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6
tllMlillMiMMW
INVARIANTS AND EQUATIONS
ASSOCIATED WITH THE
General Linear Differential Equation
THESIS PRESENTED FOR THE DEGREE OF PH. D.
3
■y
r^;
i*/^"^
GEORGE F: METZLER.
JOHNS HOPKINS UNIVERSITV,
BALTIMORE.
I89I.
s£m
PRESS or
ISAAC rRlBDBIIWAI.D CO.
■ AI.TI«0««.
57^
T?
Introduction.
The formation of functions, associated with differential equa-
tions and analogous to the invariants of algebraic quantics, has
occupied the attention of several mathematicians for some years,
because of their great value in leading to practical as well as
theoretical solutions of such equations.
Starting with the work of M. Laguerre and of Professor
Brioschi, M. Halphen, in two important memoirs,* indicated a
method for the formation of invariants, but involving very diffi-
cult analysis. He derived the two simplest invariants for the
cubic and quartic and such derivatives as may be deduced from
them. For this purpose he, by means of the transformation
Y=ye '• , brings the equation to a form having zero for
the coefficient of the second term.
Meauwhile Mr. Forsyth, starting with the letter of Professor
Brioschi, prepared a very valuable memoir,']' in which, by means
of the following transformations, he obtains a canonical form in
which the coefficients of both the second and third terms vanish.
This may be stated as follows :
When the linear differential equation
(t)'-:
^ P, = o
*" M^moire snr U redaction des ^qaationrdiSerentielles lin^aires auz
formes integrables" (M/moiru du Savanti £tr»ngtrt. Vol. aS, No. i,
301 pp., 1880). Also, " Sur les invarients d«s Equations differentielles
lintfaires du quatriime ordre " [,Acta Mcitk., Vol. 3, 1883, pp. 3>5'-38o).
f " Invariants, Covariants and Quotient DerivatiTCS associated with
Linear Differential Equations."— /VI*/m«/;Ii'm/ TrantaetiMu •ftht Royal
Sttitty of Ltndtn, Vol. 179 (1888), A, pp. 377-489.
has its dependent variable y transformed to « by the equation
V = «A A being a function of x and its independent variable
changed from x to z, where z and A are determined by
dz
3 — *-» — >
= f-'.
rfV
+
rf;r
/> ^ = o,
dx* ^ n+ I
the transformed in « is in the canonical form
(0
(2)
(/*»
(t)e-£5.©c.
d—*u
dz'-* ^
+ Qn = 0,
( — ) being the binomial coefficient ^pr^zTf] '
The coefficients P and Q of these equations are «« connecteji
that there exist « - 2 algebraically independent functions a (^)
of the coefficients P and their derivatives which are such that
when the same function »,(*) is formed of the coefficients Q and
their derivatives, the equation
e,ix) = ij^y^aiz) (3)
is identically satisfied. For this form of the differential equa-
tion
where
r=<r— 8
M^)=Q, + ^ ^£^ (-»)'«'.
rf^g>-.
<»r,» —
ff — I U - 2 1 2ff - .^Tlll
Thus e,{z) is independent of the order of the equat^n. In
this z is completely determined by equations (0 and (2). But
there may be difficulties in the way of solving (2). and thus it «
desirable to form the invariants for the uncanomcal form of the
***ForThis purpose Mr. Forsyth establishes relations between
the coefficients /> and Q for tt»e case in which . being arbjt^
is given the value x + c;«. where . is so small tnat the square
; by the equation
pendent variable
nined by
(2)
. + Q,-o,
I are so connected
int functions S {x)
lich are such that,
coefficients Q and
(3)
: differential equa-
i!'
the equation. In
5(1) and (2). But
ig (2), and thus it is
nonical form of the
;s relations between
har, being arbitrary,
lall that the square
5
and higher powers may be neglected, and ii is an arbitrary non-
constant function oix. These relations are expressed thus :
, (nut — 1 r ,!
{nis-o-i) + s + o-i}P> J^_r+'i]
(5)
These relations are fully developed in Mr. Forsyth's memoir ;
also in Dr. Craig's excellent work* they will be found, and such
a general treatment of the whole subject of differential equations
and differential quantics as makes the work an invaluable help
and guide to any student of the subject.
Then we derive
d'Q.
dsf
dxr
\i-{r + s-).!.'\-s.p/£^,
m=i imlr — m +!!''■
tir-m + l//
- *=3i— 1 r ,1
(6)
The only invariants that have been formed, so far as I know,
are ^i , *« , *. , <'« and ^, , where ^, is the invariant of the rth order
of an equation of order n.
In Section I of this thesis the general invariant 0, is con-
sidered, and it is there shown that in the non-linear part every
term is of the form ABC. Where ^ is a number, ^ is a
function of P, and its derivatives, and C ?> L\n invariant or the
derivative of an invariant with suffix diffennf from s by an even
number. When s is even C may be a number.
Section II deals with the coefficients of (f„ giving some
* Treatise on Linear Differential Equations. By Thomas Craig, Ph.D.
Vol. I.
)i^4
general expreMtons by which they may be calculated for any
value of i. . . J •
Section III treats of associate variables and associate equa-
tions, showing which are identical and which may not be.
Dr Craig having discovered that the condition for the self-
adiointness of the sextic and octic was that their invariants with
odd suffix all vanish, suggested to me the general theorem
announced in his treatise, pp. 293-295. The proof given at that
time only applied to equations in Mr. Forsyth's canonical form.
By aid of what is established in Section I, it is shown to apply
also to equations in any form. ... , ,
A fuller history of the subject will be found in the works to
which reference has been made. ^ ^ . . . u j
This paper was not only suggested by Dr. Craig, but has had
his valuable criticism.
Iculated for any
i associate equa-
nay not be.
tion for the self-
ir invariants with
general theorem
roof given at that
s canonical form.
; shown to apply
d in the works to
>aig,but hashad
-MMM
MM-
"
Section I.
The Form of the General Linear Prime Invariant t*,.
Since (i, has only a linear part when Z', vanishes, its form must
be as follows ;
[AP. 4- BF.., + (:/>:'-,+ . . . + fWi-" 1
+ [/», {a.«._, + «.^.- . + «i^'-. + • • • + «.-.A-*' I ]
+ in \ *.«.-» + *4K -. + ... + ^ -.''i'"" { ]
+ [p';s ^A-« +<:.<*:-.+... +f.-./'i—'n
+ [P\) rfA-4 +</.»:-.+... +''.-,/'i'-/'(]
+ \.K\ \
+ [etc J
+ [etc ]
etc., etc
In this (r) is the differential index, so that
The sum of the suffixes and differential indices, it will be
noticed, equals s for every term ; that is, B, possesses a kind of
homogeneity.* s is called the index or dimension number of
0,; the dimension number of /'J"'<T-« being
Denoting the terms within the square parenthesis by L, «, ^, r*
i, etc., then », = Z, + « + ^ + r + « + • . •
The notation used here will be nearly that used by Mr.
Forsyth, but to simplify the work the m's and their derivatives
arising from b=ix + sii will be dropped, that is, they will be
« Philoaophical Transactions, Vol. 179 (1888) A, pp. 391-92.
