=P Pn; “)
TT (a; — p) TT (pr — 2) TI (ps — po — 1)... HE (ps — pn — 1)
II (p; — a — 1) II (p, —a, — 1)... I (p;— a — 1)
=F 1 F(a —p, +1, ...@&—pit1;
2— pi, po— pit 1,..-pn—pitl; 2)
+(n—1) other terms analogous to the last
ries II (a, — 1) II (a, — ps) see TI (a — pn)
oa mae = ate — Ds he = Uh ose
Maa) (ae) Waa Haass pastel ee sca
a@—a,+1; —1/z)...... (28),
wherein the argument of every power «” lies between —mm and + mr, the symbol of
equality being interpreted in the sense that :—
(A) If the real part of « is positive, the error committed by stopping the series
on the right-hand side after s terms is less in absolute value than the next term,
provided s exceeds a certain number. Of the series on the left we may select the
first so that 1, p,,... pn, are in ascending order as also @, %,...@. If a is any
* The semiconvergent series for J, (x) may be readily obtained by forming the equation satisfied by e* . I, (x)
and using the analogues of (26), (27).
a
ee
HYPERGEOMETRIC SERIES. 181
fractional number let [a] denote, if @ is negative, zero, if a is positive the integer
next higher than a Then s is not to be less than the greater of the integers
r=n ren
[a —a@ —1])+ = [2, — pr], [en — @ —1) + >> [a, — pr].
r= r=2
(B) Whether the real part of « is positive or negative*, # can be taken so great
that for any assigned value of s the error in stopping after s terms is less in absolute
value than the next term multiplied by 1+ e, where e is any assigned positive quantity
however small.
It is to be noted that in the enunciation of (A) the additional factors occurring in the
numerator and denominator of the second term omitted are all positive, and the argument
of w is restricted to a range of 7, including that value which makes all the terms
omitted real and of alternate signs; some restriction on the argument or on the amplitude
of « being, as remarked in Art. 1, not merely incidental to our method of proof but
essentially necessary from the nature of the theorem.
5. We will first prove by induction that there is one solution of the differential
equation satisfied by
TE(Chg Ch cod he, fake [ary con on StH)
which can be written in the form Ce~*x*(*-») where, as @ increases indefinitely, having
its real part positive, C tends to a fixed limit. (The minus sign has been inserted
before the w as it is easier to reason about a negative quantity when we call it —#
than when we call it +.) This is true also if the real part of « is negative but not
required for the present purpose. Let us assume that this is true when there are n
a’s and n p’s, and introduce another « and another p denoted simply by a and p. The
differential equation for the new hypergeometric function is satisfied by
Mica)
ai-P i (v— ap p*"1¢ (v) dv,
where ¢(v) is the solution of the old series referred to (see Pochhammer). Writing
(v) = Cory” = Cory
and making the substitution v=2+u, the above solution may be written in the form
70)
Curae | Cemursea’ (Ie \a4) a) “te littean see vsecateccecee tees (29).
Let us assume in the first instance that p—a is positive; this is then a multiple
(depending on the unspecified arguments) of the line integral
Cre | Ceo*ur" (1 + u/x)ee> du.
0
If (1 + u/x)2*"— be now expanded in powers of u/# the modulus of the remainder
after a certain term will be less than that of the next term, and it is accordingly
* If the latter, it will also be shown that the error is less than a certain multiple of the next term
(s restricted as before).
24—2
182 Pror. ORR, ON DIVERGENT (OR SEMICONVERGENT)
evident that the above may be written in the form C’e~*a*-P*™ where, by increasing 2
sufficiently, C’ can be made as nearly as we please equal to a certain constant.
Also in case p—a@ be negative, on integrating (29) by parts the integral may be written
in the form
1 7
1 = i fatz
= Ce up-* 1 +u/z a+m—1 as Fan em ype (1 + u/z atm—2 jo"
aaa ed reer (1+ wfayem—s
1 o_oa 44/2)
L
+-(1 buje) aca cite
The expression in the square brackets vanishes at the infinite limits, and by making z
great enough dC/du may be made as small as we please by hypothesis, and the above
integral, if « be large enough, can thus be made as nearly as we please equal to
TO) =
me = : Back NdeI SE fi ees ye C-Ca+ wa) du,
from which it is evident that any limits to the value of p—a may be extended by
unity, and therefore, for all values of p—a, (29) may be written in the form
"one oe = (a-—pr)
where C’ tends to a fixed limit as # increases indefinitely.
6. The solution which does tend to this form is a multiple of
II (— p,) 11 (—p.) .-. 1 (— pa) . ae
ieai=acuee - Ay, +» Any Pir +++ Pn; — 2x)
4's" T= 2) (p= p= 1) --- (pr = pn = 1)
at II (p,—a@, — 1)... (py —an—1)
aor F(a, — p, +1, ++ My — pr +1; 2—p,, pi— pr +1,
w+ Pn— pr+1; —2),
wherein the argument of «” lies between —mz7/2 and + m7/2. This may be seen by
making in the theorem indicated by equation (28) the arguments of # in succession — 7
and +7 and subtracting the results after having multiplied one of them by e”*. It then
appears that the above solution is one which, when in it 2 is made real and positive and
sufficiently great, can be made less than a certain multiple of any specified term of the
series
2% F(m,4—ptl,...; a@—-a4+1,...;+1/2),
and therefore must be the one in question, as from equation (28), assumed to hold for the
above functions, no other solution can be of this order for infinite values of « which have
their real part positive.
7. We require to evaluate the integral
ip tl OM (Be CB nie CRT NY Oh BED [PACD ICH aGacbondenonasacacteasacc (30),
when intelligible, that is when m-+1 is positive and m—a,+1 is negative, where a, is the
algebraically least a. (When « is very great the hypergeometric series is of order #-%.)
HYPERGEOMETRIC SERIES. 183
The hypergeometric function is equal to
II (p, —1) es
I—pn (> — n—An—1 pyan—1 , ae
P(e =) eae | Og me CO ah Plaes- Ba) Oe
provided a, and p,—a, are positive. Substituting this value in (30) and changing the
order of integration, which can be shown to be legitimate since m—a,+1 is negative,
(80) can be written in the form
— i (Pn Di al yx—1 . : ‘i M+1—pu ( u—an—1 7
Mel Gale UOT (0; 51... C1} Pry «-- Pr=i3 — 0) av : apn (a — yp dz;
but the # integral is equal to
I (an pL Toa 2) u (pn —4a,—1)
II (pn — m — 2)
gittl—a
v m
and accordingly (30) is equivalent to
Tl (pn — 1) 1 (a, —m — 2) .
TL (@ — 1) IE (pn — m — 2) Jo
EH (Coss «on Onmaisy Pals ee Pasi) aU:
In the same way provided a, and p,— 4 are positive
TI (p, — 1) I (a, — m— 2) (*
Il (a, — 1) IL (p,; — m — 2) Jo
_ IT (p,— 1) Il (a —m— 2)
~ II (@,—1) I (p,—m— 2) Gy
ame dx
I a” F(a; py; —") dx =
0
Therefore by induction we finally obtain for (30) the value
r=n TI (p,— 1) II (a,—m-— 2) :
Il (m) . we Tl (a, =i) I (p, = a — 2) eee cece cee ceesececccesesecs (31),
provided, in addition to the conditions necessary to make (30) intelligible, all such
quantities as a, and p,—a, are positive.
These latter conditions may however be removed. Since
F(a, +1, Ga, «++ Any P1> Pa +++ Pn — &) —- F(a, Oy, «0. Ans Pi, Pa: -++ Pn —)
eS > F(a, Qs, +++ Any Pr, Pas +++ Pn — 2),
by multiplying both sides by z™ and integrating we obtain
| 2B (ay +1, G3, «0 ta; pis «-» Puy — 2) Ce — | OMIT (Cietewe ln Ors ea) Pas we) Cer
0 Jo
ae ret Lf
-|* FG), <. On; Piss Pai —2) | ee GOEL (Chi reael ns Pusey) oe
1 0 1 0
If the second integral in the left-hand member is intelligible, so also is the first; the
expression in square brackets then vanishes at the limits and
| a (as, see Ans Pry +++ Pn3 — 2) dx= a-t2 1 / x” F(a, +1, Qe, +e» And Pris +++ Pnjy — x) dx,
“0
0 a
184 Pror. ORR, ON DIVERGENT (OR SEMICONVERGENT)
and accordingly any inferior limit imposed on any a may be extended by unity, the
other a’s and p's being kept unchanged, provided in (30) the integral remains intelligible.
Also from the equation
F(a, @, +++ &n> Pis Pr» +++ Pry —2)—F(%,%, see Bn Pi» Pe, «++ Pn—-1» pn—1; — 2)
—a d
= ——_ —_- <: _—f 9
melas ee BOR nee ea apoy/ yA ==) borongorosoond sagedgcatonee (32),
by multiplying both sides by #” and integrating we obtain
| 2B (a, Gy, «++ Bn Pir Pos +++ Pans —o)de— a” F(a, s+» Q&r3 Pry Pos +++ Pn pn—1; — x) dx
0
= = [ame (a, Qs, eee And Pir +++ Pans -2)|
0
Pa—
m+1 [*
Sh HOI Can, con Cay [Sg Ep coo (2 SHA)CHD. conosacsdebooosuodsnse0e (33).
Pa= 0
If (30) is intelligible all the terms in this equation are finite and that in square
brackets is zero at the limits and thus we obtain
i x” F(a, Gy, ++» Any Piy Poy +++ Pry pasts — «x)dx
Se he
Be ela | GHOTH (Chg suo @h8, (Sip 606 [2R =) CMoovoocc0c (34),
Pa — Is aie
and accordingly any limit to the value of any p may be extended by unity, the other
a’s and p's being kept unchanged. The result stated as to the value of the integral (30)
has thus been established *.
8. It is to be noted that the proofs given of the results of Arts. 6, 7 for functions
of any order assume the truth of equation (28) for functions of the same order. We
now proceed, assuming the results of Arts. 6, 7 and equation (28) for functions of any
and the same order, to extend equation (28) to functions of the next higher order by
the introduction of another p and another a. We do so in the present Article, taking
the equation in sense (A) but subject to the restrictions that each of the quantities
P2— 2, +++ Pn—m, p—4 %+1—p,,...%q+1—pa, ut+t1l—p, is positive. In Art. 9 we will
extend the equation in the sense (B), and in Art. 10 the restrictions introduced in the
present Article for the sense (A) will be removed.
As indicated by Pochhammer the equation satisfied by F(a, @%, a2, -.. @n3 Ps Pry «++ Pn; +2)
is satisfied by
ai | (Oa) Swe B (0) dv eek eheceeesce Seog ee (35),
where ¢(v) is any solution of that satisfied by F(a, %,... nj Pir ++» Pn; +) and the
path of integration is a closed one such that the final value of (v— a -*1v7"¢(v)
differs by zero from the initial one.
* This result may be generalized by omitting any number of a’s, if the integral remains finite; write r=y/a
and then make a infinite.
HYPERGEOMETRIC SERIES. 185
Let the path be one which makes a circuit round the point w in the positive
direction, then round the origin in the positive direction, then round the point @ in
the negative direction and finally round the origin in the negative direction. Suppose
in the first instance the real part of # to be positive.
Such a path is equivalent to the paths ABCA, ADBA, ACBDA which may be
replaced by the four portions ABCA, ADBA, ACBA, ABDA, (Fig. 3).
—
Let A be a point h on the axis of real quantities and let ¢(v) be
CF (a, Ao, vee an; Pi> Pe, =us Pn; +)
r=n
+> Cw F(a — py t+], ... tr—pr+1; 2—p,, pi—pr +1, .-. pa— perl; + v)seeeceeee (36),
r=1
those values being taken which make the initial arguments of every power of v zero
at the point h (before multiplication by C, C,), and make the initial argument of
(v—«x)-* diminish indefinitely as h increases indefinitely.
On examining the values of the arguments at different points it will be seen that
the second and fourth portions of the path together contribute to the integral (35) the
expression
(z)
(en (a—p) — g2ria) gi—p if C (uv — ap ut F(a, Ms, -.. On; Pir +++ Pn; +2) dv
r=n fx)
iS (ei (e-P) = tien) af C,(v — £)p-2-} ya-er F (a, = pr + MS Sob, —prt+ L2 — Pr, ++
TL h
+e Pn—Pr+1; +) AV.....00..0-- (37),
the initial arguments being taken as above.
If the differential coefficient of this expression with respect to h be written down
it will be evident that in virtue of equation (28) assumed for the function of the
(n+1)th order, i.e. that which satisfies the differential equation of the (n+1)th order,
provided we take
Tl (a aa 1) (= p:) oss TI (— pn)
II (— a.) I[(—a,)... J (— a)
II (a, —p,) I (p, — 2) II (o, — p, —1)... 1 (p, — pn — 1)
II (p, — a — 1) I (p, —a;—1)... I (p,— an, — 1)
(oH (e" (a—p) _ e27ai) =
(6/3 (e7 (a—p) _ g2nt Car
this differential coefficient when h increases indefinitely will be of the order he-=-*; and
the same is true also for a complex value of A provided the argument of h is kept
between —7 and +7 and the value of the function to be integrated is reconcilable
186 Pror. ORR, ON DIVERGENT (OR SEMICONVERGENT)
with that previously taken. Accordingly provided p—a,—1 is negative, a condition
which we have for the present supposed satisfied, the expression (37) will remain
finite when h increases indefinitely, and its value will be unaltered if for h we
substitute any infinite limit whose argument les between —7 and +7 and if the
value of the function to be integrated is reconcilable with that already taken.
Now let be increased indefinitely. We have also supposed for the present that
p— is positive; each symbol of integration in (37) may therefore be replaced by
(en (p—a) __ |
z
in which the infinite limit is on the production of the line joining the origin to the
point 2.
With the above values of the constants C, C,, and assuming the theorem to hold for
the functions of the (n+ 1)th order, the expression (37) may be written
TI (a, —1) (a — (yoo WIN(CA — Pn)
art (p—a) __ E
ea TGs =a.) l(a en) Le)
| ape (v — 2pm yoo
~ Zz
F(m,%—pitl,.-- G—patl; ma—-m+1, a—a; 41, ... | —a, +1; — 1/v) dv...(39).
