficients were linear the equation to any curve of the system is S+\T= 0, X being the arbitrary coefficient and S, T determinate functions of the co ordinates of the degree m — 2, such that S = 0, T=Q are two curves of the system.
Of the m(m — 2) intersections of the curves £7=0, 4Sf+\ZT=0, 2/t(&-l) + m-2, that is m2-2m-2, are fixed. Thus only two depend on X. Call these P and Q.
Let A be one of the m — 2 fixed intersections of S + \T = Q with U=0. Replace A by any other point Al taken at random on the curve. Then we have another system of curves >S>1 + X1:T1 = 0, whose inter sections with U=0 are all fixed but two. Choose \ so that P may be one of these and let Q1 be the other.
Q1 will not be the same as Q. For a curve of the degree m — 2, satisfying all the conditions above prescribed for S + \T= 0 except that of passing through A, and also passing through both P and Q, will be altogether fixed, and all its intersections with [7 = 0 have been already specified but one. This one is A, and therefore it cannot be Ar Hence Ql and Q are different.
CURVES OF DEFICIENCY ONE. 89
The three equations U=0, S+\T = 0, £1 + A1771 = 0 will therefore enable us to express the two coordinates of P rationally in terms of X, X1? and also to eliminate those coordinates and find the relation between X and Xr
When X is given, there are two possible values for Xj, found by substituting in —^JT1 the coordinates of P and Q respectively. In the same way when XT is given there are two possible values for X. The equation connecting them must then be of the second degree in each, and may be written
This equation may be solved for \v the only irrational element being the square root of a quartic in X. Hence this is the only irrational element in the expression of the coordinates of P in terms of X, and it may be removed by a substitution for X in terms of elliptic functions.
Thus the theorem is proved.
§ 97. If the curve is not plane, but twisted, we may suppose 8 + X T = 0, Sl + Xj Tl == 0 to represent not curves but cones, of a degree lower by 2 than that of the curve. Take £7 = 0 to be a cone with any vertex standing upon the curve and S + \T=0 a cone with the same vertex, and having as a (k — l)ple edge any multiple edge of order k on U = () and also having m — 2 fixed edges in common with U = 0.
^1 + \1T1 = () may then be a cone drawn in the same way with another vertex and we may ensure that Q1 is not the same as Q as follows : —
Let the positions of P and Q when X = 0 be F and G. Through F and another point // draw a cone with the vertex that is proposed for Sl + \lTl = Q and satisfying those of the conditions that Sj+A^ssO must satisfy which are not at our disposal. Take the other m — '2
90 ELLIPTIC FUNCTIONS.
simple intersections of this cone with the curve as defining the fixed edges of the system S1 + \lTl = (). Then as G is not the same as H, Ql cannot in general be the same as Q.
The rest of the argument goes on as before, the two equations to the curve taking the place of the single equation [7—0.
The deficiency of a twisted curve is thus understood to mean that of its projection from an arbitrary point upon an arbitrary plane. In general the double points of the projection will not all be the projections of double points of the curve, but some at least will be the intersections with the plane of chords of the curve drawn from the vertex of projection.
§ 98. The simplest examples of curves of the kind in question are non-singular plane cubics, and among twisted curves the quartics which are the intersections of pairs of conicoids, and in particular sphero-conics.
If X is the parameter of § 96, and u the elliptic argument, then it follows from § 9:2 that the coordinates
are expressed rationally in terms of X and -j— , which
we may call X', and X/2 is a rational quartic in X. To each value of X there correspond two values of u and two points on the curve the two corresponding values of X' being equal with opposite signs.
§ 99. It may be proved that if a variable curve of any assigned degree meet the curve in points whose arguments are uv uz, ... , un, then
u1 + u2 + . . . + u.n = a constant.
For let </>j = 0, </>2 = 0 be any two curves of the degree assigned. Then we can prove that for the intersections of the given curve with 01H-/x02 = 0, SH is independent of /UL.
ABEL'S THEOREM. <)1
In 0! and </>.2 substitute the values of the coordinates in terms of u, and let fv /2 be the results of sub stitution.
Then u is given by the equation
da , /(//, . rZ/«
and 7 = -/9-J- -7- +/X-7-2
dp \du da
Now /! and /2 are rational functions of A and A', so that /2-J-(/i-i-M/2) is also a rational function of them, say \/r(A, A')-^x(X, X'). Its denominator may be rationalized by writing it
VKA,A')x(x, -x'Hx(x,x')x(^ -X')-
Thus since X/2 is rational in A we may write
^1 , ft, 6' being rational functions of A.
Let Ap A2, ..., An be the roots of the equation (7=0, corresponding to the values uv u2, . . . , tfcM.
Then A/C and i^/(7 may be resolved into partial fractions, there being an absolute term in the first case because A and C are of the same degree.
Hence we have an identity of the form
ft
Now of the two points for which A = A,., only one is generally to be taken, suppose that for which A' = A/. The left-hand side is therefore finite at the point for which A = A;. and A'= —A/.
Making this substitution after multiplication by A — Ar, we find pr—qr^,' = 0.
92 ELLIPTIC FUNCTIONS.
If, however, the point (Xr, — A/) is one of the inter sections we must have \it = \rt X/= - X/ corresponding to u,» another of the series uv u.2, ..., un. Then the equation 0=0 has only one root corresponding to the two arguments, and there is only one fraction (pr + gvV)/(X - Xr) for both.
But in this case the equation pr — q,.\r' = 0 does not hold, and we write
^grV + Pr X' + X/ qt\* +pr V + V
' ' ' ~ ' '
\-\r ' 2Xr' X-Xr ^X/ ~ X-X*
so that the final form is the same. The identity
ft
being thus proved to exist, we may find the value of gr in the usual way, by multiplying by X — X,- and putting u = ur.
Thus qr . 2X/ . ' Xr)
= value of /2\'
when z/,. is put for it,
= — \rdu r/dfJL.
Tliat is, </r= — ^dUr/dfjL.
Now give u such a value that X becomes infinite. Then X' is infinite of a higher order ; but as f: and f., are of the same degree, /^(fi + ftfo) is finite. Thus
and
so that 2ar is independent of //.
AN EXAMPLE OF INTEGRATION. 93
Giving /UL the two values 0 and oo , we find that ^ur is the same for the two curves 0j = 0 and 02 = 0- But these were taken to be any curves of the assigned degree. Hence the theorem is proved.
It will clearly hold also if the given curve is not plane and <p1 = ^, 09 = 0 are any surfaces of the same degree.
§ 100. The facts proved in §§ 96-8 may be applied to * integration. If y is a function of x, and the relation connecting them is the equation to a curve of deficiency 1, then any rational function of x and y may be expressed rationally by means of the functions sn, en, dn of a single variable, and may be integrated with respect to x or y by means of these functions together with E and II.
§101. Take, for instance, (l-x*
Put y = (l-a?),
so that xs -f ?/3 = 1 •
This is a cubic without singularity, so that the de ficiency is 1.
Put x+y=z*
Then z*-3xyz=l,
z2 1
The radical is therefore (40 — 04).
The real quadratic factors of z* — 4>z are
z(z-tf) and
* Here z takes the place of the X of § 96, and the curves 5=0, 7^0 are respectively the straight line x + y = 0 and the line at infinity, the point of intersection of these two being clearly a point on the curve.