6s
KM
mmmm
8
treated as unity, when the result will not be changed by doing
so. Also Z^"' = -J I will be considered = with P,
doP
The general form of the terms in Z. is
I shall now show that when * is odd each of the numerical
coefficients a,,b,,c,,dn, etc., of the non-linear part of «, equals
zero.
From page 4 of the introduction we have
identically satisfied. If in the right member of this identity the
Q'i and their derivatives are replaced by their values in terms
of the /"s and their derivatives, as expressed by formulae (5)
and (6) (page 5, introduction), then the terms of dimension ' s '
in each member cancel, those of dimension ' j — i ' furnish the
numerical coefficients in the linear part L, and there remain
terms of dimension equal to and less than s — 2 with which we
may determine the coefficients of the non-linear part.
Remembering the convention Pl"^ =. /*,, formulae (5) and
(6) are included in
^• = /'i'>ii-(r-Hj)./i'|
d^
r = o, I, 2, 3 . . . i
(7)
Also, difTerentiating the invariants, we find
anged by doing
i,2....f— 2. (a)
>f the numerical
part of ff, equals
this identity the
r values in terms
by formulae (5)
of dimension ' s '
— 1 ' furnish the
nd there remain
2 with which we
T part,
jrmulae (5) and
+ 1)
r-m + l)
1)
• + »)(
r)
t, 3-
. S
■ (7)
i
9
Then i^. = /». (I - 2t//) - ? + ' «//",
4
I?, = /»,(!- 6t/i') - i5e//'/» - 4 (« + 9) ^m'"/**
- 5 (« + 4)'/^"'/'. - *(3« + 7) '/^'•"/'. - " -;^ " 5*/^"*.
»4
'g = />; (1 _ 3.//) _ 2.m"/> - "-±i c/.'^
-^^ = />i" (I - 6*//) - gei/'P: - ^^ti."'P,
- « ^ )6jr«"iP. + (« + 5) m"'^.{ - a (« + *M^
etc. ... etc.
From these follow
iK = /»; ) I - 2r./i'} - r/>;-' "J ^ .m'«.
I
}
lO
If /^,"«*i*_-«" be a term in S., then will the term -^ ^^I,
be multiplied by (/)' or (i — siit), and
= (I + «y) I P'," (I - seja') - 9«m"^.'
«=«-• r X — 5!
|(.-x)(.-4-»») +»f i^i-'.M'"-*— *]•
In this equation the terms oif dimension ' s ' cancel and — e
is a factor of the remaining terms, so that when every term in
S, is treated in this way, all terms of dimension ' s ' cancel each
other and the remainder is divisible by — t. Denoting by HL
the remainder of the linear part A by H, the remainder of the
(ti \ ** •
— j the binomial coefficient jp(:^zry\'
also omitting the /x's and dividing by — e, we get
jiL^Al"^ />._.+ '-^.in+ I + 2is- 2)/'._, + etc.1
L "* 2.3. -J
+ Bis- I P.-1 + .'•}+'•'
...^'r.l^'irt.j-'oTfaifl;
II
1? rf«—
;» L»*'«— 4"""*'
e/t"
■^
X — 4 — »» !
J ' cancel and — e
den every term in
)n ' * ' cancel each
Denoting by RL
i remainder of the
n\
efficient yrf^ZTrl'
iget
f- 2)/»_, + etc.J
• "J T • • •
,. x=l,2,3...*
+ C [{s-2)P,-^ + ... + Y7^{ 2 (j - 2) + ij />;_,
+;i:{r^)"-^'S-(s)}(^-'' >"
for the first three terras of B,. Replacing A, B, C, etc., by their
s s.s — I .s— 2
r—i
(lO)
)
values ((a) p. 8) I, - -|^ , ^"^ •/_ ^ , etc., the (r + i)9t
term gives
,_ y s\s—2\2s—r—2\
^ ^' 2. r! J— r! J— r — i ! 2^—3!
f ( s — r \ n + I X — r~ I
\v-r) 2 X — r+i
x = r + i,r + 2,
as a remainder. By giving r all values o, i, 2, 3, ... (10) ex-
presses the whole of HL.
*«= P, [J (*- 3) «A-. + «*)*-4.<'.-« + (2*-7)<^.-*l + etc.]
r = X — 2 J ,
with similar expressions for the other parts, Urt R^, etc. Suppose
that the coefficient of /»<!», in RL is A^(n + i) + A + C
Then the general forms
(->'T iimi'^XB^) ^^ <" ^-'"'^
r = », w + I, ... X — I,
Siissii;
12
(-.)' H>-)(^Fi-^-)(r^-.>"-'-'"'
r.= w, » + I, • . . X — I.
and
(-.)-(t)&E1)(^^5^)[(t)(«-')-(.^t)]
will, when expanded, give A^, B^ and C respectively. In these
i^l.-AZL^ is the reciprocal of f^^^). Thus A is found
to be
^.=(-.)-(-:-)(SXii^)r»[
L f z^— »>—
2... 2f — X — 2
X — W+I
X— t;— I
X — V\
S — V — I
+ 25-z>-4---2Y'~''='-z^-3
j — t; — I . s — V— 2
X —V—l\ ' ^ 1.2
2S — X .2S — X — I . . . 2S — X — 2
(-1)
X S-V-l...S-x+_2 ^ ^^^
^ X — V — 2!
•]
\ (S — X + V — I ... J — I
j - ^ [ 's-v+i\
S — V — 1 . . . S — x\
X — v\
S — V — I . , . S — X
X — V — I
Use the upper or lower signs according as x — » is odd dr
even. To obtain this result expand
«* (I — ;r)— — ' = jc'-Cs — v- i)x'
+ a «*-... -(-I) i,_t,_2l^
-}
(a)
mmmmmmmm
^t — »l I
+ 1, ... X — I,
lively. In these
Thus A is found
<S—X—2
n
V — I
X — V—l
V — I . S — V— 2
~ I . 2
— X — 2
S— * \-V . . . ^— I
X — v\
. . . S — I
-i!
13 X — 3; is odd dr
•v—a\'
,..},
w-ii^_. ?-(»)
'SiB^irWriini* wnnw wiwiMni
13
and
.j_,v-.-.) == ^ + (25 _ X _ 2) -i
2f
X — V + 112S — X — 2\
Differentiating the last equation,
— 2X~
•(I -AT)-"-'-"
_ (2* - X - 2) X-* (I - *)-<»'-'-'
_ — I _ 25— X — 2
+ . . . x—V—l
+ ...
+ +
2S
2S
3 ! 2 J — X — 3 !
V — 2 !
— t/+i ! 25— X— 3!
(b)
The coefficient of Jf""* in the product of the right members
of (a) and (b) is the series of terms in square parenthesis in the
expression of Ai above, and the coefficient of x"— in the pro-
duct of the left members is the quantity within square paren-
thesis in the final value given for A^.
Bx is found by putting (i - x)— — * and (i - xy—'-* equal
to their expansions and taking the coefficients of *"-•+* from
the product of the left members and also from the product of
the right members. Then
^^~^~^^''\^)\v^)\2S-i) ~2V [_~~ X-V+ l~~
If in these expressions fory4i, Bi and G, » is made equal to
zero, then for all odd values of x
Ar = o = Bx + Q, (II)
while for even values of X
."^a^^/Ut^/ I („)
j^______^ („ + I) + ___ J j
For v= I
A, (« + i) + ^, + Q
H
and X increased by unity, ^i (« + 1) + ^i + Q becomes the
same as in (12) multiplied by ^ . Then in HL, if JFbe
the coefficient of Z'. _ , when x is even,
- W.