Expanding the divergent hypergeometric series in descending powers of v and
integrating the terms successively we obtain
II (p—a—1) 1 (4 —p)U (4 —1) 1 (a —p,)... 1 (um
II (@, — a) I (a, — a)... IE (a, — an)
(er! e-2) — 1) = Pa) oa F(@,a—p+i,...
%—pratl; m-a+1,a—4,41,...q—a,+1; —1/z)...... (39a),
the argument of a™ lying between — 7/2 and +4,7/2; and by expressing the error as
an integral, the theorem being assumed for the function of the (n+ 1)th order, it appears
that, provided @,+s and all the quantities of the type a,—p,+1+s are positive, the
error in stopping the divergent series at the sth term has a modulus less than that of
the next term.
Returning to (35), the sum of the portions contributed by the first and third
portions of the path is
: f (0)
(_-e™ ay ge CO (uy — 7 P-*1 9! F(a), ..- On} Pir --» Pn; +0) QY
r=n Tf (a, 0)
+S (1 — erie) | C,a0'-° (y — @)P-2 y-p
7=1 h
F(a —p,+1, ... @2—prt+1; 2—p,, ... pa—pr+1; +) dv...... (40),
the initial values of the arguments at the point h being the same as in (36). The
differential coefficient of this with respect to h is of the order h?-=~ owing to the
particular values assigned to C, C,, and accordingly (40) like (37) remains finite
HYPERGEOMETRIC SERIES. 187
when fh is increased indefinitely provided p—a,—1 is negative. We now expand
(v—a)y-* in ascending powers of #; the coefficient of #'-?t” in (40) is therefore
— (0)
II (a Pp ss | Corn F(a, pe ee ie < gs’ -t0))
SE Tap) I Gn)
r=n
+ Cye—m—i-er F(a, — prt 1, ...&— pr+1; 2—pr,--- pn—pr+1; + »)| Gil osess (41),
r=)
wherein all the powers of v are initially real before multiplication by the complex
coefficients C, C,.. The successive values of m are 0, 1, 2, &e.
We proceed to evaluate the integral in this when / is made infinite. By considering
that, owing to the particular values assigned to OC, C,, its differential coefficient with
respect to h is of the order }p-"-™~, it appears that the integral remains finite when
h increases indefinitely, (provided p—m-—a,—1 is negative, which is certainly true if
as already supposed p—a,—1 is negative), and as in the case of (37) this is true also
for a complex value of h provided the argument of h is kept between —7 and +7
and the value of the function to be integrated is reconcilable with that previously taken.
Accordingly by changing the argument of h to ~—7 as in the path GFEABDAEFG,
{0)
(Fig. 4), the symbol of integration in (41) may be changed into | " where all the
powers of vw in the function to be integrated have zero argument (before multiplication
by the complex coefficients C, C,) at the point in which the path intersects the
positive part of the axis of real quantities, and the initial and final limits of integration
are not merely both negative and infinite but the same. (By changing the argument
(0)
of h to +7 instead, we might obtain the symbol | with different values for the
arguments of the functions to be integrated; the evaluation of the integral would
Vou. XVII. Parr III. 25
188 Pror. ORR, ON DIVERGENT (OR SEMICONVERGENT)
however lead to the same result as that to be presently deduced for the present case,
as of course it should.)
Let us first suppose that p—m-—1 and all the quantities p—m—p, are positive;
the integral can then be expressed as a line integral, and attending to the values of
the arguments and of C, C,, it is in fact
i" [ene wsin(p—m)a I (a,—1)U(—p,)--- I (— pn)
am)
p—m—2 : ey
Sn = ‘ ani AK ial e @,) uw F(a, +++ Any Piy+-+ Pns w)
.sin (ep — m—p,)m 1 (a, — pr) II (p, — 2) Il (p,— p: — 1) --- II (p,— pn — 1)
e(P + pr—2a) mi
1 sin(p,—p)7 — IL (p,— a— 1)... I (p,— an — 1)
2 YPM pr
r
i
F(a, =p, +1, .:.@,—pr+1; 2—p,, pi—pr+1, ..-pPn—pr+1; -»)| du.
Using the values given in (31) for each of the n+1 terms of this integral and
making use of the relation II (n—1)II(—n)=cosec nz, the above may be written
a Il (a,—p+m)II(a,.—p+m)... I (a,.—p+m)
Il (p:— p +m) I (p.— p+m)... U(prx—p+m) UA—pt+m)
Apneyat SIN @ 7 SIN 4 7... SIN &, 7
sin p 7 SID p, 7... SID Py 7
Mae ee is sin (% — p,) 7 Sin (3 — pr) 7... SiN (an — pr) 7 | ee a. (42)
a sin p; 7 Sin (p — p;) 7 Sin (p; — pr) 7 «-- SIN (Pn — pr) T
But the expression in square brackets is equivalent to
sin (p — a) 7 sin (p — 4) 7... SIn(p — &) 7
sin pT sin (p — p;) 7 Sin (p — ps) 7 ... SIN (p— Pn) 7’
ef —2a) ri ‘ epmi
for it is easily seen that this last may be written in the form
e(P—2a) wi f : A 2 ae 2 38
(sin pw sin (p — p,) 7 J
?
where A, A,, etc. are quantities independent of p and their evaluation in the usual way
leads to the result stated.
The value of (41) is then
U(a—p+m) Il (a, —p+m) I (a—p+m)... I (a,—p+m)
II (a—p) IU (m)° I (p,—p+m) Il (ce. —p+m)... II (pn — p+ m) (1 —p+m)
sin (p — &) 7 Sin (p — a) 7 ... SIN (p — Gn) T
sin p7 sin (p — p;) 7 ... Sin (p — pn) 7
(e2"" p—a) __ l)r
The limit of the expression (41) when h is infinite still has the value (43) even if
the additional conditions introduced in the process of evaluation (viz. that p—m-—1 and
all the quantities such as p—m-—~p, should be positive) are not satisfied. For bearing in
mind that
Gy Ao..
l ;
ap Fm Az, --- Ans Pi: Poy +++ Pn; v)= sy (h(a a Geet Warsi Anictaaks pi bl, see Put Ll; v),
Pi1Po +--+ Pn
efi ot
HYPERGEOMETRIC SERIES. 189
and that
d
qin F(a — at, en pi tl; 2— pr, Poa—pPpitl,...pn—pit1; v)}
=(1—p,)v F(a,—p,+1,...a¢,-—p, +1; 1—p,, pp—pit1,... pr—pit+1; »),
if we write the integral in (41) in the form
fo)
| ye—-m—2 & (y) dv
wh
and integrate by parts, we obtain
1 h 1 7 (0)
—m—1 = p—m—1 of’ ,
seul” bo] aul g P (v) do.
But, attending to the values of C, C,, given by (38), it appears that the difference
between the two values of the expression in square brackets when » is equal to h is
zero when h is infinite and that —¢'(v) only differs from $(v) by having all the
constants @,...@n, pi,.-. Pn mereased by unity, and accordingly we can increase all the
quantities @. a,...2n, P, Pi,+-- Pn in the equation Lt. (41) =(43), keeping m unaltered.
In a similar manner we can show, by writing the integral in (41) in the form
ao)
[cme e(v) do,
Jh
that we can increase p, by unity, keeping the quantities a, a),...0n, Ps Pis-+-Pr—is Preis-++Pns
m unaltered, and still have the equation: —Lt. (41) = (43).
h=a
The initial value of m is zero and in that case (43) reduces to
TI (p — 2) IL (p— p: — 1) UI (p —p,— 1)... I (p — pn— 1) I (a — p)
I (p= SN pap Seah a oa
and accordingly the expression (40) contributed by the first and third portions of the
path is this multiple of
— (e27i (e-2) _ ])
e- F(a—pt+1,m—ptl, wa—-ptl,...¢ar—p+1;
2—ip; pr—p+ly2..ipn—pAls 4pm)... vec (44),
wherein the argument of z'~? lies between =F —p) and +5(1 —p).
- Again returning to (35), (36),
(x, 0, w—, 0—)
| (GOS SEA (are ayinne Ons Pion Pasieea Ona. U) QU:
ce
wherein the arguments have values reconcilable with those chosen, may be shown to be
Tl (p—a—1)M(a—J)
[1 (ep —1)
— de (*-)) sin (9p —a—1)rsn(a—1)7 s
tHE (iv chs asia Oley sayeth) Pais) eee eee a eee (45),
190 Pror. ORR, ON DIVERGENT (OR SEMICONVERGENT)
wherein the argument of 2° lies between — (p — is and +(p — 1)z, and making use of
the values of C, C,, given by (38), the integral (35) may after some reduction be
written in the form
a II (p —a—1) TI (a, —1) I (— p) 0 (— ps)... Th (— pn)
2ri (p—a) _ “ . * 4
(ate ay OS aya) ag aay FO Or Bad Ps Ps Bas
"=" TI (p—a—1) 1 (a,—p,) I (p,—2) 11 (p,— p —1) I (p,—p, —1)--- 1 (p»—pn-1)
ay >l—pr —
pact II (p, —a—1) II (p, — a — 1)... If (pr — an — 1) PE a rts
4
Q&—p,rtl; 2—p,, pi—prtl,--- pa—prt1; »)| Score been coneecmnactcaachss (46).
Equating (46) to the sum of (44) and (39a), and dividing by (e °-» —1)II(p—a—1),
we obtain an equation of the same form as (28) and to be interpreted in the sense
(A), but with an additional p and an additional a.
I do not see how to remove the restrictions imposed at the beginning of this Article
without first showing that the theorem is true in the sense (B).
9. We now proceed in a different manner to extend the theorem in the sense (B).
— ae
The differential equation for F(a, a,...@23 P,Pi;-++ Pn; +) is satisfied by
(—2z, 0, —x—, 0—)
i Grea () 2h GPO (OND). so oaseoensnososososorace (47),
c
where @(—v) is any solution of that satisfied by
IH(Cee sca GES fIxycna/Spy = OCD
(It would be more consistent with what has gone before to change the sign of v
in the above. The introduction of the minus sign has however the advantage that
the function of —v with which we will be concerned does not involve 7 explicitly to so
great an extent.)
Fig 5
The above path is equivalent to the four paths ABCA, ADEFA, ACBA, AFEDA (Fig. 5),
A denoting a point h at a great distance on the positive part of the axis of real quantities,
HYPERGEOMETRIC SERIES, 19]
It is assumed in the first instance that the real part of « is positive and the argument
of a is taken to be between —(1—p) 7/2 and +(1—p) 7/2. Let @(—v) be
AF (a;,...n} pis ++» Pn} —V)
ae > Ao F (a, — pp +1, «0. an — pr #1; 2— pr, Pi— Prt+1,... pPa— prt 1; — v),
r=1
those values being taken which make the initial arguments of every power of v zero at
A (before multiplication by A, A,) and make the initial argument of (v+)-*— diminish
indefinitely as h increases indefinitely.
On examining the values of the arguments at different points it will be seen that
the first and third portions of the path contribute to the integral (47)
y (=2, 0)
(i es*) ase ik A (u+ @)P>° U2 (a, ... Gn Pr, --- Pn3 —V) av
r=n T(—2, 0)
+ (1-e (oat? | A, (v+a)P—2-1y2-Pr F(a, — py+ 1,...; 2 — py,-.- pn— Pr+1; —v)dv...(48),
r=1 h
the initial arguments being taken as above; and assuming the results stated to be true
for the function of the (n+1)th order, if we take the differential coefficient of this with
respect to h it will, by Art. 6, as h increases indefinitely, become of the order of a
product of e~* by a certain power of # and therefore diminish indefinitely, provided
ee “ee TI (—p,)..- 1 (— pn)
— parat Qrpt __ == a (at+p)t —=
[1 — e?r*) (e27? — 1) A =] 4e7 (**?)* sin am sin pr. A Tis). seas
[(1 — etie—er)) (e2t(o— Pr) — 1) A, =] 4e7(@+P—2hr)t sin (a — p,) 7 Sin (p — p,) 7. A,
A II (p, — 2) Il (pr — p1— 1) ... II (p, — Pn—1)
II (p,—a, —1)... IL (p,— a, —1)
pe eee (49),
{
and therefore with these values this integral will remain finite when h increases in-
definitely.
In the expression (48) we now expand (v+w)?-*7 in ascending powers of z; the
coefficient of w°*” is
m U(a-—p+m) [
i ez 2 —m—2 . : ;
(-) TI (a—p) II (m) ie aa excy) A ( uP F(a, axe © ans Piy oe Pn; = v) dv
“T(-2, 0)
€ ‘a — ene-»9) A, | yr—m—1— Pr F(a, — py +1, ...3 2— py, ++» Pn—pPr+1; —v) ae] ...(50),
7=1 h
i Mi
wherein all the terms are initially real, and the successive values of m are 0, 1, 2, &e.
This too remains finite if 4 increases indefinitely for the same reason as (48).
We will first suppose that all the quantities p—m—1, p—m—p,,...... p—™M—pn,
are positive; the expression in square brackets can then be expressed as a line integral
192 Pror. ORR, ON DIVERGENT (OR SEMICONVERGENT)
taken between the limits 0 and h, and when / increases indefinitely, using the value
given by (31) for the integral (30), it is in fact
S11 (5 —1)I(@,— p+m)
x (at+p)i _- = rae
ger'e*0% sin am sin pm. ATI (p—m—2) Tl Tr ay Tp =p +m)
ae TT Wiese pr) I as — p+ m)*
= "geriat 2p) sin (a— p,) 7 Sin ,)7.A,Il(p —m—1—p, wi Aetna iL
+= (a— pr) (p — pr) (p pr) I Ht oll, Seen
Attending to the values of A, A, and making use of the relation
II (x — 1) Il (— n) = II cosee ur,
this may be written in the form
T s=n TI (as — p +m) ( SIN @7F SIN G7... SIN Ay
Il (m —p+1) sai I (pp—p +m) \sin pyr sin ps. ah Da —p+t+2)7
Jee sin (a@,—p,+1)msin(a,—p,+1)7... sin(@,—p,+1)7
a 21 8in(p,— 1) 7 sin (p,— pr +1) 7... sin (pn — pr + 1)7sin(m+1+ p,—p)7\-
But the expression in brackets is equal to
sin (a, +m—p+2)rsin(a+m—p+2)7...sin (a,+m—pt+2)7
sin(m—p+2)7sn(m+1+p,—p)7...sin(m+1+pn—p)7
(Se
for it is readily seen that this last can be written in the form
B r=n B
+ | —____—~_____,
sin(m—p+2)7 ;=.sn(m+1+ p,—p)7T
where B, B, are quantities independent of p and their evaluation in the usual way leads
to the result stated.