94 ELLIPTIC FUNCTIONS.
The roots of the equation
=0
are
Hence we put
that is, z=- Then
We therefore take
using the substitution II of § 89, since the radical
is (4>z -zrf, not (04-4^)l Then
- l)'2z(z- 2*) = -
Thus (# - 1) V(aj - y)2 = 2(2 - ^/3)2Hii2u dn%, a; - y = 2^3 J(2 - >v/3)sn u dn u
u){l - ('2 - V;>)cn u
AN EXAMPLE OF INTEGRATION. 95
From these equations x and y can be found at once. Now if v be written for 1 (1 — x3)~ dx, we have, since
Vnt
so that v = 2~§ . 3* . u + const., that is to say,
the modulus being (/v/3-l)/2/v/2.
§ 102. It should be noticed that when a is a con stant, the equation connecting snu and sn(u + a) is of the same doubly quadratic form as the one found between X, \ in § 96.
For the two values of sn(u + a) when snu is given are sn(u-j-a) and sn(2/lT — u-\- a). Their sum is
2 sn u en a dn a -j- (1 — &%n*u sn2a), and their product is
(sn2u — sn2a) -=- (1 — &2sn2M. sn3a). Hence sn2(tt + a) { 1 — fc*sn2u sn2a)
— 2 sn(i6 -f- a)sn u en a dn a 4- sn2tt — sn2a = 0, that is, A;2sn2a sn2^ sn'2( a + a) — sn2(& + a) — su-u + 2 sn(it -h a)sn u en a dn a + sn'2</ = 0.
96 ELLIPTIC FUNCTIONS.
The same holds for any other of the elliptic functions sn, en, dn, sc, etc.
This suggests another way of integrating Euler's equation (§ 40) which was given by Cauchy.
Let <j)(x, y}=0 be an equation of the second degree both in x and y, and let
Then
^ = 2( But since (p(x, y) = 0 we have
and (X0y + XJ2 = X,2 - X0X, = X, say.
Hence <p(x, y) = 0 is an integral of the equation
X~ dx + Y~*dy = 0, and X and Y are quartics in x and y respectively.
Also if in </>(x, y) the coefficients of x2y and xy2 are equal, as also those of x2 and y2, and those of x and y, then <£(#, y) will be symmetrical in x and y, and Jf will be the same function of x that Y is of y. Also the number of coefficients in 0 is still one more than the number in X or Y so that if the coefficients of X and Y are known, 0 = 0 will contain one and only one arbitrary constant, and will be the complete primitive.
§ 103. If in a doubly quadratic equation connecting x and y we transform x or y or both by substitutions of the form x = (e£-\-f)/(y£+h), the transformed equa tion is still of the same form in the new variables, though with different coefficients.
Now there are three arbitrary constants in such a transformation, and they may be so chosen as
DOUBLY QUADRATIC EQUATIONS. <)7
to make the transformed equation symmetrical, since symmetry is ensured if six coefficients are equal in pairs, namely those of x2y, xz, x to those of xy2, y2, y respectively.*
When the expression has been made symmetrical, x and y can be rationalized by a substitution for either in terms of elliptic functions, the two substitutions being of the same form and having the same modulus but different arguments. It follows however from the differential form of the equation that if u and v are the two arguments,
du = ± dv, u±v = & constant. Hence transformations a; = -4-Hr, = — 1-. can be
found such that £ and r\ are the same function (sn, en, dn, sc, etc.), with the same modulus, of arguments differing by a constant.
* With the notation of § 102, it may be proved that the anhar- monic ratio of the roots of X = 0 is always the same as that of the roots of Y = 0.
For, by putting xy = z, 0(#, y) may be made a quadratic function of x, y and z, so that the two equations xy - z - 0, 0 = 0 represent a twisted quartic curve. The cone standing on this curve whose vertex is any point of it will be a cubic cone and the anharmonic ratio of the four tangent planes to it drawn through any one of its edges is a constant. (Salmon, Higher Plane Curves, §167.) Thus if A, B, C, D are any four points on the curve the four tangent planes through AB have the same anharmonic ratio as those through BC, and these have the same as those through CD.
Now let A B, CD be the lines at infinity in the planes x = 0, y = 0 respectively, these being chords of the curve xy = z, 0 = 0. The equations X = 0, Y = 0 represent the two systems of tangent planes and the theorem follows. Another proof is given by Salmon (Higher Plane Curves, § 270).
It follows that by a linear transformation of x the roots of X = 0 can be made the same as those of Y = 0. This is the transformation wanted, for it may be verified that 0 is symmetrical if the coefficients in X are proportional to those in Y. In carrying out this verification it is advisable to suppose X and Y reduced to their canonical form, in which the second and fourth terms are wanting. (See Salmon, Higher Algebra, § 203.)
D. E. F G
98 ELLIPTIC FUNCTIONS.
§ 104. This applies to any case in which two para meters are connected by an algebraical relation, such that to each value of either there correspond two values of the other. There are two or three important cases of this which we shall now discuss.
In the first place, let P, Q be two points on a conic, such that the line joining them touches another fixed conic. If P is given there are two possible positions of Q, one on each of the tangents from P to the other conic. The relation between P and Q is reciprocal, and the coordinates of each may be expressed rationally in terms of a single parameter. Hence the parameters of the two points are connected by a doubly quadratic equation of the form we have been considering.
The same may be proved if the tangents at P and Q are to meet on another fixed conic, or if P and Q are to be conjugate points with respect to another fixed conic. It is in fact known that these three conditions are only the same stated in different ways.
§ 105. Jacobi has given a full discussion of the case when the two conies are circles, into which they can always be projected.
Take any four points A, a, /3, B (Fig. 2), in order on a straight line, and on AB, a/3 as diameters describe circles. Let the centres be Q, 0, the radii R, r, and let OQ = S.
Let P, Q be two points on the outer circle, such that PQ touches the inner circle at T. Let PTQ' be a consecutive position of PTQ, meeting it in U.
Also write 0 = BAP, <p = Then
2Rd</>.
JACOBI'S CONSTRUCTION.
99
The angle and the angle
Thus
PUP' = QUQ\ P'PQ = QQ'P'.
and in the limit
dO _d<f> PT~TQ
But PT2=OP2-OT2
Fig.
If then we write
sn =
>, k),
we have cos 0 = en u, cos 0 = en v,
100 ELLIPTIC FUNCTIONS.
Also cos 9 dO = en u dn u du,
dO = dn u du. Thus du = dv,
v — u = a, a constant.
§106. If now we put £ = tan0, ^ = tan0, the co ordinates of P and Q can be expressed rationally in terms of £ and r\ respectively, and we can find the algebraical relation between £ and r\ that follows from the equation v — u = a.
Take £25 as axis of x, and a perpendicular to it from Q as axis of y. Then the equation to PQ is
&cos(0 + 0)+y sin(0 + <£) = # cos(0-0). The perpendicular drawn to it from 0 is r. Hence
R cos(0 - 0) + S cos(0 + <£) = r, that is,
Putting ?•/( « + (&) = cn(a, k)t
the value of cos <£ when 0 is 0, we find
(R-S)/(R + S) = dn(a, k).