S — X
is the coefficient of ^_«_i.
(13)
W hen v = x — 2,\etAi(n+i) + Bi+Cihe denoted by a„ .
The following are the values of -<4, , Bi and Ci when » = x — 2 J
^'-^-"'^ [^)\x-i]{2S-3) J-xfl.J-x.4
2J — X— I . 2f— X— 2 . 2J— 2X + 3. x— i
2.6 ~
'■. = (-)-'(TX£fXiI^,)
2S—X—2. y— 2« — 2.x— I
2.6 ~
Now, when the whole remainder is considered, the coefficient
of each of the (/'i*2j^)'s must be zero. Let us now consider
those terms of dimension x — 2. They will be foiind only in
^ If
XL and /?a. The coefficient of /*._, is — oj, + —J— «i. This
equals zero, and when v=o and x = 2
«■■ = i (l)('-^)(sii)«« - 3X» + ■) + •• - 5- + 61.
therefore
The coefficient of /^_, is, by (13).
— V — a, + — ^— tf»
*— 2
2
a, = o.
o,,;
«+ 1
o*;
dien
The coefficient of P'.Lt is
6 * ^ 6 'V 2 y 4 . 2* - 3
i;,.ijpi^iiiSi4..^''!:Wi'./iM^'.^^. .• ' ■ ■
WTBi l M liiii i n i'\t ^'« ttiMOBM imtam fi ^ aM .'m
Q becomes the
iiin/?Aif IVbt
-1- (13)
e denoted by a^.
, wheni'=* — 2:
. 2S—*—l.2S~ x
2S—2* + $.x—l
— 2«— 2.x— I
.6 ""•
'ed, the coefficient
i us now consider
be foiind only in
. n + i
Ot. This
+ J»-5* + 6}.
+ (f-2).(i-3)}.
« + I
a»',
-3
— flw
J
15
Substituting for Ot and «,« their values,
«• = J^ (i)(-l ') ^ (-'"^ .X..-5)+.-4..-5}.
Calling the three terms whose sum gave the coefficient of P','_^
i, II, K, then the coefficient of /*i?., is
"^ a, + -'^ -I + -f~4ri-^ " - «» = "» + ^1 + /^. + ''i.
say. The last three terms reduce to zero ; therefore
a, = 0.
The coefficient of /*i*i, = », + ^» + /^i + «i., say
s — ^.s — 6
4 . 2J — 9
.«!— «1. = 0.
Reducing this,
*• = ='^i^ \i-6!j-6!2J-3!3!2
— 6 / *!.r— 2! 2f — 12! \, , , X, ,
+ s — 6.s — 'j\.
Similarly th may be shown equal to zero and
— 6 j!* — 2!2*— 16! , , , .. -.
«• = ^+7 2.3i^-8i.-8!2.-3l ^^^" -^ ^^^"' - 9^
+ 5 — 8. J — 9}.
Had the terms in the coefficient of PSii been denoted by
*4-^ Ot, a,, A,, Ai, and Ox,, then those giving a, would be
o
6 • 2.2J— 15 * 4.2J— 13 • *
6 . 2*— 1 1
It thus appears that A., ju,, ct have a relation between them
similar to >l„ /nt, r* and K%ih,«rx, etc.,^and if we follow the same
law the coefficient of i^ri becomes
+ 1 ,/ Ml + lf *U— 2I2J— X — 2/— 2!
2J— 2x.2f— 2X— 2.i9fl, ,
i6
where ^ = 2* — 4/ — i and
0, =t {/>(« + 0(2* -2p-l) + is- 2P){S -2P- l)K
2/ = 2, 4, 6 ... X — I or X.
Also a, would equal zero when x is odd, and when * is even
*" «+ 1 2.3l5-x!5-x!2j--3! V2 '^
To prove that this law holds, consider the series
/ = (- I)" [(« + I) 4T7-7rr:rx -TT
• — 2 ! 2* — X !
! X — 2 ! 2J — 3 1
J ! J _^ zr— X — 1 ! (25 — 2«)( J — 2x + 3)
^rr—x ! 5 — x— i!x — 2!2J— 3!
^ ! ^ — 2 ! 2.? - X — 2 ! (3f - 2x - 2)(2.y - 2x)(s - X + i)
4!^ — x!5 — x+ l!x — 2!25— 3!
5 ! J — 2 ! 2J — X — 2^ — 2 !
2S — 2X
4!x — 2^!* — x!j — x + i!2*~3!
X 2J — 2X + 2 . 2J — 4^ — I . <?/,
2/ = 2, 4, 6 ... X — 1 or X.
The first three terms are what A . ^1 and Q become when v is
made equal to x — 2. As the series is to be shown to be equal
- . S\S—2\2S — 2X — 2 1
to zero, the common factor (—1)" a\s — x\s — x + it 2s — ^i
may be omitted. Then
-;?• . 2J — X — /^ — I . . . 2S — 2X — I
77^^+ 2!
/ 2^ — x-jg- \ _ 25 -
\x-g+2J
2S-.-jr 2S-.-^- l ^ (^ + 2) = ^ (^). (X4)
x—g+ 2.x— g + 1
The series to be considered now becomes
m
■MMi
mmmmtm
%
dwhen * is even
){2S -x-l)
ieries
xj^
2 ! 2 J — 3 !
2X){S ■
— X
+ 1)
-3!
,
S — 2X
.25 —
4/>
— I.
ep
'i become when v is
shown to be equal
2 1 2J — 2« — 2 1
r — X + I ! 2J — 3 !
. . . 2f — 2X — I
.2;ir(^). say.
i) = x(jr)' (14)
17
/ (4) . 25 — X . 2J — * — I . 2J — X — 2 . 2J — X — 3 . (« + l)
— ;f (4) . 25 — X — 1 . 2J — X — 2 . 25 — X — 3 . 25 — 2x . 5 — 2x + 3
+ ;f (4) . 25 — X — 2 . 25 — X — 3 . 25 — 2x . 5 — X + 1 . 35 — 2x — 2
— ;|r(4).25— 2X.25— 2X +JI.25— 5(n(25— 3) + 5* — 35 + 3)
— ;f (6) . 25— 2X . 25 — 2X + 2 . 25 — 9 (2« . (25 — 5) + 5* - 55 + lo)
— /(8).25— 2X.25— 2X + 2 . 25 — 13 (3«. (25— 7) + 5*— 75+21)
— /(2^+2).25 — 2X.25— 2x +2. 25 — 4/— l{/» (25 — 2/— l)
+ 5» - (2/ + 1)5 +/(2^ + I)},
2/ = X — I or X.
Consider the coefficient of n,
;^ (4) [25 — X . 25 — X — I . 25 — X — 2 . 25 — X — 3
— 25 — 2x . 25 — 2X + 2 . 25 — 5 . 25 — 3]
= ;?« [8^ — 4* (2« .+ 3) + * (* + 11)] X — 2 . X — 3
=:x — 2.x — 3./(4) Ji.say,
= 25 - X ^ 4 . 25 — X - 5 Jvir (6) by (14).