Accordingly (50) reduces to
(— Il (a—p +m) II (p —m — 2) II (p —p,—m—1)... TI (p — pa— m—1)
II (a—p)II (m) I (p — a, —m—1) II (p —a—m—1)... 1 (p-—a, —m—1)
This result may then be extended to cases in which the conditions that p—m—1,
P—M— Pi eee p—m-—pn should be positive are not satisfied, as is done in a parallel
ease in Art. 7.
Therefore the portion contributed by the first and third portions of the path is,
when / is made infinite,
TI (p — 2) I (p— p:— 1)... I (p — pn — 1)
Il (p —%—1)1(p—a@,—1)... I (p — an —1)
2—p, pi—pt1, po—pt1,...pn—pt1; +2)..........-- (50a).
vw F(a—p+1,a—pt+l,...@—pt+1;
* For s=r, pg is to be replaced by unity.
HYPERGEOMETRIC SERIES. 193
Again, the second and fourth portions of the path contribute to the integral (47),
: Oe
(e?"P" — 1) wp [ A (vu + @)P91 9-1 (a, 6. An} Pry Pas +++ Pn} — V) dv
r=n ; nO)
+ (e2re—Pr) — 1) ate | A, (y + c)P-91 yt F(a, —pr+1,...d,—pr+1; 2—p,,...
r=1 vh
Pn — Pr+1; —v) dv...(51).
We now expand (v+#)-*-' in descending powers of «; the coefficient of a-*-™ inthe
above is
(—)y™ I (a— p+m)
Il (a—p) Il (m)
7 (0)
jer 1) A| ymra-1 F(a, ... An; Pr--- Pn; —v) dv
oh
+3 (enn = 1A, [ume-eF (a —Ppr+1...dn—pr+1; 2—p,... pa—pr+1; —v) ae] «»«(52).
r=
Suppose at first that the quantities m+a, m+a—p,4+1,...... m+a—p,+1 are all
positive; this then reduces to a line integral and the value of the expression in square
brackets may be obtained from the value obtained for the expression in square brackets
im (50) by changing m into —m—1, interchanging « and p and then multiplying by
—1; accordingly the value of (52) is
Il(a—p+m) UW(m+a—1)1(m+a—p,)... I (m+a—p,)
IIl(a—p)Il(m)° U(m+a—a) UW (m+a-—a,)... L(m+a—a,)
(-)rm
This value may as before be extended to the case in which the conditions that all the
quantities m+a, m+a—p,+1,...... m+a4—p,+1, must be positive, do not hold. The
successive values of m are 0, 1, 2... and accordingly we obtain for the part contributed to
the integral (47) by the second and fourth portions of the path, the divergent series
_H@—-)U @—p) =p.) ... 1 G@— pn)
Il (a —a,) l(a —a,)... I] (a—a,)
w* F(a4,a—pt+l,a—p,tl,...a—pratl;
a—@q+1,¢@—-a@+1,...a—a,+1; —1/z)... (53).
As regards the remainder in this series after s terms, the remainder after s terms in
the expansion of (v+a)-* has the origin of v for a multiple point of order s, and has,
by Art. 2, a modulus less than that of the next term provided a—p+1+-s is positive;
and accordingly bearing in mind the order of ¢(—v) in the neighbourhood of the origin
of v, the remainder in (53) may be written in the form of a line integral
Ca-*-* | pur (— 2) do,
“0
provided, in addition, all the quantities «+s, a—p+1+s, a—p,+1+s,...... a—p,+l+s,
are positive, C denoting the numerical factor, and p a quantity whose modulus is less than
unity. We are not however justified m assuming that this integral would be increased
numerically by replacing p by unity, and hence that the remainder in (53) is less in
194 Pror. ORR, ON DIVERGENT (OR SEMICONVERGENT)
absolute value than the first term omitted; for it seems possible that @(—v) may change
sign between zero and infinity which would invalidate such reasoning*; (if this objection
could be removed, this proof would establish the theorem in the sense (A) also). We will
show however that if the inferior limit to s imposed by the conditions just laid down be
raised by unity, the modulus of the remainder after s terms is less than that of a certain
n
multiple of the next term. Denoting =(a,—p,) by o, as v increases indefinitely ¢(—v)
1
becomes of order e~’v’. Suppose that of values of v lying between zero and infinity 2,
is that which gives to v*"**$(—v), which owing to the inferior limit of s being raised
is now zero when v is zero, its numerically greatest value. Choosing any positive quantity
y less than unity, find a value v, of v so great that for all greater values ¢(—¥v) lies
between the limits C’(1 + vy) ev’, C’ being a constant which we could find if desired. The
integral in the above remainder is therefore less than
Ke) (— %) + (6 al + | e-ryaitste
y
and therefore less than
v%,6(—%)+ CO A+y) I (a-1+oa+s).
Therefore the remainder is less in absolute value than a certain multiple, independent of
a, of the first term omitted.
If the real part of w is negative, the same may be proved, for a different multiple,
depending on the argument but not on the modulus of w, in a manner similar to that
in which the parallel case for the function F(a; p; x) was treated.
Since by taking w great enough, terms at the beginning may be made to outweigh
as much as we please any finite number of those that come after, and since there is in
the above no superior limit to s, it is evident that x may be taken so great that the
error committed by stopping the series at any assigned term is less in absolute value than
the next term multiplied by 1+e, where e is any assigned positive quantity.
Returning to (47)
(—*, 0, —%—, 0—)
| DP (UY -- e)Po tS UE (Ai, .-. On; Pir =» Pn; — 0) aU
“¢
with the values of the arguments reconcilable with those already chosen may be shown
to be
—4et)™ sin (p— a) sina . ae trea re: &, ++» &n; Ps Pr» ++» Pn; +2),
and
ire 0,—2—, 0—)
ai? (y + @)P-* yr BF (oy — py +1, ...3 2 — py, pi— pr + 1, .-.; —v) dv
Je
* As @(—v) is an integral of the form of that discussed therefore be proved in this manner subject to these
in Art. 5 it may be seen that it does not change sign for restrictions which may be removed as in Art. 10. Art. 8
values of v between zero and +o provided for all values (part of which had gone to press before this was noted) is
of 7 from 2 to n, p,—a, is positive. The theorem might therefore to a great extent unnecessary.
=.
HYPERGEOMETRIC SERIES. 195
to be
II (a — p,) Il (p —a—1)
Il (p —p,—1)
%—prt+1,...én—pr+1;2—p,,;p—pr+1,... pn —pr+13 +2)
— 4e'e+o—2") Ti sin (p — a) or sin (a—p,+1)7. oe F(a — pr +1,
wherein the argument of a'~*r lies between —7(1—p,) and +7(1—p,), and making use of
the values of A, A,, given by (49), the integral becomes
by ] II (a — 1) HI (—p) Il (~p,)... U (= pn)
II (a — p) II (— a) UW (— a)... (— an)
é. 1 S IT (a — p,) U1 (p, — 2) Il (p, — p — 1) IL (p, — p, Eyes Il (Pr — pn — 1)
Il (a@—p)4 Il (p,—a@,—1) I (p,— a — 1)... IL (pp — &n — 1)
(Gy, «06 On Ps Pry «+> Pn; @)
aver F(a—p,t+1,aq—p,t+1,...d.—pr+1;2—-—p,,p—prt+1,... pn—pr+1;+2)...(54).
Equating this to the sum of (50a) and (53) and then multiplying by Il (a—p) we obtain a
result similar to that indicated by equation (28) in the sense (B). It differs however from
that obtained in the sense (A), in having a and a, p and p, interchanged.
10. Before proceeding to remove from equation (28) taken in the sense (A) the
restrictions imposed in Art. 8 that certain quantities must be positive, we will first
show that if a—a, is positive and if the theorem holds for the remainder after
s terms of the function involving a, a, it holds also for the remainder after (s+1)
terms of the function involving , %,+1, the other a’s and p’s being unchanged.
Tf Wm, %,-.-G3 Pi, Ps --» Pn; £) denote either any one of the n+1 seres of the
left-hand member of (28) including the constant multiplier, or the sum of the terms
at the beginning of the divergent series on the mght (including the constant multiplier
and the factor 2") up to and including the term involving a specified power of @, it is
easily verified that
Any, Qa; Onis Ag+ ls, Pry Pay --- Ps 2)
=(a% — %) (a, Gz, +e» An-1, Gr; Pir Par +++ Pns z)
—wv(a+1, Os, +++ Any, An; Pi, Pas +++ Pn; 2).
If then R(s+1), R(s+1, a,+1), denote respectively the remainders after (s+ 1)
terms of the right-hand member of equation (28) for the function involving %, a,
and for that involving m, a,+1, and if R(s, a, +1) denote the remainder after s terms
for that involving a+ 1, a, we have
R(st+1, m+ 1l)=(m—%) R(s+1)—R(s, 1 +1).
If then the theorem holds for R (s, a), a fortiori it holds for R(s+1, @,), and it
also holds for R(s, a +1), for = [a,—pr] is not increased by increasing a; we
thus have ie
mod. R(s + 1) < mod. ai ae to ie ae ETE eat
Vol. XVII. Parr IE. 26
196 Pror. ORR, ON DIVERGENT (OR SEMICONVERGENT)
and mod. R(s, 4 +1)<(s+1) times the same, therefore provided a,—a, is positive we
deduce that
II (a,+s)... I(aq—pat+1+s) 1
TI (a,—@,+1+s)...11 (4-4, ~+1+s) II (aq —a, +s) I (s+1)ants*)?
mod. R(s+1, a, +1) < mod.
which is the result stated; this reasoning holds for all values of s including zero.
We will next show that if the theorem holds for the remainder after s terms of
the function involving 4, it also holds for the remainder after (s+1) terms of the
function involving a,—1, the other a’s and p’s being unaltered.
If (a) denote either the left-hand member of (28) or the sum of the terms at
the beginning of the right up to and including the term involving a specified power
of xz, we have
a? ab (a4) = = fea U trea = Lia fv, coruseets ce seeees eee (55).
If then we denote the difference between the left-hand member of (28) and s terms
of the right-hand member by R(s, a, x), we have
au Ri(s, a, 2) = = ociaeal £o| Cid bet Hemel aE tan sa ska tin (56),
and therefore . :
Ee R(s+1,a,-1, »| =| ATTA 3) Oh) CAB sceooanos05s76cc0" (57).
For all values of s including zero, 2» R(s+1,a,—1, 2) vanishes when the modulus
of « is infinite, whatever be its argument, since equation (28) is true in the sense
(B) and since the first term omitted when multiplied by 2*' has for its index «+s,
the being negative of which is one of the conditions that the theorem should hold
fore (Gaceeee):
Accordingly M
Repel z)=— a | w3R (6, a, 2) U2 sacecsccceieeaeestew (58).
By taking for the path of integration the production of the straight line joining
the origin to the point 2, it appears that either member of (58) would be increased
in absolute value by replacing R(s, a, 2) by the modulus of the first term omitted; but
this change would replace the right-hand member by the modulus of the (s+2)th
term in the series obtained from the right-hand member of (28) by diminishing a by
unity; therefore R(s+1, 4—1, 2) is less in absolute value than the next term:
this reasoning holds for all values of s including zero,
We will now, having in fact proved the theorem for all values of s including zero
subject to the conditions that for all values of + from 2 to mn inclusive a,—p,+1 and
p,—4a, should be positive, proceed to examine what restrictions should be placed on s
if these conditions are violated.
HYPERGEOMETRIC SERIES. 197
We suppose that a, is the greatest a and p, the greatest p.
Suppose first that a,—p,+1 is negative lying between —a, and —a,+ 1, and that
Pn—% 18 negative lying between —b, and —b,+1. Then a,—a,+1 must also be
negative lying between —c, and —c,+1 where c, is either d,+6, or d,+6,—1. For
the other values of 7 from 2 to n—1 let b, denote [a,.—p,], which for some values
of r may be zero.
The theorem then applies for all values of s to the function involving
&%+Cn, a, — bs, see a, —bdy, 1, Pir» Pes «++ Pn»
since the necessary conditions are satisfied.
We may increase the value of a,—6, by unity 6, times in succession, keeping
all the other a’s and p’s unaltered, provided at each such operation we increase the
lowest value of s for which the theorem holds by unity, the condition for the validity
of the last such process being that a,+¢,—(4,—1) is to be positive; thus when we
attain the value a,, s has to be raised from zero to b,. Then for each other value
of r in turn we may increase in a similar manner the value of a,—6, by b,, increasing
the value of s at the same time by b, also, the condition for the validity of this
being that a,+c¢,—(a,—1) should be positive, which is true since a, is the largest a.
Thus when we attain the values a, a,...a,, the lowest admissible value of s is Sao
r=2
Finally we diminish the value of a,+¢, by unity c, times in succession without altering
the other a's or p’s, at the same time increasing the value of s by cy. Thus the
n
lowest admissible value of s is [a,—a—1]+[a,—,], as the enunciation states.
2
Next suppose that a—p,+1 is positive but that p,—a, is negative; we have
now two sub-cases according as a,—4a,+1 is negative or positive.
Taking first the former: as before the theorem applies for all values of s to the
function involving
& + Cp, 2 — bo, Sinks an — Dn, ul. Pir Pa, +++ Pn>
and we proceed as before, with the result that when we attain the function involving
@, Go, .-. &, I, Pis +++ Pn>
the lowest admissible value of s is ise & [a, —p,].