Thus 1 + 2£y dn a + ^2dn2a = ( 1 + f 2)( 1 + »;2)cn2a. Solving the quadratic for q, we find
_ - f dn a ± sn a en a(l + f 2)^( 1 + /^/2^2)"
As was to be expected, this is rationalized by the substitution £=sc(u, k), and becomes
_ sn u en u dn a + sn a en a dn u cnzu cn2a — &'
x sn u en u dn a + sn a en a dn u so that BCitt -f a) = — — o - pi — 2 - 2 -- »
cn2^/, cn2a — k 2sn2a sn%
JACOBI'S CONSTRUCTION. 101
the lower sign being taken in order that the two sides may agree when u = 0. This is justifiable because <t. was found from its en and dn, and therefore the sign of sn a is as yet undetermined.
The equation just found is one of the addition- formulae. Others may be written down at once from the figure. For instance,
PT+ TQ = 2R sin(0 - 0),
that is, (72-f (5)sna{dn^ + dn?;}
= (R + S)( I + dn tt)(sn v en u — sn u en v),
sn(i& + a)cn u — sn u cn(u -f- a) sn a
or
dn(u + a) + dn u 1 -f dn a'
§ 107. When the outer circle and AB, the axis of symmetry of the figure, are kept fixed, the quantities a and k depend on the position and size of the inner circle. It is of some importance to know under what circumstances the modulus k will be constant.
Now k* = 4<m/{(R + S)*-r2}.
But if s is the distance from £2 of the radical axis of the two circles
and 2sS
so that s = 2R/k2-R.
Hence if the inner circle vary so as always to have the same radical axis with the outer, the elliptic functions will have the same modulus. The quantity a is then the argument belonging to the other end of a chord of the outer circle drawn from B to touch the inner circler
$ 108. An interesting case is that in which the inner circle has its radius zero, so that all the tangents to it
102 ELLIPTIC FUNCTIONS.
pass through the inner limiting point' of the coaxial system.
In that case ciiei = (), so that a is an odd multiple of K, if real. Let L be the limiting point. Then if PL produced meet the outer circle again in Pv the argument u -f K belongs to the point Pr
Thus u + 2K belongs to P. It should, however, be noticed that when the argument u is increased by %K in this way, 0 is increased by TT only, so that snu and en u have signs opposite to those they had before. The signs of BP and AP are in fact changed, be cause the positive direction of measurement has been changed in each case by a rotation through two right angles.
We have then sn u = BP/BA , = AP/BA,
and, travelling along the arc
Now BPl = BA sinBPL = BA sin PEL x BL/PL
= PA. BLfP L.
Thus sn(u + K) = cd u.
Also AP^PB.AL/PL.
Now AL/BL = dnK = k'.
Thus cn( n + K ) = — 7/sd u ;
and since PL .LP^BL.LA,
$ 109. The coaxial system of circles have a common self-polar triangle of which L is one angular point, the other two being L' the other limiting point and
PONCELETS POLYGONS. 103
the point at infinity in a direction perpendicular to AB, which we may call M.
The figure shows that if L'P and MP meet the circle again in P2 and P3, the arguments belonging to P2 and P3 are K—u and — u respectively, for P2P3 passes through L.
But since sc('2iK'— u) = scu, every point on the circle has two distinct (that is, not congruent) argu ments belonging to it, and the second arguments belonging to P2, P3 are respectively congruent to 2tK' + K+u and 2iK' + u (mod. 2^, 4Jf ).
It is now clear that if the inner circle in Jacobi's construction is replaced by a circle of the same coaxial sytein, but containing the other limiting point, then the quantity a is not purely real but has its imaginary part equal to an odd multiple of 2iK'. If on the other hand a is purely imaginary, its en and dn are real, so that the inner circle is to be replaced by a real circle of the system, but one which contains the original outer circle.
§ 110. By help of the foregoing we can answer the following question : Can a polygon of an assigned number of sides be inscribed in one given conic and circumscribed to another ?
Project the two conies into circles as before. Let u be the argument of one angular point, u + a that of the next, then u + 2a will be that of the third, and so on, and if the polygon has n sides and is closed the argument u + na must belong to the first angular point.
Hence u-{-na = u or 2iK'— n (mod. 2K, 4iK').
Suppose first that
u-\-na = ^iK' — u, then u+ a = 2iK' — u — (n—l)a,
u + 2a s 2 iK ' - u - (n - 2)a, etc.,
104 ELLIPTIC FUNCTIONS.
so that the second angular point coincides with the nth, the third with the (n — l)th, and so on. Thus there is no proper polygon in this case.
If on the other hand we take u + na = u we find
a = 0 (mod. 2K/n, 4iK'/ri).
This condition does not assign any of the angular points, but only shows that unless the two conies are related in a particular way the problem has no solu tion. If the conies are so related, that is, if a has one of the values included in the formula (ZrK-\-4>siK')ln, then the value of u does not matter, and any point on the circumscribing conic may be taken as an angular point of the polygon.
AECS OF CENTEAL CONICS.
§ 111. It is most likely known to the reader that the length of any elliptic arc can be expressed in terms of the coordinates of its ends by means of the elliptic functions sn, en, dn, E, and that it is from this fact that the name " elliptic " arises.
The ellipse xz/a? + y2/L>2 = 1 is the locus of the point (a sn u, b en u) for different values of the argument u.
If S is the length of the arc measured from one end of the minor axis (0, b) then S vanishes with u and
(dS/du)* = (a2cn2
So far we have not assigned the value of k. If we take e for its value we have
dS/du =
and 8 = aE(u,e)t
if x = asn(u,e),
y — ~bo,n(u, e).
ARCS OF CENTRAL CONICS. 105
This expression holds equally well for the hyperbola, but it is not so useful, as the modulus of the elliptic functions is then greater than 1 and the point from which the arcs are measured is imaginary, b being imaginary.
§ 112. In the hyperbola x2/a? — y2/b* = l we may however put
y = I cs(7i — u) = 6//sc u, x = a ns(K-u) = a dc u.
so that u vanishes for the point (a, 0).
If ti is the length of the arc measured from this point we have
(dS/du)z = <>2&'4sc
if a*k'* = b*k*t that is k= l/e.
Thus dS/du = bk'uchi if k = l/e, and S = ae { sc u dn u + k'2u — Ew}.
§ 113. The equation
Eu + Ev - E(u + v) = k2sn u sn v sn(u + v)
may be expected to furnish a geometrical theorem concerning arcs of a central conic.
We must first find what geometrical condition is expressed by such an equation as u — v = t, connecting the arguments u and v of two points on the ellipse. It will be more convenient to put
The tangents at u, v are then
106 ELLIPTIC FUNCTIONS,
and at their intersection we have
/Y» /yy
- sn a en /3 dn /3 + -.- en a cri /3 = 1 — &2sn2a sn'2/3,
- sn 8 en a dn a = , sn a sn /3 dn a dn 6, a b
whence x = a sn a dc /3,
Eliminating a, we have
o?2/a2dc2/3 4- 2/2/^2nc2/3 = 1 . Eliminating /3, we have, since e is the modulus,
Each of these conies is confocal with the original one. Thus if u±v is constant, the intersection of tangents at the points u, v traces a confocal conic.