Take from this
;f (6) . 25 — 2X . 25 — 2x + 2 . 25 — 9 . 25 — 5 . 2i
and the second remainder is
;f(6)x-4.x-5.[i25»-65(2x + 5) + x(x + 29)]
= X — 4. X — 5 .;f (6) 4„ say.
This equab
25 — X — 6 . 25 — X — 7 . ;if (8) J, by ( 14).
Take from this the next term of the series,
;f (8) . 25 — 2X . 25 — 2X + 2 . 25 — 13 . 25 — 7 . 3 ;
the remainder is "
x-6.x-7;f(8;[l65»-85(2x.+ 7) + x(x + 55)]
= 25 — X — 8. 25 — X — 9 ./ (10) J,, say.
Supposing this law to hold for all differences till the (m— i)th,
it can be shown to hold for the mth. The (m — i)th is
i8
X + 2 - 2W.X + 1 - 2«.;^(2«)[4W5*- 2»m(2x + 3« - l)
+ X (x + 4W' + aw - I)] = z(2« + 2) ^—1
2j — X — 2»» . 2J — X — am — I.
Taking from this
there remains
- 2 (W + l)(2x + 2W + I) + x(x + 4«' + 6»» + I)]
= « - 2m . X - 2WI - lAf (2»» + a) J, ;
that is, the mth difference is the same function of m as the
/^-,_ i)th is of »» — I. .. , , ,^ _
When 2W = 2/ = X - I or X the subtrahend is the last term
of the series and the difference vanishes. , Thus we see the
coefficient ofn in the series vanishes. .»/•«;.
The algebraic sum of the first four terms mdependent ot n is
X- 2.x-3;?(4)[2^--**(2x + 8) + icw(x + 1)
then by (14) it equals
Jj2« — X — 4.2J — X — 5. ;ifC6).
Taking from this
;^(6) 2J - ax . 2J - 2x + 2 . 2* — 9 • *• - 5* + 1°
there remains
„.6)x-4.x-s[a**-(« + «)^ + (i4«+25)*-«(« + 29)]
If the (m - i)th difference be
x-2». + 2.x-aw + i./(a«.)[ai*-(ax + 4«)^+{2«(2«+i)
+ 2(m - i)(2i« + i)}*-x(x + 4»»»' - 2*«- ^>J'
which we will denote by <^ (« - 1) ; then the »ith is
<&(«- i)-z(2'«+2)[2J-2x.25- ax + a. a*- 4»»- I
= y (aw + 2).x - 2m. X - 2m - i[2«»- i*(ax + 4(»» +0
4-{3x(2m+3) + 2»»(2'"+3)}*-«(«+4»»*+6»»+0J
^i^,i»ig9E..
J (2x + 2m—i)
(2IW + 2) -^.-i
I.
--1.2S — /^m — t.
+ 4»»' + 6»t + i)]
ction of m as the
nd is the last term
, Thus we see the
ndependent of n is
t + i)
2.x — 3Xt^i'^^y>
(6).
t» - 54 + 10
»-25)i-x(* + 29)]
|»l)j»+{2x(2lf»+l)
+ ^m* — 2m- i)].
tie mth is
+ a . 2* — 4*** ~" *
l)j + «»(2»»+l)}]
a* (2x + 4 (»» + i)
«+4w*+6»i+i)]
»9
This vanishes when 2m = x or x — i, and also completes the
series.
Thus the whole series has been shown to vanish whatever be
the value of x. (15)
Assuming that <!« = o when x is odd, and
— 6 *!j— 2!2J — 2x! f X / , x/ \
= W+i 2.3\s-.\s-.\2S-3\ |y("+0(2^-x-1)
+ (f - x)(f - X - I) }
for all even values of x less than 221/ + i> then it may be shown
to be true when x = 2w + i and 2W + 2. The coefficient of
/'i-tolii in HL is a,„+ti and if Mi represent the value of ««
when X is even, and N^ represent the expression
(-1)^
SIS — 2\2S — T
2\t\s — TlS — T — l\2S— ^l'
i. e. the coefficient of PlV.r in L, then the whole coefficient of
Pi*:^ii'U is
^- a^+, + «,.„+, - iM\Nf-V + MINT--,* + MvNtLV
+ . . . + i«/;jNri_..] ^^
Now <h.*m+\ is the sum of the first three terms of F, and the
following terms are those of Talso; for taking any one of them,
as
it becomes, when written in full,
6 «+i s\s — 2\2s — 4^
« + I (
2.3u-2^u-2^!2.:::i! ^*(«+')(«-^^-»)
-'+(s — 2g)(s — 22— i)]
S — 2gl S — 22 — 2 \2S — 2W—28 — Z
2\2W — 22 + l\ S — 2W — l\ S — 2W — 2\ 2S — ^ — ^\
S\ S — 2\ 2S — 2W — 22 — x\ . 0,
=s — , , i 3 5 f 2S — AW
\\ 2W — 22 -^ l\ S . 2W — l!*— 2ZC!2f— 3! ^
X 2J — 4W + 2 . 2J — 4« — 1,
d
[ill ■ 1
II !
PI
i ,;i
90
which coincides with the last terms of 1' when « = aw + i and
m:=/>. Thus the coefficient of P!^i.t i consists of ^^-g— «.. + 1
plus a series of terms which vanish by (15) ; then
The coefficient of /*!!!'», -1 is
(16)
+ . . . + ^;.^.'-i-] = o-
r gives all the terms in this expression when x = 2a/ + 2,
excepting the first or "^ a,.+.. But the last term M\J^]-,.
is the second last in F when x = 2a; + 2, 2/ = 2, 4 • • • aw + 2.
Taking T from the above coefficient, -g- Of+i 5- ^•''+«
is the coefficient, since r = o always. And as this must vanish.
Thus (16) shows that if for any odd value of « and all lower
odd values a. = o, then «.+. = o. and (17) shows that if for
any even value and all lower even values a,= M'n, then
«« + ! = J^'n + t'
On pages 14 and 15 it is shown that «. = o for x = 3. 5. 7 and
a ^; for X = 2, 4, 6. 8. Therefore it follows that (16) and
(17) are true for all values of o'.
It foUows. then, that in 9, the row of terms designated a, of
which Pt is a factor, contains no invariant or derivative of the
This is also the case for the terms entering in the row desig-
nated /9 and of which Pi is a factor, for the term /\^.-4 is found
only in Ra and Hfi. Its coefficient is
2*4 + (2J — 7)<»«;
then
A _ g^-7 -
ai
X = 2W + 1 and
tS OI — z — <»iif + l
tien
(16)
hen x = 2w + 2,
= 2, 4 ... aw + 2-
} this must vanish,
C17)
[>f X and all lower
I shows that if for
= J/;, then
> for X = 3, 5. 7 and
3WS that (16) and
ns designated a, of
: derivative of the
(18)
J in the row desijj-
irm /».^.-« is found
Any term as /'■^i-';", * being odd, could appear only in i?a
and i?/9, and as it does not appear in Ha it cannot in H^,
The coefficient of Pt^ir^i;" is
or
a*„ + (« - i)(aj - a« - 3) a», = o. (19)
The terms of dimension s— i and of form Pt'^Z, can appear
only in 7P/9 and Hy, and when * is odd no such term appears
in /?/9 ; therefore it does not enter into Xr.