Taking the latter sub-case, the theorem now applies for all values of s to the
function involving
CR CRS UN meas I, hg Se. fare
198 Pror. ORR, ON DIVERGENT (OR SEMICONVERGENT)
as before, for each value of r+ in tum we may increase the value of a,—b, by },,
increasing the value of s by 6, also, this being legitimate since a,—a,+1 is positive ;
thus when we attain the values a, @&,... a, 1, pi,--- pn, the lowest admissible value
r=n
of sis = [a,—p;].
r=2
Next suppose that a,—pnt+1 is positive and p,z—a, positive, but that for some
values of r, p,—a, is negative. This is similar to the sub-case last considered and as
r=n
in it, the lowest admissible value of s is = [a,—p,;], the term [@,—p,] being however
r=2
zero.
Finally suppose that @,—pnr+1 is negative, and p,—a, positive. The theorem
applies for all values of s to the function involving
% +[pr—%—1], dt — bo, Boo Cin ale Pis «++ Pn-
As before, for each value of 7 we may increase a,—b, by b,, increasing s by b,
at the same time, this being legitimate since 4+[p,——1]—4%,+1 is positive, p,
being greater than a, and a fortiori than a,. Finally we reduce the value of
a,+[pnr—a—1] to a and thus when we attain the values a, %,... Qn, 1, py, .-. pn the
r=n
lowest admissible value of s is [p,—%—1]+ = [a,—p,], the term [a,—p,] being zero.
r=2
These several limits are all included by the statement that s is not to be less than the
r=n r=n
greater of the two integers, [p,-—a—1]+ = [a,—p,], [an—u—1)+ = [a,—p,|.
r=2 r=2
11. We may in fact obtain a limit to the error even when the real part of
x is negative. The reasoning of Art. 3 suffices to show that for the function aad (a,
a —pitl; +1/x) if the argument of «2 be +¥, y being <7/2, the modulus of the
remainder is less than that of the next term divided by (sin(@+-y))'*J (cos @)s—ts+.
where 6 and @+¥ are each less than 7/2; and by changing in the integral (37)
the point h to the point at infinity on the production of the line joiming the points 0,
x, we see that the same statement holds for the function of the (n+1)th order if
for all values of r from 2 to n, m—p,+1 and p,—4, are positive. Also a reference
to the method by which these restrictions are removed shows that in the most general
case the index of sin(@+ +) may be replaced by the greater of the integers [a,]+[a,—a—1],
[mJ+[pn—%—1], while that of cos@ is left unaltered in form, affected only by the
increase in s, s being the number of terms taken and subject to the same restrictions
as before. We must bear in mind that every p is greater than 1, pn the greatest p,
r=n
a, the greatest a, and p, the p omitted from the sum > [a,—p,}.
r=2
We may investigate the numerical value in the case of the semiconvergent series
for the Bessel functions. In this case we may write a=}—n, pj,=1—2n. Hence
[mJ is 1, ma—p,+s+1 is }+n+s. The divisor is thus sin(@++y) (cos 6)#"*s; to make
this as large as possible, @ should be nearly zero unless y be very small, and we
deduce that the error is less than the next term divided by siny; if y be very small,
HYPERGEOMETRIC SERIES. 199
the greatest value is greater than when y is zero, in which case it is (4 +n+4s)id+™”.
(}+n+s) t+" > this tends to equality with (f+n+s)“e+, even for moderate values
of s, and the error is thus less than the next term multiplied by a number which is
nearly (4+n”+s)!e. The multiplier thus obtained when y¥ is zero is considerably larger
than that given by Weber (Math. Annal. xxxvut.) for all arguments, which is about
shar cos nm.
12. By reasoning similar to that by which it is shown that Lt. (1—«/a)*=e* we
may show that by writing #=y/h, a=—h and increasing h indefinitely we can diminish
the number of a’s in equation (28) successively by unity; we thus obtain very general
results. From the theorem that as tr increases indefinitely the ratio_of II (7) to
er’ V27r has unity for its limit it follows that if « and @ are positive quantities and
t be increased indefinitely Teas 78 has unity for its limit, and making use of
this result we see that the general theorem may be written in the form that if m}n
VT Cees Og re ee ere
So Rel (cys Cs, <
MEO Mer eran seo
2 Il (a —pi) UW (pi—2) Tl (px —p2.—1) 0c Il (p, = Pn—1)
11 (py — a — 1) il (p;—a,—1)... I (p) —am— 1)
++ Am}; Pi, Pa» +--+ Pn; (-)"—-™ a)
a F(a, —p, +1, a—p,+1, Beni ny — Pict Ls
2— pi, p2— pit, ee Pnu—Pitl A) eat)
+(n—1) terms analogous to the last
BN (Ge 1) 10 (cx) p,) | E = ag = Ny — Nx ==) — a aay SU eeeaacene (9),
and three of the type
2, a? LEG feed ol tC Ih rE
E, pda 7 ta ie da ert dady *~ n, idee are (10)
In the present instance the equations of type (10) are identically satisfied and need
not concern us further.
When the material is symmetrical round the z-axis we find in place of (9)
ES eee d? eas 7) 20+9) .@ -
TCE) een oleae dady
oie ye Spee TE a* > Re 1 @
dz ("> = Be) eC E a (= — nxx — Nv) — = dydz y= Ose qi ).
@ (/z—qw 14-\ , 1 1 @ =e
dz ( (a a 2) + E Be- a) n dxdz”
* See Todhunter and Pearson’s History of Elasticity, Vol. 11. Part i. p. 74; or Love’s Treatise on Elasticity
Vol. 1. p. 122.
Vou. XVII. Parr III. 27
204 Dr CHREE, A SEMLINVERSE METHOD OF SOLUTION OF THE
§ 4 The greater complexity of (9) or (11), as compared to the corresponding
equations for isotropy, does not affect the type of solution; and, as im the papers (A)
and (B), we may assume
we = A, + A.w? + A,y? + A,’2’,
m= B+ Bat + Bly + B’2,
z= C0, + Ca? + Cry? + C.'2,
w =2QLyz, 2 =2Maz, z= 2Nery|
Here A,, A,... NV are constants to be determined from the body-stress equations
(5), the surface equations (6), and the equations of compatibility, the latter of which
alone vary with the type of material.
Fortunately there is an immense economy of labour owing to my having in papers
(A) and (B) expressed all the A, B, C constants in (12) in terms of the three L, M, N.
In effecting this simplification I employed only (5) and (6), equations which, as pointed
out above, apply to all kinds of elastic material. We are thus enabled at once to
replace (12) by the following equations established in the two earlier papers :—
= Se? + Ty + UP 48 [age + 7)(1-2-F—2)
GB UP
a b? a b2 Ce
Ba Se + Ty + UP+e | Rp + U)(1- 2 -F-2) |
By? 2 3a yx 2
+L -3-$- )+ar(-3-¥-2)|
gw =QLyz, m=2Maz, w=2Nay
§ 5. The results (13) apply to all kinds of elastic material, whether possessed of
2 or of 21 independent elastic constants; but the values of LZ, M, N vary of course
with the material. The expressions for the strains corresponding to (13) vary. Thus,
for material symmetrical with respect to the three coordinate planes, we find from (3)
82H, = (1 — 92 = Ms) (Sa? a Ty + Uz) + {a? (4Pp Ep S) — M2)" (4Qp +T)
\
ee nal
|
+(@M —mbl) (1-2-4 —*2) 4 @N — meth) (1-2 -F -2)
2 aye 2
= (nd? N + nyc*M)(1 — ae
Oyz = 2Lyz/n,
EQUATIONS OF ELASTICITY, AND ITS APPLICATION, ere. 205
The expressions for the other four strains may be written down from symmetry.
If the material be symmetrical round the z-axis we have
1 ,
se = ( —t _ j) (Set + Ty + Ue) + |
a (4 Pp +S) — 1/0 (4Qp + 7)) |
m,n i
— Te (dRp + uv) (1-5-4 -2)
ie (“oe (1 es BURN =) a (Gr ney ¢ Be OY s “)
E’ E
Sy = expression obtained from that for s, by interchanging a with b, # with rE
y, L with M, P with Q, and S with 7, tite (15)
Es, = (1 — 2n) (Sa® + Ty? + Uz*) + {c? (4p + U) — na? (4Pp + S) — nb? (4Qp + T)}
|
a@ 8 ¢ |
2 2 2 2 |
—n(@M +¥1)(1- 2-2) 4 (en naeN)(1— sy |
+(eM—qbN) (1-30-42),
we ob
ye = 2Ly2z/n, o72,=2Maz/n, oey=4(1 +7’) Nay/E’
§ 6. Results which depend only on the form of equations (13) are true irrespective
of the nature of the elastic material. For instance* if S, Z and U vanish, or there
be no surface forces, the resultant stresses across parallel tangent planes at their points
of contact with the system of confocals
@/a?+y/b+2/C=r
are all parallel; and their intensity varies as 1—X.
§ 7. We have now to consider how Z, M and N are to be determined. Substi-
tuting from (13) in the equations of compatibility, whether (9) or (11) as the case may
be, we obtain three simple equations of the form
ay + aM +a,N =a,
QoL + de + desN = Ge,}....---- ja pusedsch saat ts dyeorcaaeews (16),
sl + a3M + a,N =a,
where @,,... 3... are known functions of the elastic constants, the bodily and surface
forces, and the semi-axes of the ellipsoid. Representing by IJ,,, II,., &c. the minors of the
determinant
Qi, a, Gs |
=| Cisse Gans! gall ws Wee et co eee ce nou eee (17),
| |
|-Chs, ez, Ass |
* Cf. (A) § 2.
206 Dr CHREE, A SEMLINVERSE METHOD OF SOLUTION OF THE
we have from (16)
L=(oa,Ih, + oI. + asIl,3) = I,
(Gam Pee en ory UP ord MI) lM 550 ec oseeqoce esuseuoee00° (18).
N = (@, 0h; + @olles + ws 115) + I
§ 8. When the material is symmetrical with respect to the three principal planes
of the ellipsoid
otal a ed |e ili (19)
= Fr = atte 2 — abt — ca? a
and the other a’s can be written down from symmetry, the relations (4) being borne
in mind.
For the same kind of material
2 i? E, \
=, =(3Pp +S) E (m2b* + msc?) + EB, [4% (nc? — b?) + re (1 — 9) E, aw v|
= E.
+5 | tRp (nab c+) +0 {nad ge |, |
be
m= | BP (mt — 0°) + 840° — 10) a] +042) 7p Cme+ me) | te
ar Fe [2p (qn@ — &)+ Ula Ce nm) 4p 7Ar ‘ol |, |
“Lie (nb — at) +815 — na) et |
ee |
2 yale B; —b? ee 2 2
+ 5 | 89p (naa? = 0) +7 forme) GU} | + Gp + 0) pe (ret + nab
Under like conditions
8 4p 4¢3 bict
II, =8 i ak: BE. (9 = MN) +o (9 = MMs) +9 E.E, qd — NesNs2) -
ab? /(3 Ame ae (3 4ms Free WEL 2ms\ (1 2me se 2n =
> oR (= 2)+aG rae ae \G— eas me) ee vy
abc? (= = 2ms = ahs 43 a*b*ct B | 1 ae 2m. is “e)
E, \t. E, E, 3) Tt, Fy, DR Ib? (21)
bic! 3 :
Tl. = EB, is US — 532) + Tel 35 ame — Mtn) — 3 aq5 ae (lie Mss)
+8
bic? ames rs pete 2)
+ =
S By EF, % /
a*b>c* (3 4 mse 2 ese, (a 4 LOmsths _ *)
E, E, - ra E, \ Ep E, %
EE (n2b* + nasc*) +
Ey
The other four minors of the determinant (17) may be written down from symmetry.
EQUATIONS OF ELASTICITY, AND ITS APPLICATION, ere. 207
* In obtaining (20) and (21) free use has been made of the relations (2), by means
of which various alternative forms can be obtained.
The full expression for II is too long to write down. In practice, after determining
the minors as above, one would determine I from such an equation as
lier, snp tan Waals yg Wiigecterece sens aseheccsrente at 9 F Re / ie , /
Sa,/a = a | | 2 (4 — 5y — 397) , ( WAL +7) ai
mayne E BE’ E® nb’
a et ee
st BE ( — E lp Meee ee (46).
I have manipulated these expressions so as to facilitate comparison with the corre-
sponding results (30) and (31) for the influence of gravitation.
If in (45) and (46) we suppose #’/E very small, we have
w’pa? (1 — 7’) (2d—") 1 \
ied 0 jane 27 nj’
Sa,/Sa, = expression of order E’/H and so nevliaible!
6a,/a =
The similarity with the corresponding results (32) for gravitation is noteworthy.
If on the other hand E/E’ be very small, we find, remembering that I’ is
approximately equal to — 16y*/H°,
ba,/a = — w*pa’n/(3E), )
Sa,/Sa, = expression of order #/E#’ and so negligible) SaaS G8):
If we suppose both (39) and (40) to hold, or the material to be absolutely incom-
pressible, we find
Leet ds 32,
bala = Sohn (gar upto)
sea pad es Oa allan 1)| cooeeten teeta eee (49),
: SEI \EE’ 4E62' nE ME
where II’ is given by (43).
In the case of rotation, unlike that of gravitation, a slight departure from incom-
pressibility has very little effect; we may thus regard the results (49) as close
approximations when the material is slightly compressible. In particular, if the material,
though absolutely incompressible under uniform pressure, is slightly compressible under
other circumstances, we find under combined gravitation and rotation from (44) and (49)
72%;
which can hardly fail to be universally true, Il is essentially positive.
ROTATION ABOUT SHORT AXIS.
§ 17. When the flat ellipsoid rotates with uniform angular velocity » about its
short axis, P=Q=o*, R=0, and the equations (16) take the form
QyL + doM + a43N = $ wpb? fa? (m./ L,) — b*/ £3},
QoL + dy M + do, N = $ w*pa? |— a?/ EB, + 6? (ny) £,):,
hy L + dM + a3;N ==10' ly a (a? — Hyb®) + ne (b? — na? ah,
where a, &c. are given by (51).
214 Dr CHREE, A SEMI-INVERSE METHOD OF SOLUTION OF THE
Referring to the values found above for the determinant IT and its mimors, we find
on reduction, remembering (2),
Pelion (Zot. 20 ae
L=M==5 Tl’ (ote sans nw) (52);
sea pays Pickcuietiow Sara s> uc eR Re oe ;
Ne-G(RtE- oe) |
where
,_ 8a* Sb ao(2 4Me :
II STi oe & = infalataletelstslaieterelsietstaletaleistcteteleinistersintaiatete (53).