§ 114. At a point on the tangent at u whose dis tance from the point of contact is z we have
x — a sn i& — cuu z
a en u ~ — b sn u ~ a dn u so that x = a sn u + z cd u = a sn u + z sn.(u + K), y = b en u -f- z cn(u + K).
It is hence easily found that the lengths of the two tangents at (a ± j3) measured to their intersection are
a sc /3 dn a dn(a ± /3). Call these tv f2. Then
#! + *2 = 2a sc /3 dn2a dn ^/(l - A:2sn2a sn2/3), f t _ t2 = - 2ae2sn2^ sn a en a dn a/(l - 7,l2sn2a sn2/3). Now by the addition-formula for the function E E(a + ft) _ Ea -E/3 = - tfsn. a sn @ sn(a E(a -ft)- Ea + E& = /.'-sn a sn /3 sn(a -
GRAVES' THEOREMS. 107
and by addition and subtraction
E(u + /3) + E(a - /3) - 2 A«
sn a en a dn a/( 1 - £2sn2a sn2/3)
= - 2fc2sn2a sn £ cii 0 dn £/(! - &2sn2a sn2/3)
If then a + /3, a — /3 are the arguments of the two points P and Q the tangents at which meet in T, and if B is the point from which the arcs are being measured, we have, when T traces a confocal ellipse, so that /3 is a real constant,
arc BP - arc BQ-TP-TQ = Si constant, or TP + TQ - arc PQ = a constant ;
and when T traces a confocal hyperbola, so that a is a real constant,
arc BP + arc BQ - TP + TQ = a constant - twice arc BR,
if R is the point of intersection of the hyperbola and ellipse between P and Q. Thus
TP - arc IIP = TQ - arc RQ.
§ 115. This applies also to the hyperbola, but since in that case b is a pure imaginary the relation
TP+TQ- arc PQ = a constant holds when T moves along a confocal hyperbola, and
when T moves along a confocal ellipse.
For geometrical proofs of these theorems, which are due to Dr. Graves, see Salmon's Conic Section* (Lap. XIX.
108 ELLIPTIC FUNCTIONS.
It is noticeable that the system of confocal conies is the reciprocal of a system of coaxial circles with respect to one of the limiting points, so that this case is closely connected with that of §§ 107-110.
A CASE IN SPHERICAL GEOMETRY.
§ 116. Another case of a doubly quadratic relation between two parameters is afforded when an arc of a great circle moves on a sphere so as always to have its two ends on two fixed great circles, its length being constant.
Let PQ, P'Q' be two consecutive positions of the movable arc, GPP', OQ'Q the two fixed arcs (Fig. 3).
Fig. 3. Let OP = 0
POQ = A,
Then the integral equation connecting 0 and 0 is cos 0 cos (p + cos A sin 0 sin cp = cos a.
To form the differential equation, since PQ = P'Q', we have PP'cos OPQ = Q'Q cos OQP in the limit, that is,
(I — sinM cosec2
+ (1 - sinM cosec2a sin2#)^0 = 0.
THE AMPLITUDE. 109
We may then put
sin 9 = sn u, cos 0 = en u, cos OQP = dn u,
sin 0 = sn t>, cos 0 = en v, cos OPQ = dn t>,
the modulus being sin A cosec a, and we have du + dv = 0, ?£ + v = constant = iv, say.
Then it; is the value of v given by supposing u and therefore 0 to vanish, so that
en w = cos a,
and we have en iv = en u en v — dn w sn it sn v, that is, cn(it + v) = en u en ?; — sn u sn v dn(u+v ).
This is one of the addition -formulae. We have also
cos 0 — cos a cos <£ + sin a sin 0 cos OQP, or en u = cn(u + i>)cn v + sn(i& + f)sn ^ c^n u» and en v — cn(u -f v)cn it + sn(u + t')sn u dn v.
These three equations may be solved for
If the modulus is to be real and less than unity and w real, we must have A obtuse and a -{-A greater than two right angles. We may then write
sin 0 = sn u, cos 0 = en u,
sin 0 = sn^t; — u), cos 0 = cn(iv — u),
w being a constant.
§ 117. In this case we have
dO/du = dn u or du/dO = (1 - tffcin^)'*.
The function 0 of u which satisfies this condition and vanishes with u was called by Jacobi the ampH-
IK) ELLIPTIC FUNCTIONS.
tude of u, it being the upper limit on the right-hand side of the equation
u
= f'(l-fc2sin
It was also customary to write AO for ( 1 — Thus snu, cuu, dnu were conceived as the sine, cosine and A of the amplitude of u, and in Jacobi's notation were written sin am u, cos arn u, A am u, the amplitude 6 being denoted by am u. The shorter notation, sn, en, dn, was suggested by Gudermann.
The function am u is of no importance in the theory of elliptic functions, but it sometimes presents itself in the applications of the theory. In the case con sidered we may, for instance, write
0 = am u, $ = &m(iv — u).
\J APPLICATIONS IN DYNAMICS. THE PENDULUM.
§ 118. There are certain problems in dynamics whose solution can be expressed by means of elliptic functions. The simplest is perhaps that of the motion of a pendulum.
The equation of motion is
10= —<j sin 0,
where 0 is the inclination to the vertical of the plane
through the axis of suspension and the centre of inertia
and I is the length of the simple equivalent pendulum.
A first integral is found by multiplying by 0, it is
$l02 = cj(K + cos 0) = g(l +/c- 2 sin2|0), AC being a constant. To integrate this put
THE PENDULUM. Ill
so that sin JO = sn u,
cos J (9 = en u,
Then u* = (l+K)g/2l,
and u = t{(l+K)2l}^ + const.
§ 119. Let A, B be the highest and lowest points of the circle described by the centre of inertia of the pendulum, P its position at any time, h its distance from the fixed horizontal axis, and let
Then
if the time is measured from the moment when P is at B.
If PY is the perpendicular drawn from P to a horizontal plane at a distance /c/i above the axis, that is, at the level of zero velocity, we have
PY=(l+K)hdu2nt,
Let BA , produced if necessary, meet this plane in (7. Then let a circle be described having CY as its radical axis with the circle APB. The tangent from P to
such a circle varies as P Y , that is, as dn nt. Hence the figure is the same as that in Jacobi's construction (§ 105 above).
§ 120. The application of the addition-formula will then give us the following theorem : —
The envelope of the line which joins the position of the centre of inertia at any time to its position at a fixed interval afterwards is a circle of the coaxial system which has for radical axis the line of zero velocity, and includes the circle described by the centre of inertia.
112 ELLIPTIC FUNCTIONS.
When the pendulum is performing complete re volutions K ^ 1, and the elliptic functions have a modulus ^ 1. Thus if the fixed interval is half the whole time of revolution, the straight line joining the two positions will always pass through a fixed point, namely, the inner limiting point of the system of circles, whose depth below the radical axis is
Further, the envelope of the line joining two variable positions of the centre of inertia, which are separated by equal intervals of time from any fixed position (one before, one after) is a circle of the same coaxial system ; and if the revolutions are complete, and the
fixed position is at a depth h(K2 — I)2 below the line of no velocity, the line always passes through the outer limiting point.
The velocity of the centre of inertia varies as the tangent drawn from it to any fixed circle of the coaxial system, or in the case of complete revolutions as the distance from either limiting point.