When X is even, the coefficient of /'i^i-M*' is
»'-M(i^)<'-">-(s^)}'"=°'
or
5^,. + (2K - 3)(5 - « — 2) *„ = o. (ao)
In this way it is' easy to see, by taking one row after another,
that the non-linear part ofB, contains no term having 0i,'lK as a
factor when x is odd. (21)
From this it follows that if all the invariants of a diflferential
equation with even suffix vanish, the linear part of each vanishes.
The same is true for those with odd suffix. (22)
I
mmmmmm
Section II.
Thb Coefficients of B,.
tf, has, as we have seen, a linear part expressed by
rmt —
"f NlP^r,
Then follow a series of terms
expressed generally by
(24)
6 — [Sr] j|j-alaj-4«i )
{x(«+iXaJ-2«-0 + ('-2x)(*-a*-i)}»i!li;")
'nil meaning the greatest integer in '-^ . Then follow
F>, {dA-i + *.*"-• + *•**-• + -"^
+ Pli {cA-4 + (**>"-* + c^sY-, + . . .}
+ /n" {efi>.-, + *.«i'^. + «..*r-» + • • •}
+ •• V
These are expressed generally by
Hy.e^fi."-*' + /'i'-'" 2:" " q^B^tu'-*^
= a, 4, 6 . . • etc.
riBiMli
mmmmmm
i9MMrffNNHMi«HNHM||iH
ised by
-3
.. + ...}
P':ir. (as)
(24)
Then follow
+ ...}
+ ...}
+ . ..}
-•]
= 2, 4, 6 . . • etc.
23
If any two coniccutive rows be considered, for which (w = /*),
the remaii'.der arising from them will contain a term ^
found nowhere else, because all rows preceding these have /T'
as a factor where v < n, and rows following them have a re-
mainder in which the index of *,_„ cannot be as great as
(ax — At — 3). This remainder is
r *~' i
+ ^/>jM-.) + . . . +1 "r «„<><•==•'-"
.«..["-r%-r{(^>-">-(;i^)}«--
r= 2* — fi— 2
+ . . . + terms of lower dimension 1 £ ^ ^w^i-wc" "
.^.^.{-r^/:r{(^)(.-.).(^)}«r...
r = ax — M — 3 •
Equating the coeiBScient of the term Pir^eirLu"'*^ to zero we
obtain
1
.
•vr""-"."**™
24
In this X is any number and fi any of the values of v, so that
the coefficients g^ of any row may be expressed in terms of
those of the preceding row, viz. ««.
(25) when simplified gives
/»+ I
(4 + f^)9
_ (2X — M — 2)(2S —2x — ti— 3) ■■
Making m = o, i, 2, 3 • • • this gives
4. 1 .*,. = - (2X - 2)(25 - 2x - 3)a„
5 . 2 . f« = - (2X - 3)(2J - 2* - 4) *«
6. 3. if„ = — (2x - 4X25 - 2x - 5)f„
• •• »..•••
■
(m+i)(4 + /')?.« =- (2«-/*-2X2*-2«-/*-3)«««-
Equating the product of the right members to the product of
the left gives
ft+l!M + 4l ( ilu^i 2X-2J25-2X-3! ^^g^
?«• J\ '^ ■' 2x— /I— 3 ! 2*— 2x— /i— 4 1
The ^'s being coefficients in the row multiplied by PiT+J^ it
is seen that the coefficient of any term of the form Pi«'»i*ii*-"
may be expressed in terms of the a'a. Writing this coefficient,
for brevity, W^"*-" , then
2x — 2 ! 2f — 2X — 3!fiJ — 2!2J — 4x'
"{<()
d\d + ^\2x—9—2\2S—2x — S—i\2S-3\s—2x\S—2xl2
6
' • (27)
There still remain terms of the form
Here a, 6, c, d, etc., are indices expressing powers of the
factors to which they are attached. (a*^^«Oir' » the coeffi-
cient of the term having such indices, powers and suffix s — 2x.
values of v, so that
ressed in terms of
2x — /* — 3)
- 3) «««
«.«
J2J— 2x— /t— 3)«»ic.
rs to the product of
""3' ^. a>.. (26)
tiplied by Pr+" it
e form Pi«'»i*^*-"
ting this coefficient,
IM.
IxlS — 2x! 2
6
' • (27)
«+ I .
')iJ'«i*u.
sing powers of the
■•i*e')ir' is the coeffi-
•rs and suffix s — 2x.
as
Throughout the whole invariant the order of the factors will be
taken so that _ _ _ _
«<)?<>'<*<«. etc. (28)
2x = »» + a(a + 2) + *(/S + 2) + tf(r+2) 1 (^q)
+ rf(a+2)+<f(« + 2)+ ... J ■ ^"^
The numerical value of (a'/S»r°^cOir» is found by equating
the coefficient of />i"'"-PiP''/'r"/*i*''/'i"*"*^i^«»n*e remainder
to zero.
It is
+(«-'i9'r-^.-»e+«+2)i:' ^^nrrlr' (^•+^+«)
+ («'^-y^e->e + /S + 2)i:'*^^f±|f (2e+6 + i9)
+ («'/?r-'*'e— e+r+2)i:> yffilf (2e + 6 + r)
+ («-|9»r'^-V-'e + «+2)i:>^;t^' (2c+6 + «)
+ (a^^fS't' - •2e + 2)}:' /'."^/;, (3* + 6)
+(«ry'«'e-»)i;+'+
e!e + 3!
., »» + e + 2
WIe + 3!
{(e + 3)(*-2«) + »»}
=0. (30)
5^ 2'(ori.+.2'(«rr-«*«-*)'"^7r^
(31)
jT — e + 2 Y =0.2.4.6. ..2x
— 2(a + * + c + rf+ *)'— 2» + 2 \\\
(t)ri.+i is the numerical coefficient of /»i"»i^,_._,.
oACidiitx take all values consistent with e^ < e, and
ai + *i + ^i + </i + #1 = the constant (a + * + ^ + rf+ * — i).
ft
I
I, .!
36
(<^^f9*t—^y^ stands for the numerical coefficient of
(^J•)•/>J^)»/>iY)•/»i«l*/>i•>-')
r = 2x — a (fli + *i + A + <^ + *i + »)
- « - a,a - *i/J - Ar - «^* — (^» + 0«-
In the coefficient (a'./S».r'«3*>«'0ii*— -r ^ is to be changed to
,_,_„_. 2. When * - 2x = 2 the terms that must be added
are easily recognized. r iHaiu-») t«
For an example, let us find the coefficient of Pl»lr!L*, - in
Then
|- («+l)(o0ft''-'"+O + O + O+O+(</2)8r-"" —
2x — lO . 2x — II
» + I
-(3*-4«-i2)
+ ?_+i [(o).((/)li-r."'+(o)i«(oP)«r-%">+(o)i*>(oP)li«-.">
+ ... + (o)i5'-.i"(tf)w] = o.
This states that
n + I times the coefficient of /1»i*iu"'
+ twice the coefficient of P\PWlu*'^
^ a«-io.2«-ii (y_^_ 12) times the coefficient of 7n<?il»..