We may reasonably regard Z, M and JN as essentially negative.
In our subsequent work the following result will be found useful,
3(L1+M)+2N=2(3L4 N)=2 (8M 4+ VN) =— ep... eee eee (54).
Putting in (13)
S=T=U=R=0, BQ or,
we have
ve/a? =(40°79 + M+ N) (1 = 22/u? — 77/6’) — (40% + 3M + VN) 2/8 — 2Ny?/b.
Having regard to (54) we see that the coefficient of z*/c? vanishes, and deduce
we = — 20°D (1 — 2/2 — y?/b*) — 2Ny?a?/b?,
where Z and WN are given by (52) and (53),
Similarly we find
= = 200 (1 — 227/a? — 2y/b? — 2/c*),
But we have been treating terms of order c* as negligible and so may regard = as
vanishing.
Again we have
ve =2Lyz, 2 =2Mzr;
or these two shearing stresses are of order c, and so though they are large compared
to = we may neglect them for a first approximation, The complete stress system re-
maining may be written
ae = — 20D (1 — 2/0? — y?/b?) — 2Narb*y?/b4,
wy =— 2670 (1 — a2/a? — 97/6?) — 2 Na2b7a?/a, | ....0.ceeececeececeeees (55),
zy = 2Nay
with Z and N given as above by (52) and (53).
§ 18. If w be the inclination to the a-axis of one of the principal stress axes in
planes parallel to wy, we have
cot 2p = 4 (xe — w)/ x7
= cot 26 — (a? — b*) (1 — a°/a? — y?/b*) + (2N ay) .....cceee eee (56);
where ¢ is the inclination to the a-axis of the normal to the confocal
e/a + y/P =r,
which passes through the point a, y, 2.
EQUATIONS OF ELASTICITY, AND ITS APPLICATION, ete. 215
This gives very readily the angles w and 5t¥ made with the a-axis by the two
principal stress axes which lie in the plane parallel to xy. The third principal stress
axis is always parallel to the z-axis,
Without any reference to the values of Z and N we see from (56) that
v=,
for all values of w and y if B= a3,
and for all values of b/a if e/a? + y?7/b? = 1.
We thus see that in a flat rotating spheroid, whatever be the relative values of
the Young’s moduli or Poisson’s ratios, any perpendicular on the axis of rotation is a
principal axis of stress at every point of its length.
Again for any shape of flat spheroid the principal stress axes at the rim in the
central section z=0 coincide everywhere with the normal and tangent to the bounding
ellipse.
The stress along the rim normal vanishes in accordance with the surface conditions,
while the stress @ along the tangent is given by
T= p* {(y'[b) & + («°/a) w — 2 (yja’b?) 3},
where p is the perpendicular from the centre on the tangent at «, y.
Referring to (55), and remembering that
e/at+ a/b = p>,
we easily find
tet =k ON a2 | ate te eats na sls oeddctae emcee eae anea oe (57);
or, writing in its value for N,
Ey ec at 6! 2a%b%,.\ . (8a* S8b4 a2 (3 _ +m e
@ = 2w°p (a°b?/p) ae E. \+ iF - E, + a’b ae =) senognyo: (58).
The stress along the tangent to the rim in the central section is thus a traction,
which varies inversely as the square of the perpendicular from the centre on the
tangent.
§ 19. The strains which do not vanish are, as a first approximation,
Sz = — 2(L/E,) (a? — nyb*) (1 — 22/0? — y*/b*) — 2 (N/E,) (@y?/2 — fgstlie
Sy = — 2(L/E,) (b? — nna?) (1 — a*/a? — y°/b*) — 2 (N/E,) (b%x*/a* — nna*y*/b*), | (59);
s, = 2(L/E,)(nua? + nub?) (1 — a2/a? — y?/b*) + 2(N/E,)(na?y?/b?+ neb/a’), em ;
Czy = 2Nay/ns
where L and N are given as before by (52) and (53).
216 Dr CHREE, A SEMLINVERSE METHOD OF SOLUTION OF THE
To the present degree of approximation, the strains, like the stresses, do not vary
with z; and at the rim in the central section z=0 they depend on the constant N only.
Along the axis of rotation the strains are constants given by the simple expressions
= — 2(L/E,) (a? — nb’),
ey ny (60) ;
sz = 2 (L/Es) (na? + nssb*)!
where Z, as shown by (52), is a negative quantity.
An 7 in excess of 0° is at least highly exceptional, thus supposing a to be the
longer semi-axis we may regard s, at the axis as essentially positive, or a_ stretch.
On the other hand s, at the axis is positive or negative according as
b/a > or < Vn.
For the changes in the lengths of the semi-axes we find from (59), by integration
and substitution for Z and JN,
2 ees Spel lt 12) A abt
Balam 5 parm) Lg tg + 8 (GB) Oma
/ 2 s 4 2 2) The 2 oy Nat
36/b= 5 a atc - nat) | + pte (-- R)t- (a? — mb?) | Ae 56D)
2 2b*) a3b
de/c rat nt (n@ ar Nsab* ) (Get si ie te = )
where II’ is given by (53).
In passing, the following elegant relation may be noted
3 (a2da/a — b°6b/b) Me 28c/c
as/E, — b4/ EB, (Qa@ + sb? VE,
Regarding II’ as essentially positive, we see that 6c/c is invariably negative; or the
short axis, about which the rotation occurs, necessarily shortens. The two perpendicular
axes if similar in length in general both lengthen. If 6, however, is much smaller
than @ it will usually shorten.
For instance, if 3/3 apy aneisane sess eceeestesievsciscckcinesien sence (63),
we have 8b/b = — 2@°pa? (m2/E£,) (1 — mann) + (8 + 4meta + BI Hi/Ns). 00. eee eee (64).
The relations (60) and (57), it should be noticed, supply simple physical meanings
to the constants Z and WV of the solution.
EQUATIONS OF ELASTICITY, AND ITS APPLICATION, ere. 217
SECTION IV.
THIN ELureric Disk ROTATING ABOUT THE PERPENDICULAR TO ITS
PLANE THROUGH THE CENTRE.
§ 20. In a previous paper* I have shown that in isotropic material the first
approximations to the stresses and strains in a thin elliptic disk may be derived by
applying the constant multiplier
{4a* + (3 + 7) a®b? + 4b} + (Bat + 2a*b? + 3b4)
to the values of the corresponding stresses and strains in a flat ellipsoid of the same
(central) section zab, and equal axial thickness 2c or 2/. A similar result holds when
the material is of the more general type dealt with in the present paper, the constant
multiplier being alone different. To find the suitable constant multiplier we may pro-
ceed as follows:
The mean values of the stresses, as | showed in an earlier papert+, are given by
simple formulae of the type
| | [Fava ydz = If} Xadadydz + If d dior PS epne cee pee acCoO REC CTEL (65):
where X is the a-component of the bodily forces per unit volume, and #' the z-com-
ponent of the surface forces per unit surface. The volume integrals extend throughout
the entire volume, the surface integral over the whole surface of the solid.
In the present case we thus have
i |[Feaeayae =n) |[[ednayae ape NOE roo hee Moa (66).
Supposing C to denote the constant multiplier required for transformation from the
flat ellipsoid to the thin disk, we find for the disk from (55)
we = — 20 La? (1 — 22/0? — 7°/b*) — 20 Narb*y?/b',
where Z and WN are given by (52) and (53).
Substituting for x: in (66) and integrating, we tind
—C.2mrabe.a?(L+3N)=o'p.27abe. a*/4;
whence =—'p/(4L + 2N),
=(83L+ N)/(2L4+N) by (54).
Referring to (52), we have at once
5 (Sat She _ 4m =a . (6a! es Rise 1 2ms\) me.
Cale te tee (oe) ae + 2a%b ‘= E)} hp eee (67)
Thus, writing for shortness
6a4 6b aa fe 2ma\
pt pt 2e8 (| m= eanes musica (68),
* (A), p. 49. + Camb. Phil. Trans. Vol. xv. equation (109), p. 336.
218 Dr CHREE, A SEMIINVERSE METHOD OF SOLUTION OF THE
we have as first approximations to the stresses and strains in the thin rotating elliptic
disk :—
2n92f,974 Dhs 2}2 x 2 4 4 3\ yf?
2a [EFF Deleon]
a 42 1g 1 2 1
—~ «wpb? [/2at 2b' a?b* ( oy te UE ie Gaaee a e ==
oS THE IG i E, 33 =) ; a* =) FN, bee at E,/ a |’ | sisisiess (69) ;
~ 2wip/at , =) =
ame (atz- ET)
=z=r = yz =0
_ op 5 Zar 2b aad ay \
a 7 1G [( = mab ) (Fe ar E, at Ng ) (e a 5)
9 (& ee ope ™2\ (VY? _ bra*\
+2 (p+ 20) (Gemma) | |
arp: a Ay ea 2b! —a*b*) Sieh ) |
= PM’ lo UP ) & a5 E, ats Ns ) ( fie i |
4 4 2 2 2,2 |
2 os 0 2a: ) (OS a)
rie G i ey ae aaa) (70).
|
2
ee ee eee
et BM" | (rs + Mab Nes a E, * Ns (2 a &
22 20g (2
hal Sy Be = E,
Cgz (Cuz 0
arb?) LY,
BE) 4
§ 21. The position of the principal stress axes in the disk is given by the same
equation (56) as applies to the flat ellipsoid. Again, over the perimeter of the disk
the normal stress component vanishes and the tangential component, in the plane of
the cross-section, is given, cf. (58), by
4 4 242. 4 4
F = 2etp(b'|p') (G+ pp) + = + + 20% (7 — 2) ee (71),
1 2 Ns
where p is the perpendicular on the tangent from the centre of the ellipse.
The increments da, 6b and él (J=c) in the semi-axes of the ellipse and the axial
semi-thickness are given by (61) when II’ is replaced by II”. The relation (62)
applies equally to the disk.
Employing 6/ as above, we find from the value of s, that the displacement yv
parallel to the axis of rotation is given by
(OUND (Ul SHANE = IAIN). coonscohoedqoond sce dncscooa sence (72);
EQUATIONS OF ELASTICITY, AND ITS APPLICATION, ere. 219
where
» (20% 2b¢ ab? ( 2a* 2b* arb “a
A? = (N30? + nob?) | i a = ) + ym (= +h +2) +n (; + 2 ) ;
- 1 2 3 3 od |
» (2a* 2b' ah? 1 4ny' 2a‘ 2b* a%b*\)
a 2 5? a res bss! 2) aA nd pate
bP = (naa + mb?) (Fe ay as ) jm (7+ E, ja +1 (Fe Pers y}
2 Ns
Since 6/ is negative, assuming II” positive, (72) shows very clearly how the originally
plane cross-sections parallel to the faces of the disk become paraboloids whose concavities
are directed away from the central section, and whose curvature increases with the
distance z from that section.
The curvature at the centre of an originally plane section is greatest in the zy or
in the za plane according as
a@f> or b, we easily find a2>6,? if either
Ns2/ Qa = 1,
or (HD REST PA G Pate | la ssen6 | pecnenasnacta ane poe nne een re (74).
Thus the curvature is greatest in the plane containing the shorter axis of the ellipse if
Ns: and ns, are equal or if (74) holds.
Whilst the reduction in the thickness of the disk diminishes as we retire from the
axis of rotation it remains a reduction right up to the rim. For it is obvious from
(70) that ys, the value of y over the curved surface, is given by
a Oe pea) { ay? bat
te ee) (mae te =) Pht RAS (75),
2 =— 2 (2e%p/ E11”) (
a bf a*b*n
a 2, E,
It may be worth noticing that the reduction in the rim thickness is greatest at the
ends of the minor axis or at the ends of the major axis according as
can hardly fail to be positive.
any > Or in
the present case are only of the order /* of small quantities, our solution is presumably
an exceptionally favourable specimen of its class. Still it would not be legitimate to
apply it without further investigation to the species of anchor ring which arises when
a—da’ is comparable with /.
At first sight, it might appear better to have omitted the terms in F? and 2
altogether; because in their absence * would vanish exactly over both rims. If, how-
ever, we omitted those terms, we should be unable to satisfy all the internal equations.
Such a failure, in the absence of special knowledge, is much more serious than failure
to satisfy a surface condition. For in dealing with internal equations we get, through
differentiating, contributions of like magnitude from terms that are of widely different
importance in the displacements and stresses. It is thus almost impossible to judge whether
failure to satisfy an internal equation is trivial or absolutely fatal.
In the present case, while the terms in / and 2 serve mainly to save the pro-
prieties and silence criticism, they fulfil a useful purpose in indicating the degree of
approximation reached and the circumstances modifying it. For instance, the solution
29—2
222 Dr CHREE, A SEMIL-INVERSE METHOD OF SOLUTION OF THE
becomes absolutely exact if
7=0, or H/E=0;
and it is the more exact the smaller 7 or E/E is.
On the other hand if £’/Z be large the solution has a very limited application.
§ 25. When E and £’ are of the same order of magnitude we may omit the
terms in /? and 2 in ordinary practical applications. When these terms are omitted I
shall use the notation rr, u, &ec. When the material is isotropic the values of u, w, &c.,
constitute what I have called elsewhere* the ‘Maxwell solution, as being the solution
to which Maxwell’s treatment of the problem would have led him in 1853 but for
some small inaccuracies in his work,
It is noteworthy that rr and ¢f depend on no elastic constant other than 7’,
while w is independent of 7 or #. Thus the stresses and radial displacement are
exactly the same as in an isotropic material whose Young’s modulus in £#’ and Poisson’s
ratio 1’.
The longitudinal displacement # on the other hand depends on 7 and £, but
even in its case the law of variation with the axial distance depends only on 7’.
For the increments in the radii a and a, and in the semi-thickness at the two
rims, we find
(Sa/a)=(w%p/4B)A—a)ae+ Benya) (79);
(a’/a’) = (w*p/4E’) {(3 + 9’) a2 + (1-9) a} J
(81/2) pa = — (w*pn/4B) (1 —9') a + (3471) a4.)