§ 121. In the case when the pendulum oscillates, 1 — K is positive, so that the modulus of the elliptic functions is greater than unity. The expressions may be transformed by the usual formulae ; putting g = lm'2, we have
the modulus being now 2~-( I +*:)'"• The velocity varies as en mt.
The general theorems derived above from the addition- formula still hold, the system of coaxial circles having now real intersections, namely, the extreme points reached in the oscillation. The limit ing points are however imaginary, and the line joining
MOTION UNDER NO FORCES. 113
positions separated by an interval of half the period is always horizontal, as is also that which joins two that are separated by equal intervals from the lowest. The coaxial circle, which is the envelope in this case, consists of the radical axis and the line at infinity, and the tangents to it pass through their intersection.
MOTION OF A RIGID BODY UNDER NO FORCES.
§ 122. Another interesting case is that of a rigid body in motion under the action of no forces. The centre of inertia will then move uniformly in a straight line or be at rest, and the motion of the body about its centre of inertia will be unaffected by the motion of the centre of inertia, which we will therefore suppose to be fixed.
Let cov o>2, o>3 be the angular velocities of the body at any time t about its three principal axes of inertia, and let A, B, C be the three corresponding moments of inertia, and suppose that they are in descending- order of magnitude.
The equations of motion are then
The form of these suggests a substitution
wx = a en qt, u>2 = — /3 sn qt, «3 = y dn qt,
since the sign of C—A is negative and opposite to those of B-C, A-B.
Making the substitution we have
-Bq/3 = (C-A)ya,
D. E. F.
1H ELLIFl'IC FUNCTIONS.
6
The equations are therefore satisfied if
-(7)(.i -
where
g
and the arbitrary constants of integration are jf, the modulus k, and f0.
The following two important equations are easily found either from the equations of motion or the integrals : —
A X2 + #V + ^ V =f\tfAB + WA C-BC)I^=G\ say.
§ 123. Suppose now that (/, m, n) are the direction- cosines of a straight line fixed in space. We then find
and wv a).,, co3, are now known functions of t. If these equations can be integrated the problem is completely solved.
The equations give
and therefore I'2 + m2 -f n2 = constant. The value of this constant is known to be
MOTION UNDER NO FORCES. 115
Also A &J, + Boj2m -f- Cw3h
= 1(0— #)w.2o)34- m(A — CVgWj 4- n(# —
Hence A lu>l + Bma>2 + CnwB = K, a constant.
This equation expresses that the line (I, m, 71) makes a constant angle with that whose direction-cosines are (AwJG, BcoJG, Oo)3/6r) and shows therefore that this latter is fixed in space. It is easily found that the equations are actually satisfied if
Z = ^Lo)1/G, m = Ba>2/G, n =
§ 124. We may now simplify the problem by sup posing the line (I, m, n) to be perpendicular to this known fixed line, that is by putting K = 0.
Let (X, //, v) be the direction-cosines of another line perpendicular both to (I, m, ri) and to
so that G\ = Cma)3 — Bnw.2, etc.
Then since (X, /*, v) is also fixed in space we have
X = and l\ — \l =
--l.
Also l* + \* + A'2a>i2/G2 = l.
Hence ^ arctan l/\ = G(T-Auf)/(G* - 4 V)- Thus ^ = X tan v,
if
-
This integral can be expressed in terms of the function H, for the subject of integration is a known function of t.
116 ELLIPTIC FUNCTIONS.
Then I, m, n are given by the equations
A Ico! + Bma)2 + Cnu>3 = 0, Gl cot v — (7wo>3 + -8^ft>2 — 0,
771
-ACc 1
G cosec <B To find X, /UL, v we need only change v into v+-^-
in these expressions.
Referred to the three fixed axes, the direction- cosines of the principal axis of greatest moment are (AwJG, I, X), those of the mean axis (BcoJG, m, /x), and those of the third principal axis (CcoJG, n, v). Hence the orientation of the body is completely determined at any time.
The actual value of v is found to be
if su a = t{A(JJ-C)/C(A -B)}*,
the values of en a, dna being both positive, as well as that of — isua. VQ is the value of v when t = t0, and it varies according as different straight lines in the " Invariable Plane " are considered, a is a purely imaginary constant depending on the nature of the rigid body, k may be any real quantity. If it is numerically greater than unity the formulae may be reduced by the usual transformation to others in which the modulus is less than unity.
The values of arctan rti/f/. and arctan n/v might have been found in terms of II functions instead of that of
ATTRACTION OF AN ELLIPSOID. H7
arctan l/\ ; the formulae thus found must however reduce to those we have by means of the formula for addition of parameters in the function II.
A further discussion of the motion, with references, may be found in Routh's Advanced Rigid Dynamics (Chap. IV.).
ATTRACTION OF AN ELLIPSOID.
§ 125. The potential of a solid homogeneous ellipsoid at any point may also be conveniently expressed in terms of elliptic functions.
The expressions
x2 = a2aW2/(a2 - b2)(ct? - c2),
7/2 = 1*W'*/(12 - C2)(62 _ a2),
for the coordinates of any point in terms of the semi- axes of the three conicoids of a confocal system that pass through it, suggest that we make x,y,z constant multiples of S, C, D respectively where
S = sn 1&! sn u2 sn u3 = sl s2 s3, say,
C= en u± en u.2 en u, = c^c2 cs,
D = dnitj dn u2 dnu3 = d^d^d^ Since JM\Z8* - ^ClV + dfd* = V*, we have Vlfl&/sf - ^C2/lr2 + D'2/d* = k'2,
where r = 1 , 2 or 3. This equation is the relation that connects S, C, D when ur is a constant. If then we put
I being any constant, the locus of (x, y, z) when ur is a constant will be a conicoid whose semi-axes are the square roots of
118 ELLIPTIC FUNCTIONS.
The differences of these quantities are constants, so that the different conicoids are all confocal.
§ 126. For an ellipsoid the imaginary part of v.r must be an odd multiple of iK '. It will be more convenient to have ur real in this case ; we therefore put ur + iK' for ur throughout, and we have
x = W/kS, y = l. iD/kS, z=-l. C/8, the squares of the semi-axes being now
When ur is constant and real, we now have an ellipsoid, when its real part is an odd multiple of K a hyperboloid of one sheet, and when its imaginary part is an odd multiple of iK' a hyperboloid of two sheets. In other cases the surface ur = constant is imaginary.
Since then one surface of each kind passes through any point, we may suppose uv i(u2 — K), u^ — iK' to be all real.
The semi-axes of the focal ellipse are found, by putting ur = K, to be Ik' and Ik'2 ; and, as I and // are arbitrary, these may be made equal to any lengths whatever, so that any system of confocals whatever may be represented in this way.
§ 127. We must now transform the equation V2F=0, that is,
&y 32 F 32F
3a2 + 3?/2+302~~
Now, in the first place, if V is expressed in terms of S, C, D,
3F 3F
ATTRACTION OF AN ELLIPSOID. HO
'* ^1C1S2^2^3^3 0"^1^2S3\^1 i "* ^1 /
with symmetrical expressions for 32F/9?/22, Thus
all the other terms disappearing.