+ ^+JL times a number of terms = a
Any one of these last terms, as (o)^' Wi?-"."' . « "S* «S « f""
thus: The coefficient of /Vi«. times the coefficient of /1»i«u+'«
in the invariant »i*i.. , -,„t^
As another example, find the coefficient of /»J/>i»> /^i" '^-•«-
Here
2x = m + 23, tf = 2,3 = 3.f = 2,
o = o, i? = I, t = 3, rr = 1 . 3 . 5 . • • a« — *7.
tefficient of
r=jf + It —
m.
r — d,9 — (,ei+ i)«.
; is to be changed to
\ that must be added
Qt of Pjoi^r."'. In
? = 6,
y sz 2* — 12 — ff.
10) Jl
3
- 12)
coefficient of 71<?i^»..
JT."' , is written in full
Efficient of /1#i*iu+'.
of/»J/>i»>*/'i*''^---
, 5 . . . 2x — 17'
27
Then
^^-^ (o^i'3')i:' + (o^»*3 . 5)i:' (2 . 6 + o)
3
+ r4i(3-6 + 0(<^i'3.6)J:'
+ ^, (2.6 + 3)(o^i'8)i:'
+ HLZl^ (6i - lox - 23)(6'i'3X+-^*
Mt I Ol
+ ^[(3)i" {(o»i»3)&ri»+(|)c(o^i'3)l?!-.
+ (rfi«2)ffL.} + (3)S{(<^i*3)l!?rr
+ ^A) C(o«i»3)ir-." + (f) C(6'i'2)irrr
+ (-^) C(o*i3)irr." + [■^)c(.c^i'2)&=i>
+ (A) c(o'i*)irr.« f (-J) C(tf3)i?'-.
+ (-2-) C(o*i2)irL. + (f ) C(o»i*)i:L,}
+ (3)8' { i^i*3)Szii 4- (^) C (o»i»3)i?rJ|
+ (f ) C(rfi'2)i?rj| + (^) C(o*i3)i?rA'
+(f ) C(tf3)i?-a + (-^) C(o«i2)tfzS
+ ("f ) c (tf lOirr ji + (f ) c (o^2)&=ii
+ (-5-) C(o*0&r}i> (^) C(o*i)!i-u}
+ ■
+
+
(3)r--u»' {(o^i)'*' + (oT'} + (3)ar-»i'» {(o*)}]
• =o. (33)
H
a8
In this r varies, being =y\-it — m always, and C also
varies. The term (3)?4(|-)(o^2)&-" ^ea^ t^* coefficient of
/»i"<>i*lu times the coefficient of /»JPi'«i*l.*Vu in the invariant
^riu multiplied by (-2-) C- r = m - 5 + 9 - w = 4. and C
is the numerical coefficient of P\PfP^^ in *
^(PSmand(^)=;^
1'
lilt
Thus every term in the invariant #, has been considered, and
by (23), (24) and (27) every coefficient has been expressed by
simple formulae in terms of * and n excepting those represented
by (30), and they are expressed in terms of preceding coefficients.
xrays, and C >1bo
the coefficient of
•!n« in the invariant
) — m = 4i and C
•
rn-
een considered, and
been expressed by
ig those represented
■eceding coefficients.
Section III.
Associate Equations and Associate Variables.
In the memoir previously referred to, Mr. Forsyth shows
that in connection with any differential equation Ai of order
n there are n — 2 other equations, A,, A,, A^, . . . A,^i, whose
variables are formed as follows : Let ih, »,,«,,...«. be solu-
tions of ^], then if we take any two «a, »^, the determinant
I uxu^
!<<
is a solution of At. Generally if we take any x of the «'s and
form a determinant
Uy . . . U,
l^f • • • **»
«ir-« »],«-" <-**
= ««,
where a, /9, ^ . . . x are any x of the numbers i, 2, 3 . . . n, then
a. will be a solution of A,. As there are f — j combinations of
n tiiingB X at a time, there will he [— ] variables a. satisfying an
equation A, of order ( -^j. A, will be called the (x — i)th
associate equation, and the variables a. the (x — i)th associate
variables. These variables a« are particular and linearly inde-
pendent solutions of ^.. v4,_i is the Lagrangian adjoint equa-
tion. a« may be written (<^/' . . . »<"~")f or, as we are not
concerned with which suffixes are taken, 0123 ... (x — i), then
5" ... n — i'"-*') or 01234 ...(« — 2),
30
a.= («/9') = di, a« = (a/8'r"0=oi23.
The number of these « (-^) •
«._i=("3 4 5
while (i2'3"4"' • • • »»""') °f '234 • • • (» - >)
is the non.vanishing constant J. To illustrate what follows I
shall first take a particular case, n = 5. Then i4i will be
«"> + lof ,»" + 5f 4«' + ffi« = o- (34a)
«. «..«..««.«. are the five independent solutions ; then «,=oi.
o and I being the differential indices of the diagonal of the
determinant formed with any two of the u'a and their first de-
rivatives, then
^ = fli =02,
Oi' =03 + I2,_
a'," = 04 jf 2.I3,_ _
fl^ = 3. 14 + 2. 23 + OS-
Substituting for u" in 05 its value from (35),
«i«) = 3 . 14 + 2 . 23 — io<p,62 — 5f .01,
ar + lOfiPS + S*"*®! = 3 • 14 + 2 . 23 = J4, say.
Difierenriating, __ _ _
5 . 24 + 3 . I5J= ^. = 5i:*4 -.3 • lo^'." + 3<P.oi,
Ji-3«».«« = J»=5-24-30f«i^ _^ _
il= 5.34 - 30(^1" + W) + 5- 25_ _
= 5 . 34 - 30 (fii2 + f,i3) + 5 {5fi^2 + *'»o2}>
5i _ 5^,«i = 5 . 34 + (25«^* - 3^.) " - 3PV»]Z= '*' say-
ii = (25fi - SOfi') " + (25f « - 30W 13 _
— 30f,Ci4 + ^) + 5-35
= (25fi - 30p'i') la + (50v»*^6o<pi) 13
-3o^.(r4+^^^|^)+5^.K-r2)
+ aSfiC**— 3'H). _
y.~io?.5.-5f.«i' = -(5f.-25?'* + 30fi')i2 _
+ f50f4-6ofi) 13-60^,14-
!3-
.(«— 2),
rhat follows I
, will be
(34a)
; then«i=oi.
agonal of the
their first de-
*, say.
i5_ _
f^2 + f.02},
,13 = *•• say-
?i)i3 _
+ 23) + 5- 35
pi) 13
+ 5V.(«i'-")
kI) 13 — 60V.I4-
iV^Fif ■ii^fPif^«RW/?lvs-
31
Let
-YS - (s^P. - 25^« + 30fl'). >'=(50f«-6o^i), Z=-6o?..
and ' ,
4 — tOftSi — 5f »«i = h'
Then
J. = -X'i2 + yi3 + ^14, (35)
y, = A"i2 + (A'+ r)i3 + (K+Z')i4+ r23 + Z(24 + T5)
' — "7 ** — -f (4 + 2f,at) J
y. =. [x" + 2 ^) 12 + (2^' + r '+ 1') r3
+ (-^+^+'?")H + (-^ + n23
+ (-^' - ^(M + 15)
(36)
(--D
(Ji + 2?',fl,).