(81/l),-a = — (w*pn/4E) (3 +7'/) a +(1—7'/) a} )
From these we deduce the following elegant relations
(8l/l) rma = — 0 (E/E) (8a/a) =—n" (8a/a),
(81/l),-a = — 9 (E'/E) (8a' Ja’) = — n” (Sa Ja’), beveeeee entree cess (81).
(81/0) rma + (81/Yenu == 0*p (n/E) (a? + 0’)
The arithmetic mean of the reductions in the thickness at the two rims of the
disk is thus independent of 7’ or #’. The reduction in thickness is invariably greatest
at the inner rim.
Originally plane sections parallel to the faces he during rotation on paraboloids of
revolution, the radius of curvature at whose vertices equals H + {w*pn (1+7’) 2}.
The curvature increases as we approach the faces z=+/. The general character of
the phenomena is the same as when the material is isotropic (see Camb. Phil. Soc. Proc.
Vol. vu. pp. 201—215).
* Camb. Phil. Soc. Proc. Vol. vu. p. 209.
ae
EQUATIONS OF ELASTICITY, AND ITS APPLICATION, ere. 223
SECTION VI.
ELONGATED ELLIPSOID, c/a AND c/b VERY LARGE.
§ 26. Retaining only the highest power of ¢ in each case, we have for three-plane
symmetry
dy = 8c4/E3, dy = 3c/E;, dy=3 (a4/E,) + 3 (b4/E,) + a*b? {1/ns — 2m,/ E}},
Qe= C/E, ay=—c (80°95)/B; + b°N5o/E5), des =— Cc? (a*ny + 3b? Ns2)/ Es,
‘L 2me Wms
Tl, = (ct /B)| Oa nuneoe E; “G — Matha) + Ba°b? (—- -F- ee NE
1 1
( 1 2 2nxMs
Ila = (c*/E;) Es (1 = mss.) + z “(9 — NxM) + 3a*b? ( ie a 7 i =|,
|
|
Ts = 8¢°/E,?, Noes (82).
[
3a‘ 3b 10nae 7) on | |
Ths = (ct/E3) |- E (i= MMs) a E, (1 = nex) + rl Somat E. — zx) | ’ |
II,, = 8c°a*n,3/(L,E;), |
IL, = 8¢°D*n.,/(H,E:), |
2m2 — 2m
is 8 (c°/E; i) EB (Ui sade E, 2a = nate) + abe (2 — EE |
ROTATION ABOUT THE LONG AXIS 2c.
§ 27. The values of the a’s in equations (16) are as follows:
CS op= 4o%pc* (an + b'ns2)/ Es,
@;= 40" \e (mb? — a?) + = (qne — v»h
Substituting the above values of the II’s and o's in the equations (18) we have
the values of the constants LZ, M, N of the general solution. Thus for NV we find
4 bs 22h?
2@"p | a d > Ma) = E, d =e Nossa) oF Ee (me s= nm)
= 3a4 hs we [oS
cB (1 = msn) + E, (1 = neq) + 2b = = E =
It is unnecessary to record the values of LZ and M as I have eliminated these
quantities by aid of the following relations, which are not very difficult to verify :—
o (L — M)= (nua? — neob*) N,
2 (L + M)= (nna? + nb?) (N + jw’p))
whence CL =n aN + to*p (naa? + et (86),
OM = 7h N+ top (nt neh) a :
224 Dr CHREE, A SEMI-INVERSE METHOD OF SOLUTION OF THE
Retaining V (given explicitly by (84)) for brevity in the expressions for the stresses
and strains, we have
ie [Ger +N) (1 = = Be a) =(ho*p +30) |
ed
w= »|- (Joi + 3N) 2 + hotp +) | fie ,-5)| é
Qn? Dy? 22 Ba Ae (87);
z= (ne =e Ned?) (4@*p _ N) ( a: be <) se (qa = Nob? \N € == ) > |
xy = 2Nay, |
y% and = of order a/c or b/c, and so negligible J
cea am Daye 27)
54 (0'p/B) (amb) (I= FZ) — dena mal) (1 — fe - Sh |
2 Qa? 2) 3a2 2 2 |
+(N/E,) ie (1 — mss) (1 — =— = > = — b? (m2 + ms92) (2 — we -% = a) , |
Gey Z Z 2a? 2 2 :
ee — mt) (1-5 BS) — dom tnd) (1- eat | ee
; é ae oDee Yee ee de Be
=P (N/E,) {ma — NxNs2) ( = Go : wal “)- a? (2 2 + Nasi) (15 Fags 2 -*)k ’ |
8,=— 4 op (nna? + Nab*) (1 — 2°/¢)/Es, |
Oxy = 2Nay/ns, |
oy: and oz, of order a/e or b/c and so negligible J
§ 28. For the displacement parallel to the long axis we have
1
y= — bo%p (nat?-+nab)2 (1-5 5)/ B, asaisse seesle se sonwe eee (89).
Sections perpendicular to the axis of rotation thus remain plane. The shortening of
the long semi-axis is given by
dc/e=- 4@"p (730° + Tee) Be. “cv asic sdhws'tocemcstnceesesteeeeeeree (90).
Using undashed letters as immediately above for the case of the long ellipsoid,
and dashed letters for the case of the flat ellipsoid of Section III., the velocity of
rotation, the material and the axes 2u, 2b being the same in the two cases, we find
from (90), (61) and (53)
(Befe) = (Be'fe)=1—4 inte ne + ab" (= + +b) (P+ +S) ae (91).
E i, £,
Thus the shortening per unit of length in the axis of rotation is less in the
elongated than in the flat ellipsoid.
2 si Pe OP TT A
2 vt See
EQUATIONS OF ELASTICITY, AND ITS APPLICATION, ete. 225
For the increments in the other principal semi-axes of the elongated ellipsoid we have
lw 2 N
3 > {a — Mb — 4s (Mad + Nab*)} + 3 a (1 — 37x),
; LD eh oe. (92).
1 wo 2 N
db/b =54° {b? = Nyt? — tn (nat? + Nb*)| ap 3 EB. B* (1 = gesno.) |
ba/a=
3 2,
The values obtained for 6$a/a and 6&b/b when its value (84) is substituted for NV
are somewhat lengthy even for a spheroid (b=a).
As, however, the influence of the elastic structure is very clearly exhibited in the
case of a spheroid, I shall record the value obtained for the difference in the expan-
sions of the two semi-axes taken along the directions of the two principal moduli Z,
and £,. It is given by
da — da, {3 3 1 2 ' 2 ae
“wipa z iz (1 = mss.) 1 (1 = aos.) + ata i i 7
PAL 1 1 1 1
- 3 \P, c z) IE (1 — 37x) + E, (1 = mos32) + ad
: 2 ; ; d 1 1 = 6m» 67:2
+ 12° E, (ns? — Nao”) te (1 = msn) + E, (1 — 32) — Pa E. E, Nere tel (93).
By supposing equality first between £, and £,, and secondly between n, and np»,
we readily see how 6a,—6a, depends on the difference between elastic moduli and on
the difference between Poisson’s ratios.
SECTION VII.
Lone ELLiprtic CYLINDER ROTATING ABOUT ITS LonG AXIS.
§ 29. By a long elliptic cylinder is meant one whose length 27 is very large com-
pared with the diameters 2a, 2b of the cross-section. The solution for the elliptic
cylinder—terms of order a/l or b/l being neglected—is obtained from that for the elongated
ellipsoid by simply omitting all the terms in 2. We thus have
= a 2 . j)
as [oa (1-2) - Gop tay) f], |
2 2
ae |- (ftp +3N) = + (Sa% + WV) (1 = | |
2. 22 9. 2 u : ee 2
= = (Nn? + Noob?) F wp +N)( + = p) + (7310? — N32b*) N Vas ) ,
xy = 2Nay,
= =%=0 )
226 Dr CHREE, A SEMI-INVERSE METHOD OF SOLUTION OF THE
1 * 2a? 2 \
sz= se {a- 7120") (1-2. = a = P) - $m (ni + Ns2D*) (1 sale ioe = \ |
2 952
+7 fe (l= MMs) (1 = <- -) —F (m2 + ™3Ns2) a-* ey, eas =—¥) > |
_ lop a ae -*)
a= a — a0) (1 == Bos (i 0? + nab*)( )} | aie cad Ane (95);
+ z {i (1 — ns 732) (1 = = = r) — @ (x +12) (1 2 ant
8, =— to’p (nna? + 2b*)/ Es,
Cxy= 2Nay/nsz,
ne ES ES
xz = Tyz=0
where NV is given by (84).
§ 80. The conclusion that the above solution applies to an elliptic cylinder may be
justified as follows:
The terms containing z in equations (87) contribute nothing to the body-stress
equations because d=/dz, &c., are of the order of small quantities here neglected; thus
the expressions (94) for the stresses will satisfy the body-stress equations. (This is
easily verified of course directly.)
Again over the cylindrical surface
e/a+ p/P =
we have from (94)
az = — 2Na*y?/b, yw =—2ND'a/a?, ty = ANY «2... ereeerneeonee (96);
whence
(w/a?) zz + (y/b*) zy = 0,
(w/a?) ay + (y/b*) w = 0.
The equations over the curved surface are thus completely satisfied.
Over the terminal planes z=+J/ the normal stress # does not vanish everywhere,
as it strictly ought to do, but instead we have
| 2 dady = 0.
Thus, according to the theory of equipollent systems of loading, the solution is satis-
factory, except at points in the immediate vicinity of the terminal sections.
§ 31. The increments in the semi-axes a and b are given by the same formulae,
viz. (92), as apply in the case of the elongated ellipsoid. The reduction in the half
length J of the cylinder is given by
81/1 = — Lwp (0? + Tab?) / Bes ...c cee eec ese ce nec eeceeeeeeeewees (97).
Comparing (97) with (90), supposing 1=c, we see that the shortening in a long
cylinder is greater than the corresponding axial shortening in an elongated ellipsoid
EQUATIONS OF ELASTICITY, AND ITS APPLICATION, ere. 227
in the ratio 3:2. This arises from the reduction in the strain s, near the ends of
the axis of rotation in the case of the long ellipsoid.
If undashed letters refer as above to a long cylinder, and dashed letters refer to
the awial thickness of a thin disk of the same material and elliptic section rotating at
the same speed, we find comparing (97) with (70) of Sect. IV.
+ RDIFIN — L{a* 6 eye 2m2\ _ (2a* , 2b , ar?
(3Y/t)+ (BU /l)=1-5 {+ + eb Ge Te) + (F + mae —) ChCP’: (98).
Thus the reduction per unit of length in the axis of rotation is invariably less in
the long cylinder than in the thin disk.
As in the case of the disk, the tangential stress # in the plane of the cross-section
has a very simple form at the surface. For if p be the perpendicular from the central
axis on the tangent plane at a point w, y on the surface, we easily find from (96)
Bie 50 Nati [pie tes aera eek ee ee (99),
N being given by (84). At least as a rule WV is negative and @ a traction. The formula
(99) differs from the corresponding result for a thin disk only in the value of WN
(cf. (71).
We can easily attach a simple physical significance to N. Thus let % and # represent
the minimum and maximum surface values of @, occurring respectively at the ends of
the major and minor axes of the elliptic section, then
NV = di Ge tte ye (GPA eeaaes aed Sdeeat s andico eee (100).
SECTION VIII.
LONG ROTATING CIRCULAR CYLINDER OF MATERIAL SYMMETRICAL ROUND THE AXIS.
§ 32. When the cylinder is solid, the solution can be obtained by putting
b=a, H,=H,=E£', &,
in the results of last section. When the cylinder is hollow, an independent investigation
is necessary.
In obtaining the following results I made use of the solution* published in 1892
for the case of isotropy, recognising that the type would remain unchanged. As the
method adopted is practically identical with that applied in Section V. to the circular
disk, I pass at once to the results. The origin has been taken at the mid-point of
* Camb. Phil, Soc. Proc. Vol. yt. pp. 283—305.
Vou. XVIL. Parr TIL. 30
228 Dr CHREE, A SEMI-INVERSE METHOD OF SOLUTION OF THE
the cylinder’s axis, and r, ¢, z are ordinary cylindrical coordinates. The expressions for
the stresses are as follows:
7 = wp (3 + 9 — 2n?(E’/E)} (a?- 7°) (1 — a2/r*) + {8 (1 — 7° E/E),
$ = w'p [{3 + 9 — 2m? (E’/E)} (a2 + a? + aa? /r?) — (1 + 39 + 29? E/E) r*] = (81 — E/E),
2 =o'pn(1+7) (a+ a2— Wr) + {4(1 — E/E),
6 =7rze = bz: =0 }
Se Eve decee se (101).
The displacements w parallel to the z-axis, and w along 7, are given by
w=—o'p7n(a?+a")z/(22),
= 41 ep (Can) Sane ee (ye - 2) 3
W577 || Re BiiAe Leap ae tall) Naren as) aoe (102).
+ ast (3 + 1 — 277k’/E) wa" |
Be
An alternative form for uw, worth recording, is
_1 wp
“8 El —rE/E
w ) [1 -—7')(B47')(@4+ a*)r—-(1—9*) 84+ +7)84+7/) @a?/r
— 2? (E’/E) {2 (a? + a%)r—-(+7)r+(14+7'/) @a*/r}]...... (108).
I shall assume 1—72H'/E to be positive; if it could be zero the expressions for
the stresses and displacements could become infinite.
§ 33. The solution, except when »=0, is dependent on the theory of equipollent
systems of loading, in so far as we have to substitute for the exact surface equation
z=O0 over z=+1,
| 2arzdr=0.
a
If we put a’=O0 in (101), (102) and (103) we obtain the correct values of the stresses
and displacements in a solid cylinder of radius a. The stresses and strains, however,
near the inner surface of a nearly solid cylinder are, as in the case of the disk, totally
different from those at the same axial distance in a wholly solid cylinder of the same
external radius.
Comparing (101), (102) and (103) with (77) and (78), we see that when 7 or H’/E
vanishes the formulae for the stresses and displacements in the long cylinder and thin
disk become identical. This is true irrespective of the absolute values of 7 or EZ’.