If then we put x = W2k'S, y = lk2C, z = liDy we have
If now we change ur into ur-\-iK', this becomes
The equation V2F=0 is therefore to be replaced by
+ S22(s32 - 8*)& Vfdu* + s32(-s12 - s22)92 F/9^32 = 0.
§ 128. Now it is known that the equipotential sur faces of a thin homogeneous homoeoid (shell bounded by two similar, similarly situated and concentric ellipsoids) are the confocal ellipsoids that lie outside it, that is, the surfaces represented by ^ = constant
120 ELLIPTIC FUNCTIONS.
if our confocal system is that to which the surface of the shell belongs.
If V is the value of the potential it is a function of u^ only, satisfying the equation just written, which now becomes
Hence V=Qu1 + R, Q and R being constants.
Now V vanishes at infinity and at very distant points is in a ratio of equality to M/r where M is the mass of the shell and r the distance of the point from the centre.
Also at infinity U1 = 0, and for small values of i^ the surfaces may be regarded as spheres of radius lk'/sr Hence when u is small we have
that is, R = 0, Q
The potential of the hornoeoidal shell is therefore
§ 129. If now we have a homogeneous solid ellipsoid whose semi-axes in descending order of magnitude are a, b, c and whose density is p, it may be divided up into thin homoeoidal shells, to each of which the fore going will apply. To get the different shells we need only suppose I to vary in the above expression from
0 to p sn vv its value for the outside surface, v1 being
the constant value of vl for the outside surface referred to its own system of confocals.
The sum of the volumes of all the shells up to any value of Z is
|7r£3//3cn v xdn ^/sn3^,
so that we substitute for M the expression l . 7/3cn Vdn sn3
ATTRACTION OF AN ELLIPSOID. 121
which is the differential of this with respect to I multiplied by p. The potential of the solid ellipsoid at an external point is therefore
and i&j is given as a function of I by the equation + 2/2sd2 Ui
(x, y, z) being the coordinates of the external point. We find at once
k'H di = O
Thus if now we write u^ for the value of u^ at. (aj, 2/, 0) in the system of confocals to which the outside surface belongs we have for the potential
vldnvl P1 r 2 + y^^llf + ,^nc%} sn ucnudnudu *VJ I
cnt'1dnvir x „ 9 „ 19
73 Uj^sn2^! + y sd2ux -f a
- (o;2sn2it-
o Also by definition of uv
JUi u 1
sn2u du = 3 — TZ^UV '' o
1 sni^cni^
r
0
r(
1 sni^dnu, 1 = -,- /9 - *— - — j^ k2 en t A; 2
122 ELLIPTIC FUNCTIONS.
Hence the potential
Here
dn vl — b/a, en vl = c/a,
and u^ is the least real positive argument that satisfies the equation
#2snX + ?/2sd2i61 + sfiacX = a2 - c2.
If the point (a;, y, z) lies on the outer surface, we have itj = v1
§ 130. If the point (x, y, z) lies inside the ellipsoid, the above formula ceases to hold. We may however describe through (x, y, z) a similar, similarly situated and concentric surface, and use the above expression for the volume contained.
If \a, \b, Ac are the semi-axes of this one, its potential is
1/2 /; 2cn
EXAMPLES X. 123
We have then to deal with the outer shell. This may be divided into thin homoeoids as before. The potential of each is the same at all points inside it, and equal to
This is to be integrated with respect to I between the limits \a sn vjk' and a sn v Jk', and added to the potential of the inner part. The integral is
2?rpa2(l — A2) . v xcn i^dn vjsn. vv
and the potential of the whole ellipsoid at an internal point (x, y, z) is found to be
cn?;,dn v-,\~ — 5- l \ v BU'V,
, k -en ^
The expression is the same as for an external point, but that the constant vl takes the place of the variable ur
EXAMPLES ON CHAPTER X.
1. Prove that (1 — 2#2cos 2a + x4y* can be rationalized by putting
x + - = 2 ns( 276, cos a) ,
Qu
and that then x — = — 2 cs(2?t, cos a),
124 ELLIPTIC FUNCTIONS.
IV x2 - 2 cos 2a + • ) = 2 ds(2tt, cos a),
0
2. Discuss the spherical figure of § 1 1 6 in the case when sin A > sin a and show that in that case we may put
sin OPQ = sn(u, sin a cosec A),
sin OQP = sn(w — u, sin a cosec A ), where TT — A — am w.
3. If cos 0 = cos 8 dn w, tan <£ = W ^ sc u,
r sm /3
where cos a = /;' cos /3, prove that the point whose polar coordinates are (ft, 0, <£), R being a constant, traces a sphero-conic whose semi-axes are a, /3 and that the area of a central sector of this sphero-conic is
dn u du
4. Prove that the chord joining the points u±a on this sphero-conic touches the sphero-conic whose equa tion is
cn2a = cot2/3 dn2a cos2^ + cot2a sin20,
and that this has the same cyclic arcs as the former one.
5. Show that the sector bounded by the semi- diameters to the points u±a differs from
2^2 A dn u(sn2a -f cn2q cos2/3) -f dn a cos /3 sn a en a sin a sin /3
by a quantity independent of u.
Prove also that the area of the spherical triangle
EXAMPLES X. 125
formed by these two semi-diameters and the chord joining the points u ± a is
sin a sin B sn a en a dn u
2/c-arctan r — . 2 , -, ? — —5,
1 — sii2a sin2a + dn u dn a cos p
and that the area of the segment cut off by this chord is independent of u.
6. In the same sphero-conic
(cot*0 = cot'2a sin2</> + cot2/3 cos20) prove that by the substitution
tan 0 = tan a cot /3 sin a cosec /3 cs(i&, &), where // = sin /3 cosec a,
the expression for the arc is reduced to
R tan a tan B sin B I — «- .
J tairp sn-u + tan-a cnrtt
7. Prove that at tlie intersection of tangents to this sphero-conic at the points u±a (as in Ex. 0, not Ex. 3)
cot 0 _ cot a sin 0 _ cot /3 cos 0 en a dn t& ~~ dn acnu ~ /-'sn u
and that as u varies this point traces the confocal sphero-conic
cot>2$ nc2a = cot'2u sin>20 nd2t6 + cot'2^8 cos'20.
8. The length of the tangent at u + a in the last example is
T, tan a tan 6sna sin -8
/tarctanr — ~ — — r-
taira cnucu(u + a)dna + tan-p snit sn(it -f a)
Find the differential coefficient of this expression with respect to u in the form
7£ tan a tan /3 sin ~
tan*a
L. 1
tan-3 snHtJ
126 ELLIPTIC FUNCTIONS.
and prove that the sum of the two tangents exceeds the intercepted arc by a quantity independent of u.
(Compare Salmon, Geometry of Three Dimensions, §252.)
9. Verify that when
sn% tan2/3 + cn% tan2« = 0, then
sn u = ± , cos ft, en u = ± , sin ft cot a, dn u — ± sin ft,
and the above expression for the length of the tangent becomes R arctan±t.