Let
, /v/. ^ ^^' YZ\ —
WMaaMMi
3»
-\-^ + 5 ^ »5 lo 30
Now we have four equations, (35). (36). (37). (38). by which
(12), (13) and (14) can be eliminated, leaving
(38)
7*
*' ~5 30 '
-« + 5 10
y,
y + x.
r-^
r' + 2A" + ^. Z"-Y'^\X
*Wl
30
r"+3A"'+
zr-i
3
ZY
30 -
rz'"- !>:::_ 3^1
=0.
2
z»
30 J
(39)
an equation in «., its derivatives, and ^^f^^J^'^^f,^?^^
the a)efficient8 of (34a). It » <>f *»>« ««"* ^"^^'"^ *«"* ^*"**''
iMlllliil
^
12
" + 2X'
.)
. (38)
14
;8), by which
_ 21
3
' _ y - - A"
2 2
30-1
(39)
derived from
:r and linear,
^»»
^ti
^ti
'■mi
33
and is the first associate of (34a). To obtain the second asso-
ciate, let uf represent the second associate variables. Then
w =6l2,
u/ =013,
«^' = 014 + 023,
a^" = 2 . 024 + 123 — ioy>, 012,
«^" + lofjzw = T, = 2 . 024 + 123,
rj = 3 . 124 + 2 . 634 + 2 . 5^4 0I2,
tJ — iof«ze> = 3. 124 + 2. 034 = T«, aay,
^« = 5 • '34 + 3 . 125 + 2 . 035,
tJ + 3f»»' — lOf^ie/ = 5 . 134 + 20<P, 023 = T,, say,
^» = 5 • 234 + 6oy, 123 + 2oyl 023
_^ ■— SVtit/ + loy.r,,
^•+ 5f.«^ — lOf.T, = 5 . 234 + 6oiP. 123 + 2oyi 023 = r,.
Proceeding thus, four equations are obtained from which
024, 023 and 124 can be eliminated. The result is
A'{' +
"A'{''+
ZxZ\ y,z.
4^.Z{'
30 •
z.rh
5 10
30
=0,
n + Xu
z,z\ -^
3
z.r.
30 J
l'l" + 3A'i' +
7/ ^
Z'i -Y[ — ^
2
"^w_ Jl yii_ 3X'r\
' 2^» ~r
_z\
30 J
where JT, = 5^?, — 20<»J', K = 50^?^ — 140^; , Z = 6of,.
(40) is also of the tenth order and linear.
The third associate is the adjoint equation. It is
(40)
34
The first associate of this adjoint equation may be obtained
from (39) by writing in it
— Vt for v%>
5^4 — 20f i for 5^P4 .
A little examination will show that these transformations among
the coefficients, which change A, into A and A, mio A,, also
transforms A, into A, and A, into ^,, and in particular,
s^ St, s„ Su, X, VandZ
into T„T„T.,T,„X,, K.andZ,
respectively and vice versa. Then for the quintic at least it
follows that the rth associate of an equation is the^h associate
of the adjoint equation when
Preparatory to extending this theorem to the flth.c, it will be
well to consider it in a different way.
l(a,A, represent the first associate variable of the third asso-
ciate equation, and a.^. the (r- Ost associate variable of the
(i — i)st associate equation, then
. 1 (I2'3"4"')(.-6'S"9"')
If » = 5, then 6. 9, 8 will be 2, 4. 3. say, and the above
'^'°'"'' (23'4")(i5'2"3"'4'^) = --.(^3'4"). _
where a, is the non-vanishing constant. Thena,-*^^ = CotA^, C
is a constant. Take « = 6. ^ is the adjoint Then
(I2'3"4"'5''), (12'3"4"'6"). (I2'3"5"'6")
12'3"4'"54. (12'3"4"'6'7. (I2'3"5"'6'^)'
(12'3'V"5'T. ("'3"4"'6")". (i?'3"5"'6'T
= fl;(i2'3") or «i^i^*»
a,/4.
35
obtained
ins among
o j4i , also
ar.
at least it
ii associate
(42)
:, it will be
third asso-
ible of the
^5'6"8"'9")
the above
= Ca,Au C
en
'6")
'6")"
IS
then
a,/4, = CotAi,
where C= the constant J'. The general theorem
for all values of x and k for which x + -l= »; that is, the *- i)at
assocate variable of the adjoint equation is a constant multiple
of the (A-i)st associate variables of the original equation when
««/^,_, is
C23'4"5"' . . . »"-"). (ISV'S'" . - «- I'— ', «(— ))
(I3V'5"'6" . . . »- ,<— ), «<-«),.. .
(I2'3" . . . x_ x'«-«', * + i(«-i) ^ ." J,(.-i))
(23'4"5"' . . . «'—')'. dsV's'" ...» - i<— >, «'—))'.
( )' . / ' y
(23'4"5"' . . . »'—•)''. (13V5'" . . . «"-')"'.
(23'4"5"' . . . «'— ')'«-", ...
(I2'3" . . . , _ i(«-«», , + i(.-.) ^ ^ ^ „(.-,,)„-„
This is a determinant of order *. In the third and lower
rows each constituent equals the sum of a number of terms all
but one of which will contain «<•'. and substituting for thii its
mulLrnf'^' '•?•'"'"*"' equation, the terms arf seento^L
multiples of preceding rows and may be omitted. Each con-
deSJis ir ''"" '"* "'""^ °^ 1' ""^ '^^ --i"««»«
I'— ', 2'-», 3(-«), 4(.-.)...,,.-.,
I'"-', a'—', 3'—'. 4'—' . . . «'"-'
i'"~"'. a'— ', 3'"-'. 4'"-"...«'"-«'
agi
3«
Having found a proof showing that <i,/^._, = fl.-.'^.J""'
was not, in general, true, I used it for the case when v=i,
when it is true that a.^._. = a._.^.J— . But this follows
immediately from Section 6, Chapter V, of Determinants, by
R. F. Scott. Then we conclude that for all values of * the
(x — i)8t associate variable of an equation is a constant multiple
of the (« — X — i)8t associate variable of its adjoint equa-
tion. ^ ^ (45)
When yii is self-adjoint, A,-i = A, and then
or all equations of complementary rank associate to a self-adjoint
equation are equal. (^o)
The associate equations A» and /f,-, are said to be of com-
plementary rank.
The question arises, does this hold for other associate equa-
tions of complementary rank, i. e. for any equation does
Turning to equations (39) and (40), make
y, =0 and fi = 5ff«»
then (39) reduces to an equation of the ninth order, there being
a linear relation between the a's. But /«, or (40) does not
reduce.
a,/4, is now a non-vaniahing constant and cannot be a solution
ofAr Therefore a,^, does not equal ai.4,. (47)
i
. I
\
« ^
Sech'0/i IV.
Conditions for the Sblf-Adjointness of Differential
Equations.
Any equation is sdf-adjoint when its invariants with odd
suffix vanish.
Let r be the order of the equation. The relations which
exist between the coefficients are
(47a)
+ (^)/'iV+... « = i.2.3....r J
These relations follow from those given by Dr. Craig in his
treatise, pp. 490-493. For example, take the sextic (y), p. 491,
and (r)', p. 492. In order that it may be sdf-adjoint,
P. = P^3P, + 6P'>,
or generally,
(-.)■/>... =--r(-»-(^)/'fi.-..