§ 34. The stress system (101) possesses several features of interest. The radial stress
7 is everywhere positive, or a traction, except at the surfaces, where it vanishes; it
has its maximum value where
r=Vad.
The orthogonal stress $$ is everywhere a traction. Its largest and smallest values occur
EQUATIONS OF ELASTICITY, AND ITS APPLICATION, ere. 229
respectively at the inner and outer surfaces. Distinguishing these values by the suffixes
; and ,, we have
$0; = h@%p (a? + a") + Fw*p (1 +77’) (a? — a)/(1 — ih'/E),\
ay , 4 EA 4, pee ets hdddeh doce (104).
Bo = hep (a? + a) — fo'p (1 +-0') (a? = a'*)/(1 — 9 2'/ EB) |
This shows very clearly how 43; and 6% approach equality as the thickness of the
cylinder wall diminishes.
The third principal stress 2, parallel to the axis of rotation, is a traction inside
a pressure outside the cylindrical surface
r=} (a?+a").
The surface values of 2, using suffixes as above, are given by
4 = — 2% = tw'pn (1 +7) (a? — @)/(1 — 9B /E)...... 2 ec eec eee eeees (105).
The numerical equality of 2; and 2, seems curious.
The following relation is also a neat one
Big — Bag = 7) ($b; — $80) icac soe ceececeesscavdscnesccstesesese (106).
It somewhat reminds one of the results (81) established for the annular disk.
§ 35. Coming to the displacements, we see from (102) that the cross-sections—
unlike those of the disk—remain plane. Further, if 6/ and él’ denote the changes in
the length of a hollow and a solid circular cylinder of equal length, the material, section,
and velocity being the same, we have
OU OU AGS Ga occecacccnass come aemetracasereanee se (107).
The influence of rotation on the length thus increases notably as the wall of the
hollow cylinder becomes thinner. Comparing the first of equations (102) with the last
of equations (81) we see that the change per unit length in the length of a long hollow
cylinder is the exact arithmetic mean of the changes per unit thickness in the rim
thicknesses of a thin disk of the same section and material rotating with equal velocity.
For the increments in the radii of the two surfaces of the long cylinder we find
from (102)
da/a = (w*p/4B’) {1 — 7') a? + (8 +7’) cal
8a'/a’ = (@'p/44’) {(3 + 9’) a + (1 — 19’) a}
formulae in exact agreement with the corresponding results (79) for the annular disk.
A variety of interesting relationships exist amongst the different displacements. Thus
if A represent the cross-section 7 (a? — a”), and ¢ the wall thickness a — a’, we have from (102)
(da/a) + (8a’/a’) = wp (a? + a")/ EB’,
— (dl/l) = {(8a/a) + (8a'/a’)} =— nH" /(2E) =— 319", bee eceeeceeceenececeees (109),
(6a’/a’) — (6a/a) = w’p (a? — a”) /(4ns) |
(Gal Een CHES UIE) (CVI) Sesaceeceeceer ebcenccce (110),
(8t/t) = wp {(a — a’ P— 9 (a +a’) (AB) oe cece see eeeee (LED):
230 Dr CHREE, A SEMI-INVERSE METHOD OF SOLUTION, etc.
If the increments in a, a’ and J could be measured, the relations (109) would give EH’,
7!
7
and n; immediately.
From (110) we see that the area of the cross-section of the material is always
increased by rotation; while (111) shows us that the cylinder wall becomes thicker or
thinner according as
7) 2f+29+ 2h,
Dz? Dy? Dz (tufvow") 1
ata anes S 2 daw FET (Qu (ju, be
1
Gi! gat... rainy) (Dayasivst) eee
1
FTI Omtestnt™ Ont) con onennescesseeeceee (3),
234 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
where summation extends to all zero and positive integral values of fi, fo, --. 91, ge,-+
hy, hy... 2 My VY, i, fa» M1, --- Which satisfy the conditions
Aitht--=ff Atget.-=9 ntht...=h
A+ DAA + Tri’ + ThA” = p, Atptv [py Goats feGnales Aske Bal Any apa base WoC le eee (5).
Again, on differentiating the equation
Uf 1
DHE PT amn (tap. BE 9B
with respect to & we find
UU: sy fi. EBA Bun Ein
Csi fide (Gaye) «-- FA girs C2,
But
TU! VOWS, = SD_? DA Dz (toute) g saa
therefore, by comparison of coefficients of powers of &, 9,
D2POD,ADZ (tud v9 wuz
@=ny “a a D! a) pin = hm Wy CAre TE CA LOS WS [in 8 Wg fg VAR Cred bad(B))
>
And similarly
DPD, D2 (tuf v9 wuz ;
@=D! Gem! oa as pia Sgn [p.9.73 ig h3 Ar oY; A, Ma, 1; &e.]
DPD, AA Dz (tut 094" yWz)
C=) G=—DuiGS hig iG i
ARCH S 1A GEOR Ws [PA Do: Wn [Mg VAS S20] bococossanosee (8),
= 5. Sf. Sgyn! . Shy”
where the limits of summation are the same as in (4); and similar expressions can
be written down with 2, y, z interchanged.
It is obvious that, if p+q+r<2(f+g+h),
D,?D,2Dz7 (tufv%w") = 0,
since the coefficient of &?n2¢" in TUS V2W* will be zero. For similar reasons
D?D,2D7 (tuf wuz) =0, if p+q+tr s 2f4+29 + 2h+1,
D,?D,2Dz (tufvIwuv,) =0, if p+ qtr = 2f+ 294 2h4+ 2, &e.
INDEPENDENT VARIABLES. 235
§ 2. We now proceed to establish the formula for the change of the independent
variables in a partial differential coefficient. It is first necessary to state the theorem
of Jacobi on which the method is based. Let v, v, w be three quantities given in
terms of three independent variables & », € by equations of the form
UV = E+ AaoE? + Gono? + Coons? + duofn +... + dank? +...
=&+X, say;
V = 1 + Dao& + «.. + Dyn +... = 9+ Y,
© = C+ Cok? + ... + CmeE+... =C6+Z,
where the as, b’s and c’s are any quantities independent of &, », € It is important
to remark that the linear part of each of the expressions must consist of a single
term. If these equations are solved for & », € in terms of v, v, w one set of values
will vanish when v, v, » vanish, and can be expanded in series proceeding by integral
powers of v, v, w. Supposing that & 7, € have these values, we can then expand
a general function /(&, 7, €) in powers of uv, v, @, and Jacobi’s theorem states that the
coefficient of v'y"w" in the result is equal to the coefficient of &~¢> in the
expansion of
0 (vu, v, w) 1
OCR (ee OAC Oak (Sas
where the expansion is effected by first arranging (£+X)-'*», &c. in powers of X/E,
Y/n, Z/€ and then substituting for X, Y, Z and multiplying together the various terms.
F (Em, ©)
Now let wu, v, w, ¢ be given functions of three independent variables w, y, z; and
let it be required to change the variables from a, y, z to uw, v, w and express
aitmtnt/owdv™dw" in terms of differential coefficients of ¢, u, v, w with respect to a, y, 2.
To this end let x, y, z receive increments & 7, €, and let the consequent increments
in u, v, w, t be v, v, w, t. The first differential coefficients of u, v, w, t will be
denoted by special letters according to the following scheme
, ”
U0 = 4, Un = 4, Un = ,
, uu
Vio = b, Uno = 0, Voor = 5”,
ae = , = ”
Wi = C; Wu=C; Wm =C,
ho = d, too = d’, boo = a’.
Then
v=aéE+an+a’E4+ U,
v=bE+ b+ bE + V,
o=cE+cen+c'54+ W,
ae
where U, V, W, TZ have the same values as in the previous section.
Vou. XVII. Part III. 3]
236 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
But if t be regarded as a function of wu, v, w, we have also
gitment vip™e”
ep se Ua Oi
Ouldv™w" lL! m! n!’
T=
and therefore d'+™+¢/du'dv™dw" may be determined as the coefficient of v'v™w/l! m! n!
in the expansion of 7 in powers of v, v, @.
To apply Jacobi’s theorem we make the transformation
EF =aE+ant+a's,
‘= bE+b'n +b'6,
=cE+ cm +0°S;
so that JE= AE + Br’ + CL,
Jn = A’E’ + By + C8,
Je= AE + Bly +08,
where
and A, B, C,... are the first minors of J. We now have
v=£40U', v= 41- Vi, o=F+W’, gel,
where U’, V’, W', T’ are the values of U, V, W, T in terms of &, 7’, &.
To express these values take D;, D,, Dg as in § 1, and write
Des (ADs + A'D, + A" Dp),
De > (BD; + B'D, + B’De),
Dee : (CD; + C'D, + C’Dp,
and denote D:!D,”"D,2U’, &e., by w'imn, &e.; and wWimn/l! min! by @’ima, &e.
Then Vii aioe
Vi= 2D ' por EP 090",
Wi eC eo meer
TL" = Xd por EP 20",
where in U’, V’, W’, p+q+r¢ 2, and in 7’, pt+qt+r<¢l.
INDEPENDENT VARIABLES. 237
If also we take D,, D,, D, as in § 1 and write
= 5 (AD, + A'D, + A”D,),
ine + (BDz + BD, + B’D,),
D, = 5 (OD, + OD, + O”D,),
then
De? Dy Det (T'U'1V'OW") = DPD AD, (tulv7w"),
provided that products of the operators D,, D,, D, are formed by mere algebraical
multiplication; that is to say, in the product D,D, it is supposed that D, does not
operate on the coefficients of D,, D,, D, which occur in D,.
Therefore by Jacobi’s formula
1 gltm+n ¢
I! m!n! dulddv™dw™
is equal to the coefticient of & ny’ in
y 0 (y, Vv, @) a5 1
ia} (€, n, &) (E+ (Oar (7 a Vij G Je Wir d
or
ee ee pan (Eth) (m+g)i(n+h)!
Seer et UW flmiginth!
0U’ ov’ ow’ PAO VEE
| 1+ OF , oF’? of | ern nmtoH c/nthti?
Ua eave ah RO
| On’ » ) On’ > dn’
i, neue ov’ ow’ |
GE AN BR? Nik ae
that is, in
et pees egen L+F)i (mm +g)! (n + h)!
Sap nadie Ufimtgtnth!
Spenane | Leer al OVO oye ee eee
; | Us, 1+, We piqint
Us, VW, 1+ws
where w%, %, W:, We. stand for Du, D,», Dyw, &e.
To obtain the term containing £7’ ¢’ we take p=/+f, q=m+g,r=n +h.
31—2
238 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
Hence
+-m+
OMIM Soe, 9-0, h=2 (—1)/t9th DAS Dnt Deth
= 27=0, 9=0, h=0 1 2 A
ouldu™ aw"
1+%, 0, w,| whyIw
—— (10).
‘ Sots A Tuatha ee
ins V3, 1+ ws;
Although in this expression summation extends to all values of f, g, h, it is obvious
from the remark at the end of § 1 that terms for which f+g+h>l+m+n will
be zero. In particular, if J/=1, m=0, n=0, the only term not zero is that for which
FSU, G=Q, CSO.
It is easily verified that
| 1l+u, , W, A/J, B/J, C/J ’ G+tuz, b+%z, CHUWz
Us, 1+; W> Al Is (Bide (CHS) a +Uy, B+, ¢o +,
ls, vw, L+w,; IPL fy al 2x Fuel OZ Bf a’ +uz, b+, e+ wu, !
_1| atu, b+, C+Wz
J Gi tetas (Deal Uy se UCiact Wall sencetene sasncnemeec erect es (11),
a’ +uz, b’ +, c+ w,
Ug, Uy, Uz, — I 0,
| Uz; Vy, Uz, 0, a ts 0
Wz, Wy, We, 0, 0, -1
| A B C
=| Ul, 0, 0, af ? iz i DOCOOCOCDNODDUCOUOOOUOOOGOG (12)
A Dad
0, ul, 0, ap > IT ’ It
A” B" Q”
0, 0, I, ai ? ei ? i
It must be understood that when the determinant (11) or (12) is substituted in
the expression (10) the operators D,, D., D, affect uz, Vz, Wz, &e. but not a, b, c, &e.,,
or A, B; C, &e.
§ 3. To obtain Sylvester’s expanded form of the result we use the form (12) for
the determinant and expand D,'+/D,”"*9D,"** in powers of D,, Dy, Dz. This operator is
equal to
s +f)! (mtg)! @+h)!
~ py! po! ps! qi! qo! gs! 71! 72! Ts!
(AD ay (AD Ne (ACD Del aan ie = (ena (ee
Eareace: (5 Ti Tj a fh Tita
INDEPENDENT VARIABLES. 239
where summation extends to all zero or positive integral values of p,, p., ps, &e. for
which
Ptpotp=l+f, At+t@m+tg=mt+y, r++ =n+h.
We now re-arrange the grouping of the terms and transform (10) into
ol+m+n t
Sac. 082, haw (= 1)/+9+h-3 DI Dmg Dpto
oulau™ Own = S=0, g=0, h=0
(uD;), (uD,), «DD, D, 0, O ut vr
(vDz), (vDy), (wD), 0, Dy, 0} Sigh?
(wD,), (wDy), (wD, 0, 0, D,
ee. ben woe
| Te eu ectie
Rema a ies Dt
where (wDz), (uDy), (uD) are the same in effect as D,, D,, D, but operate on wu only,
whilst (vD,),... operate on v only, and (wD,),... on w only.
We can now make use of the results of § 1 and obtain finally
(L+f—1)! (m+g—1)! (n+h-—1):!
Glimtn ¢ si/=2, 9=2, haw ~
ee (= yes
Butoum™oum — —Bemoa=o (— 1Y Pi! po! ps! = ul ga!qs! = ry! Ta! 15!
x =(p: +@a +7 -—1)!(pt+q@t+rn—- 1)! (ps + Gs +73 —- 1)!
x J—emintf+gth) AP: A’P2 APs BX B'S BY’ ON C2 O"s
X[(P+G+N), (Pot G+), (Pst stra); fg hs As wv; My By 1; &e.]