10. Prove that the following equations give the motion of a heavy particle constrained to move on a fixed smooth spherical surface : —
cos 9 = cos a sn2o)£ + cos ft cii2wt, du
rdu 1
cos'2|a sn2^ + cos'2^ cn2?^ J '
o
where 9 is the angular distance of the particle from the lowest point, a, ft are the greatest and least values taken by 9 during the motion, r/> is the angle made by the vertical plane through the centre and the particle at time t with its initial position, t being measured from a time when 9 = ft, I is the radius of the sphere, and
A;2 = (cos2 ft — cos2a)/(l + cos2 ft + 2 cos a cos ft),
1 ((/(cos ft — cos a)\^ ft) = M~ — 21 — j ' n2 = 4 sin2a sin2/3/( I + cos2 ft + 2 cos a cos ft).
EXAMPLES X. 127
I J . Reduce the above value of 0 to the form n{w>t cosec2/3-f Isc a cosec J- a cosec J/3II(a>£, a)
+ Jsc b sec \ a sec |/3II(a>£, &)}, where tin a = sin J a/sin J/3,
tin b = cos | a/cos J/3. What is the general character of the motion ?
12. On a curve of deficiency 1 and degree n, the sum of the arguments of its intersections with a curve of degree m is a: Show that if n > 3 the fact of the sum of the arguments of mn points on the curve being <j does not ensure that the points lie on an mic, but that if 7i = 3 this condition is enough.
13. If the curve of intersection of two conicoids is projected from any point of itself on any plane, the projections will all be projections of the same plane cubic.
[The anharmonic ratio of the four tangents drawn to any of the cubics from a point on itself is the same for all. It may be expressed as a function of the elliptic modulus.]
14. Verify that the expressions found (§§129, 130) for the potential of an ellipsoid satisfy Laplace's and Poisson's equations, and find the components of the attraction at any point.
15. In Jacobi's coaxial circle figure (Fig. 2, §105), prove that when a = iK', 0 is at B, and when <i = K-\-iK', at A. In general when 0 lies between L and L', so that the variable circle is imaginary, the real part of a is an odd multiple of K.
16. The arguments of the circular points at infinity are ±.iK't and of the other common points of the coaxial system K±iK'.
17. If I, m, n are in descending order of magnitude si low that the two ends of a chord of the circle x2 + y'2 = m2 which touches the ellipse x-
128 ELLIPTIC FUNCTIONS.
have for their coordinates fcmsn(u±a), mdn(u±a),
where
22 — n2) n , n
= T, ana = — ,
— 97/2 - 9\> T, — ,
m\l2 — n2) I m
and u is a variable parameter.
18. If x + iy = su(u + Lv), the points on the curves u = const., v = const, at which the tangents are parallel to the axes of coordinates, lie either on one of those axes or on a rectangular hyperbola whose axes they are. (See Appendix A.)
19. If
x + iy = sn\u + iv)
or cnu iv or nu-ii> or
the curves u = const., v = const, are confocal Cartesian ovals, and for one value of each the oval becomes a circle. Distinguish between the outer and inner ovals. (Greenhill.)
20. Examine the curves u = const., v = const, when
x + ty = sn( u, + iv)dc(u + iv).
[The distances of the point (x, y) from the points (±/£, ±//) are found to satisfy two linear relations. Hence the curves are bicircular quartics having these points for foci. In the particular cases when u = ± ^K, or v= ± \K' tliey become arcs of the circle x2 + y2= 1.]
APPENDIX A.
THE GRAPHICAL REPRESENTATION OF ELLIPTIC FUNCTIONS.
§ 131. The nature of the elliptic functions unfits them for representation by a linear graph as in the case of functions of a real variable. We may however get some idea of their variations by means of Argand's Diagram.
Let x + ty = 8ii(u + 1 v),
x, y, u, v being real, and let us examine the curves u = constant, v = constant ; we need not consider values of u outside the limits ± 2K or of v outside ± K'.
Call the point (x, y)P and the points (1, 0), (-1, 0), (I/A;, 0), ( - l/k, 0), A,BtC,D respectively. Then
Hence
= (en iv — dn tv sn t*)*/(l — &%n*u> su2iv), BP2 = (en t v + dn iv sn u)2/(I — k2sn2u su2iv), k2CP2 = { 1 - k tm(u + iv)}{l- k sn (u -iv)}
= (dn iv — k en iv sn u)2/(l — k2sn.2u sn2iv), DP* = (dn tv + /,' en iv sn u)2/(l - tfstfu suhv).
BP-AP BP + AP DP-CP k(DP +('!>}
en iv CIHVSD.U dmv
D. E. F. I
130 ELLIPTIC FUNCTIONS.
Thus the locus when v is a constant is given by BP-AP = (DP-CP)dciv,
or the equivalent
BP+AP=k(DP+CP)cdiv.
The locus when u is a constant is given by
BP-AP = k(DP + CP)sn u, or BP + AP= (DP-CP)nsu.
The curves in each case are bicircular quartics having A, B, C, D for foci. They are symmetrical about both axes.
The curves v = const, are found to be a series of ovals enclosing the points (±1,0) but not the points
The ends of the axes of these ovals are the points ( ± cd t v, 0) and (0, ± i sn iv ).
When v is indefinitely diminished the oval shrinks up into the straight line between A and B. As v increases in magnitude irrespective of sign the oval swells out. The points on the axis of x are points of undulation when 2 cd2tt' = l-f 1/&2, and for greater values the oval swells out above and below the axis of x, and is narrowest at the axis. In the limit when v = ± K', it becomes the part of the axis of x beyond ( ± 1/k, 0), together with the line at infinity.
The curves u = const, consist each of a pair of ovals, one enclosing the points (1, 0)(1//', 0) the other the points ( — 1, 0)( — l/k, 0). Each of these cuts each of the curves v = const, orthogonally.
Of the two ovals, the one on the positive side of the axis of y belongs to the values u and 2K—u (u being positive) and the other to the values — u and —
GRAPHICAL REPRESENTATION. 131
When u= ±K the corresponding oval shrinks into the straight line between (±1,0) and (±l/k, 0), the upper or lower sign being taken throughout. When u = 0 the oval swells out until it becomes the axis of y with the line at infinity.
The curve v — \K' is the circle whose centre is the
origin and radius fc~*.
§132. Since
dn(u + iv, k) = ttsn(v — lU + K' — iK, /.:'), the figures for the function dn will be of the same general nature as those for sn. The foci
(±1, ox±i/M>)
are replaced by ( ± k', 0)( ± 1, 0)
respectively, and the single central ovals are now the curves u = const., the pairs of ovals belonging to the system v = const. The curve u = \K is a circle of
radius fcT
In the case of the function en the figures are different.
Putting x + iy = cn(ic + iv), we have
x = en u en iv/(l — k2 sn% sn2iv),
y = i sn u sn iv dn u dn iv/(l — k2 sn2i& sn2£t> ).
The curves u = const., v = const, are still bicircular quartics but the four real foci are not collinear. They are the points (±1, 0)(0, ±k'/k), each of these pairs being collinear with the antipoints of the other.*
Each of the curves consists of a single oval. The curves u = const, enclose the foci (0, ±k'/k) and not (±1,0). The curve u = 0 consists of the parts of the axis of x beyond the points ( ± 1, 0), the curve u= ±K
* This may be compared with § 131 by means of the formula
cii(it, k) = sn(k'K-k'u, ik/k'), which follows from equations (20) of § 26.