If the equation had been written with binomial coefficients
this would become
If we call 6 — X, f» and divide (— j it becomes
(- lyPm = /*. ~ mPi,_^ + ~, etc.
It is not difficult to see that this will hold for any equation.
38
First, let » be odd, then
o=2/'.-«/'i_. + (^)n'-.-(-|)/'l"-.+ -.etc. (48)
2^.= 2/>.-,./>._. + ^^(-f )/>:'_,
_ n-2\2n-5\ (n\
2« — 3\2/ 2« — 3\2/ L
•Thus it is seen that (48) — 2^, contains neither P, nor Pi-i,
and that (48) - 2©,— (— ] ~~z- ^--« '* without the first two
pair of terms m Pn, P'n-i, Pll-t, Pi!'-,, and from
(4») -». - (t) ^3 «- (f)M(i^) "'-•
the first three pairs of terms disappear. By subtracting certain
multiples of the invariants and their derivatives from (48) the
terms continue to disappear in pairs. The multiplier of ^il'iv
would be 2 (^)(^)( ,^ _';_, ) = ^K> say.
From what precedes, especially (22) and (23), we know
the coefficient of Pj^„ in (48) is (j~\ ,
the coefficient of /*i'!L'i« in 2i%^, is
the coefficient of Pi'l'„ in 2M^J,'_, is
*"««■
.etc. (48)
+ ...
.4 + • . • •
, nor PUi,
lie first two
5) -*
ting certain
)m (48) the
ierof^i'l'^
know
• • •
• • •
<\
39
the coefficient of P<«'« in 2M„ev^„ is
'^' U - 2^K^;r^ri^^\ 2n-^^-^ = M,C, say.
It will now be shown that
M„C„
Let
then
(-^^) = i>^" "■'=-^7zy
i%C_
(i)
= t^aCa,
m r — " — I ! 2« — 2X — 2 ! 2M — 1 . 2X !
"•oto — j— i — r
2»\n — 2X—l ! 2M — I ! '
Witf, = »— ii2"-2«-4!2«-5.2;t!
2 ! 2« — 3 ! 2X — 2 ! « — 2x — I 1 '
generally
m^„ = .' Jl") «— I ! «-2 . «-3 . . . n—2x . 2«-4<r-i
\ 2cr / 2«— 2<r— I . 2«— 2(r— 2 . . . 2»— 2<r— 2x— I *
When « = I, m,c„ has a zero factor in the numerator for all
values of <r except «r = a The series reduces to
tftoCo =
— 2x — I !
— 2x — I !
= I.
For n = 2 the series has no zero factor, if ir = o or i, and
reduces to
— 2* — 2 1 3 2x — 2 ! 2x . 2z — I
+ ; , _ ,^
2« — I ! 2
3.2.I.2X— 3
Similarly for « = 3, 4, 5.
For « = 2x the series is m,c, _ 2x — i ! 2
For n=- 2x — I the series is
2x — I I 2
— W«<', + tftK^iCn^i ■=:
2X — 2 ! 2X !
= I.
2x
2* — 2 ! 2* — I ! 2 2x — 3 ! 2
2!
n-r = I.
iwiliMtt
40
For « = X — I,
2x ! 2x — 3 , ,^, X — 2l« + ll
tttnPo
2X — 3 ! 3 !
2X ! 2x —
2x — 2 ! :
_ 2x 1 2x ! 2« — 1 1
*«•''• - " 4!2x-4!7!2x-7
2x
«.^ , 2x!2x-7.2x!
*«''^'-*'2!2x-2!2x-5!5l^ ^^'
(- 1)-.
• • •
• • • •
• ■ • •
• • •
2x ! 2x ! 2x — 9 , .,_,
.,.....-, = «(f)(^)^x- 5 (-i)-S
»io«'o — »«,_i^,_» +• i^Ci — »»,_|f«_, + ntjCt 1- , etc.
forms a series which is equal to unity. This is seen by taking
the coefficient of y+* from each member of the equation in
which (i —yy 4- (i + J')*" is written equal to its expansion
(-^)— ^-(Ty-(lV-(Tk-(fy
+ . . . +(f )y-* - ( j)y— + ( j)y--2«y-+y".
+ ... + 8(|:)y + 7(f)y + ... + 4(^^)y + ...
The coefficient ofy *^* in the product of the right members
IS
{^(?)-=(t)(?)-*(?X?) -'(yXi)^--
i»
}
1.^
mm
which is the series [£«££-'(. ,).. The coefficient ofy + • in
2*(i _j,)(i — y)'«-« is
Therefore
»=«
.io*"'''='' (49)
Then for all velues of « in like manner the same result will
follow, and thus the coefficient of />i«!'^ in (48)
= ^Mfin + 2Jif,ei_, + 2Af,e]r_, + .,. + 2jif,ei,*i>^, (50)
of ^.!.'°K*'-*"* °^^----' '" th«J«« series is found from that
of /»<«'„ by giving a the same values and changing 2x to a« + i.
and therefore this also equals ( " ]
ouH"- '^''*^ " /r°' *•** 8^*"*'"** ""^'^t'O" between the coeffi-
cients IS expressed by
.re^ '[.(50
thitVcxV^ '™'^ *"" ?*' *^* *''"° « ^ ^^J**' 't "^y be shown
(50 = #._.+ ^.ei'_. + JV^eiL, + . . . + AT _,»i.-.,. (32)
wh^n V" • '"^*"»°*« '" (50) and (52) have odd suffixes. Then
when the invariants with odd suffixes vanish (48) equals zero!
''*^— ■"*"""T'W[n « i»iii i B[w[mrw»f iiii
■>»Mfta»fctiruw«iM<c*y^Maittiaw'.»aMflMa : QUN^b^rMdhUSi^iM'a
42
and also (51) equals zero, and the conditions for self-adjointness
are satisfied, and the proposition with which this section begins
is established.
It is to be noticed, however, that an equation may be self-
adjoint when its invariants with odd suffix do not vanish, but
satisfy the linear relation expressed by equating the right mem-
bers of (50) and (52) to zero, which is equivalent to saying that
(47a) and (57) are satisfied.
U
^
««ii»i*.-,«(#fi'.«ssa«asB»»«*SBS»«' -«?»!^i
ointness
(1 begins
be self-
lish, but
bt mem-
ing that
\>
U
Biographical.
George Frederic Metzler, the son of George Frederic and
Nancy Ann (Shannon) Metzler, was born July 17, 1853, at
Westbrook, County of Frontenac, Ont., Canada. His early
education was received at the Odessa public schools and at
different high schools. His collegiate education was received
at.Albert College, Belleville, Ont. (now consolidated with Vic-
toria College and federated with Toronto College in Toronto
University). At Albert College he took the degree A. B. in
1880, and the degree M. A. in 1883. He has taught going on
two years in public schools, two years in high school, one year
as head-master, and was called to teach in Albert College in
1881. He entered Johns Hopkins University October, 1884,
remained one session, entered again 1887. He taught in Ma-
rietta College, Ohio, 1889-90. The present year he spent in
Baltimore preparing for the degree Ph. D. His studies have
been in mathematics, astronomy and physics.
Baltimobb, Md., 1890-91.
^j^,U...iiiii(|]|HiHi.,