132 ELLIPTIC FUNCTIONS.
of the line between the points (0, ±k'jk). As u de creases numerically from ± K to 0, or increases from ±K to ±2K, the oval swells out. It has points of undulation on the axis of x when
if
When cn2i6 is greater than the value thus given the oval is shaped rather like a dumb-bell, and the two ends of it expand to infinity as u diminishes to 0 or increases numerically to ± 2K. Since
k cn(i6 -f- 1 v, k) = — ik'cn(v — tu+K' — i K, //),
the general form of the curves u = const., v = const, is the same if one set is turned through a right angle. There will be points of undulation on one of the curves v = const, if k"2>k2, that is if there are not on any of the curves u = const.
§ 133. These bicircular quartics are shown in figures 4a, oa, 6a, for sn, en, dn respectively. They have been drawn to scale with some care for the value ^/'2 — 1 of k, and for values of u and v which are successive multiples of \K and \K' respectively.
In each case the curves u = const, are drawn thick, and the curves v = const, thin. The figures 46, 56, 66 show on the same scale the corresponding variations in the argument, corresponding lines in the two figures being numbered alike. Only one period-parallelogram has been drawn for each function. In each case the centre is at the origin.
The figures 46, 56, 66 are reproduced on a smaller scale as 4c, 5c, 6c the parallelograms being divided into the regions that correspond respectively to the four quadrants in 4a, 5a, 6a.
In figure 6a the curves v = 0, r= ±\K't v= ±$K', v— ± 2K' are too small to be shown.
GRAPHICAL REPRESENTATION.
133
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Fig. 4.
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ELLIPTIC FUNCTIONS.
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Fig. 5.
GRAPHICAL REPRESENTATION.
(a) 5
135
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APPENDIX B. HISTORY OF THE NOTATION OF THE SUBJECT.
§ 134. The notation used by Legendre was as follows : — *
F(k, 0)= { (l-k*sm*ey*d6,
o
E(k, 0)= Ul-AAdn^dd,
0
F(k, l>7r) = Fl(k\ E(k, ±7r) = E1(k))
, n, 0)= f (l-
Jacob! and Abel proposed to take F(k, 0) as the independent variable. Putting u for this, Jacobi called 0 the amplitude of it, or shortly am u. Then sin 0, cos 0, AO were the sine, cosine, and A of the amplitude of u, or as he wrote them,
sin am u, cos am u, A am u.
* The expressions I\k, 0), E(k, 6), 11(1; n, 6), were called the First, Second, and Third Elliptic Integrals respectively.
HISTORY. 137
He used the symbol coam u for &m(K — u), and also tan am u, sin coam u, etc.
He changed the meaning of the symbols E, H to those we have given (Chap. V.), and also brought in the function Z.
It was proposed by Gudermann to write sn, en, dn for sin am, cos am, A am, and the notation sc, cd, etc., was introduced by Dr. Glaisher. Sometimes tn is written for sc, ctn for cs. The function gd (see § 75, note) is the amplitude, the modulus being unity. For the notation of Weierstrass see Chap. VII.
In the further development of the subject other symbols are wanted. Jacobi used the Greek capitals 9 and H; the functions On, HM may be defined as follows : —
/f" Ou = expf Zv dv
= exp(f.
Hw = ^/k . Qu . sn u.
The arbitrary constant in the value of 9 is not determined until a later stage.
Some of the properties of the function Ou have been suggested in the examples to Chapter VI.
MISCELLANEOUS EXAMPLES
(FROM EXAMINATION PAPERS). 1. Prove that
cu(x + y) =
sd # en ?/ - sd y en #'
2. Show that
sn a sn
= en a en ff - cn(a + ff) = dnadn/3 -dn(a + /?)
3. Two sets of orthogonal curves (Cartesian ovals) being defined by the equation
= sn2
show that the polar coordinates of any point (w, v) are given by
cos 0= -cn
t k)dn(v, k') '
k'*sv(u, k)su(v, k')
u, k) -cn(w, k)dn(v, k')
, k') + du(u, k)cn(v, k')'
MISCELLANEOUS EXAMPLES. 139
4. Prove that the functions
(cs u cd u en u - k'2sc u sd u sn u)2 and (ds u dc u dn u + &2&'2sc w sd w sn it)2
have periods K and iTif'.
(x f n r ~\^
j ?LL£ i </A,
e
where tt, i, c are positive quantities in descending order of magnitude, then
e sn-M = a cn-w - c,
the modulus being {(a - b)/(a - c)}i / 6. Show that
2 + u^ + ?/4)sn
x sn Uu, - ua + u-., - uA}sn }>(u, - u* -u* + uA]
'^^ X* /Ujx*****^ «*-*- •
7. Show that the form assumed by a uniform chain of given length whose ends are at two fixed points is re presented by the equation
/,-'-// = 2kb sn--
when its moment of inertia about the axis of x has a stationary value.
8. Prove that
- C)sn(C -A) + dn(£ - C)sn(A - B)
+ sn(B-C)cn(C-A)dn(A - B) = Q, 9. Verify that
2(tt - 6) } { 1 - Fsn2(c + c?)sn2(c - d) } is a symmetric function of a, b, c, d.
140 ELLIPTIC FUNCTIONS.
10. Prove that
?i / i i7-/\ cd — t(l+k)s l+ks'2 v
d - iksc c - Lsd ~ c + isd ~ d + JcsJ
where s, c, d denote sn u, en ut dn u respectively. 11. Prove that
D-kC
D-kC • C- where S = sn 2ut C = cn 2u, D = dn 2 n .
1 2. If #x/x denote sc(u\ - w^Jcs^x + Up) then
13. If A;2 = - w (where w2 + o> + 1 = 0) then
1 - sn(w - n>2)u 1 - sn uf 1 - w sn u\ 2 1 + sn(w - w'2)?t 1 + sn u\l + w sn z^/ '
then (Q'u)'2 = 4:Q3u + 4(</2 - 15^2w)^/i - 14^.^00 - 22^3.
15. Evaluate |(#m — $>v)2du, and express |(fW-|N
in terms of I (<pu - $v) ~ ldu.
J ,
16. Find Indwdtt,
Prove that
3 dn4w du = 2(I+ k'2)Eu + k2sn u en u dn u - k'2i / J
1 + sn u
f K I , Jc
/'2k I sn u du = log =. I °i-k
MISCELLANEOUS EXAMPLES. 141
17. Show that V^'5"*' c .y^
x f"log sn udu=- \TrK - }2KloS t,
0
, [ * log cnudu=- \irK' + i^Tlogf , «
x[Xlogdn*<fe-Jjriog#.
0 c: / ^7
(In the first put am w = ^ and expand in powers of L)
18. Prove the formulae
CK
log(l -
J
0
PA-
log( 1 + dn u)du =
+
0
19. Prove that
, a) + n(v, a) - U(u + v, a)
JA- sn"w c?w in ascending powers of kz ; and
o
thence, or otherwise, prove that
dk dn
(Compare Ex. 12, Chap. IX.)
21. In Weierstrass' notation, if /is the absolute invariant as given by the equations
J-l !
142 ELLIPTIC FUNCTIONS.
then the periods satisfy the differential equation
22. Verify that the expression of Ex. 19 agrees with that of Ex. 15, Chap. VI., and with that of § 67.
23. Find expressions for the arcs of the curves,
GLASGOW t PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE.
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