>A, Sita, tAn, l+f, 0, 0
GTA pli Di 0, m+4q, 0.
x Shim", Shyu", Shan", ° We AIT wie (13),
Ba+ratn, 0, 0 pP, Q> ry, |
0, po+Qa+7e, 0, Pay qa Ts |
0, 0, Pst qs+7s, Psy Is i
where the limits of summation are given by (4).
A little consideration is needed to see the truth of the last result. It is obtained
by regarding the determinant as expanded, then expanding the various terms by § 1
and grouping together all the terms which give rise to the particular term denoted by
[i+ Qtr) (Pat Gtr), (Prt Gtr); FAI ry wy Ys My My 13 &e]
The last result agrees with that given by Sylvester but differs in sign from that
-obtained by Cayley when the number of independent variables is odd.
240 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
In the case of certain terms, such as those which occur in the evaluation of
ot
ou?’
explains, there is no difficulty in deducing the proper value.
the coefficients as given by (13) take an indeterminate form, but as Sylvester
§ 4. In order to obtain a symbolical expression for the result of the change of
variables we return to equation (10), and by a re-arrangement of terms write it in
the form
= — >a fy ae (= 1y+ gth Di Dry Dpto
D,—(uD,), —(vD,), —(wD,)
Syd
—(uD.), D,—(wD.), —(wD,) | t pas “ Ae, ee ee (14),
—(uD;), —(vD;), D,;—(wD;) |
where, as in the previous section, (wD,), (wD.), (wD;) are equivalent to D,, D,, D; but
operate on w only. This re-arrangement is effected by grouping together terms homo-
geneous in D,, D,, D;, (wD,), (uD,), ...-.--
The operators required for the purpose of expressing the result symbolically will
be considered in the following sections.
§5. Taking U, V, W, T as in § 1, let suffixes 1, 2, 3, 4 indicate that (& m, &),
(&, m2, &),-.. are substituted in them for &, 7, €; so that
Ty = Gogg E,? + Aono? + ---
Let also U;, U;,,... denote & Ue “ U,,.... Let {U,U;} denote an operator formed
by replacing terms such as &?,7¢" in the product U,U;, by = . The particular
brackets { } will be used to indicate this operator and to distinguish it from a mere
algebraical product. Similarly let {U,V:,) be an operator formed from U,V;, by replacing
EP .267 by eer | Let also {U,W;,} and {U,7;,} be formed by replacing &?,%&" by 203
Oper : ; 0pm
See
and &?n, 26" by rele
We shall also suppose eight similar operators {V,U;}, {V.V¢},... to be formed in
like manner, and twenty-four others by writing 7 and € for &.
Written at full length for a few terms the first operator is
fa) () é 0 fe)
{ U, Us} = 2A» Oalann mae Aq20 A190 Daless ae Aono 5 Adlon a BA Ahr Oden = (2@a00@oa0 a5 Go) Bam
Baers a
+ Bo Cin Oden + (2dh099 Goo2 + Bin) 5— a ae (Apo9 G01 + Gon G0) 5— Oden
+ (Ahi Move + Godin) ait (2299 Mon + 220i) 5—
‘i
AF oss
INDEPENDENT VARIABLES. 241
Similarly
a a a : a
{U, V;,) = 2a. boo Osos + Gyo Dr9 Doan + dobro ODoos + (doo D130 + 2419 b aon) ODav Fieve
If it is desired to work with wpor, Upgr, &e. instead of apr, Dpgr, &c., the operators may
be formed in similar fashion. Thus {U,U;,} is formed from
a 2 ft 3 2
(1 e “ts ee 57 + Udon 51 + th bm + «+. + Us00 2 a S) (uaE: + thio + tan G1 + Uso = are. =}
by replacing £7,276" by p!q!r! ze :
OUpgr
And therefore the operators may be also expressed in the following manner
{U,U;,} => D,?D,2Dz (uuz) = 7
Upgr
{U,V;,} == D,?D,'Dz (urs) — é
par
{U,W,,} = % D,?D,2Dz (uw;) ce ,
OWpar
(U0, =ED.”D,2Dz (ut) =,
‘par
&e. ; &c.,
where summation may be supposed to extend to all positive integral values of p, q, 7,
though in the first three operators the coefficients of g ; g : a will be zero if
OUpgr’ Opgr OWpgr
p+q+r<3, and in the fourth the coefficient of a will be zero if p+q+7r< 2.
The operators actually required will be nine formed by combinations of the above, viz.,
1a; x} = {U,U;} a5 {U,V;,} ar {U,W:,} + {U,7;},
(U, y} = {UU} + [U2V2, + {Us W,,} + (U.T,,},
&e., &e.
§ 6. The first theorem to be established with regard to these operators {U, 2},
{U, y},... is that they are all commutative with one another. But before proceeding
to the proof of this theorem it is necessary to make a few preliminary remarks.
Let F(&, 7, €) be any integral function consisting of terms £?y%f" such that p+q+r<¢2
and let {F(&,m, &)}, {F(&:, m, &)},... be operators formed as in the preceding section.
242 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
It is then obvious that
{F(é, N> &)} U;, = F(&, > &),
{F(é, ™m> &)} UZ SIF (E- Ne» &),
a
ag, 22
é
= OF, {F(&, hy &)} U;
{F(&,, ™m> &)} Uz, = {F(&, ™m> &)}
é
= 9g PE. ™)> o),
{F(&, > &)} U:,= apf (Eo 12, &2);
{F'(&,, ™m> &)} V,= 0.
A number of other similar results might be written down by interchanging suffixes
and the quantities U, V, W, 7, & », €& Again
{F(é, My» &)} U,U:, a Us, (F(&, m, &)} U,+ U, {F(é, ™) f)} U:,
Ee Gee ee U, se Fb ea
{F (&, No» &)} U,V«, =U, {F (2, m2, §2)} ve
rs)
— ise P(E m> &).
Many similar results could be obtained, but these will be sufficient to indicate the
mode of procedure about to be adopted for forming the products of the operators
Ee ee 10 OR eee
§ 7. The essentially distinct cases to be considered are the products (U, 2} {U, y},
{U, x}{V, x}, {U, x} {V, y}. We will take these cases in order.
Let {U, a} {U, ys={U, a} -{U, yi +{U a *{U, y},
where the first term on the right is the result of algebraical multiplication, and the
second is the result of operating with {U, a} on the coefficients of {U, y}. It is only
the second term that can possibly be unsymmetrical. We have
AUR IUUATSUAALSUALALSUA IE IUAME SUA AES LAAT SLA a
={U,U,} * (U,0,} + (0,0) *{U.V_.} + (UUs) * (UaW,} + (Wie) * (0.2,}
+ {U,V_} #{U.V_} + {T,W} * (UxW) + (UL) # (UL)
= (Uy, Tse} + {Us 5d a dh + WryVeVe} + (Wr a0) + (LUT
3 pirditetenss Q
+02 Wah + {Ue UW adh + age leh
INDEPENDENT VARIABLES. 243
= (U, (O10 gy, + 2U¢,U,,)} + (U2 (UsVen, + Ue,Vn, + Uy Ve,)}
+ {Us(Us We», + U¢,We, + Uo, We} + {Us (UL in, + Us,T, + Uy,T%)}
={U, y} *{U, a},
from the symmetry of the expression with regard to — and 7.
Again we have
{U, a}*{V, a} =[({U,U2} + (ULV ¢,} + (Us We} + {OL e}] * Vi Ue} + {VV} + {VW} + {V2}
= {U, Uz} * {ViU,} + {U.Ve,} * {Vi0g} + {UV e,} * {Vee} + {U2 Ve,} * {VW}
+ {U,Ve,} PRA ie oe oe
=| ize (UU e)} + (Ty UiVe) + (VeUeVeh + + |Vase (UV e)}
E,
+ (Ue) + (MUM) + Voge (OW adh + {Vase (Tled}
= {V,U, U2, + ViU%, + UU, Ve} + {V0 Veg, + VU, Ve, + U.V%,}
+ {U;VsWee, + We, (VsU¢, + UsVi,)} + {UV Tee, + Te, (UVe, + Vie}
={V, «}*{U, a},
from the symmetry of the result with respect to U and V.
Finally
{U, a} *{V, yJ=[(CUe} + (UV) + {Us We,} + (U2e3] * [Vi Ui} + (VeVn} + {Vs Wa} + {VL 3]
U, Uz} * {VU,,} + {UV} * {ViU,,,} + {UaVe,} * {VoV,,} + {O2Ve,} * {V;W,,,}
+ {U,V ;,} * {V.T,,} + {Us We} * {V5 W,,} + {U7} * {V.T,,}
(
= {Vi (WUs)} + (0.0.2) + (Vo.aVe) + {Voge (Ve) + (Wy UV
0 a )
+ (DUN) + | Page OWa)t + {Ve Wate)
= {OiV, Us, ar Vy U;,U,, 15 U, O;, V, J st {U2 V2V en + V.U,,Ve, at U, ne WV, ‘a
ar {U, V; Wenn, ar We, V; Te + We U; Vz, 1 {UV Tey, a5 Ts, V, U,, a se U, Vi}.
This expression is unchanged when U, V and &, » are interchanged and likewise
the suffixes 1, 2. Therefore
{U, x} *{V, y}={V, y} * {U, a}.
By interchanges of U, V, W and a, y, z, the products of all pairs of the nine
operators can be reduced to one or other of the three preceding. It has therefore been
fully established that all the nine operators {U, z}, {U, y}, {U, 2}, {V, 2}, {V, y}, {V, 2,
{W, wt, {W, y}, {W, z} are commutative with one another.
Vou. XVII. Parr III. 32
244 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
§ 8. Now, taking U, V, W, T as in § 1 we have
(U,U,} Ul = fU! (U,U,} U =U. 00; = (0) AU eee, (15a),
Se 5g Us U;,} US -5 (Ux 1 A eae (15 b),
(U0) = Ul =e (UUs) Us =7 (U2) Re ee ae (150),
(U,V;,} Vo =gV0 (UV;,} V=gV9 UV; = Us ott ee eke (15 d),
LAA i Vo= “e (U.Vs} Vo = we (Ug Pale fhacertne eee (sie)
{U,V} x vo=s o Ueh Ve 2 (ux) Mbox Ate e (15 f),
(U.0s} T= ue pth Doerr atl ceed st af Semin (159)
By comparison of the coefficients of &?7%" on the two sides of these equations the
following formulae are deduced:
(U,U_} DePDyeDzul = DePDytDz (wo wl) ceccceeesneesserrenres (16 a),
3 a ne
{U,U;,) DPDyDi = wl = D,?D,2Dz = (u = w) Ls te SOR (16 b),
[U.U,} DerDeDs 2 w= Dedede 2 (uZ w’) sddnceeecnegagntc (16 c)
1 é, x yu Zz oy x y Zz oy On ee ae A
(UV. DED EDs = DP DaDen” po eae ee (16 d),
( &, y 7] On
AWeet
(U,Vz} Dz?DyDe “ w= DzPDyiDe 5 (uz. v’) RR te (16 6),
U.V;,} DDD © v9 = DD, eds 2 (u = i) ce (16 f),
= ay oy
at
[UsLe,| De?D,Dz t= DzrDeDz (we) Mere cr any (16 g).
The form of the above results shows that they may be generalized by replacing
D,”D,!D7 by any function consisting of integral powers of D,, D,, D:. These examples
seem sufficient to show the effect of the operators. It will be noticed that the effect
is to introduce a solitary w and to make certain alterations in the symbols of differen-
tiation. The effects are perhaps best seen by examination of (16 e) and (16/).
With a view to the application of these formulae to the result in § 2 it is con-
venient to re-write them in another form. Let (wD,), as on previous occasions, represent
D, when operating on w only, and let [wD,] act only on the solitary w which is
introduced into the last set of formulae.
INDEPENDENT VARIABLES. 245
With this notation (16 a) becomes, if F(D) represents any function of D,, Dy, D,,
{U, Uz, F(D)w = F(D) {(uD,) — [wDz)} wu. wv.
Moreover, since {U,U;} does not operate on V, W or 7, the equations (15 a, b, c)
0
still hold if the functions operated on, viz. UY, ag U ..., are multiplied by powers of
V, W, T and their differential coefficients. Thus from (15 b) we have, for example,
= a a a
(Tis) . 5p. VOVWW,T = = (Ux U) VoV.WW, 7;
and corresponding to (16 a)
{U,U;} F(D) $ (wD, wD, tD) ufviw't = F (D) 6 (wD, wD, tD) \(uD,) — [uD,z)} uw. ul w"t,
where @(vD, wD, tD) represents a function of (vD,), (vDy), (vDz), (wDz), ....
Similarly
{U,U,,} F(D) $ (wD, wD, tD) wi w't = F(D) ¢ (vD, wD, tD) {(uD,) — [wD,)} wv. ut vw" t,
{U,U;} F(D) $ (oD, wD, tD) uv wt = F (D) $ (wD, wD, tD) {(uD,) — [uD,)} u. uf wt,
and therefore
Ei (TU) +4 (00, } +4 (0, Us} F(D) (wD, wD, tD) wrw't
= F(D) b(vD, wD, tD) {(uD,) — [wD,]} wu. wi w"t........ (17 a),
whilst similar results hold for D, and D,.
Again, from formulae of which (16d) is a type are deduced formulae exemplified
by the following
A”
A cAg
[Fara + 7 (OMG
tan) F(D) ¢ (uD, wD, tD) ufv9w"t
= F(D) 6 (uD, wD, tD)(vD,) vu. wiv9wt....... (17 5).
And, from formulae of which (16 e) and (16 f) are types are deduced others which
are exemplified by
A”
J
= F(D) $(wD, tD) {(vD,) + [uD,)}} (vD,) uw. wet wt te... (17 c).
5 {U.Va} + (U,V) + (U.Ve)| F(D) $ (wD, tD) (vD,) uSoowt
If in this last formula (vD,) is replaced on the left by (vD,), then on the right
{(vD,)+[uD,]} must be replaced by {(vD,)+[uD,]}; and if on the left A, A’, A” are
replaced by B, B’, B’, then on the right the second (vD,) must be replaced by (vD,).
§ 9. Return now to the expression (14) in § 4 and write, for brevity,
D,— (uD,), — (wD,), — (wD,)
—(uD,), D.—(vD.), —(wD,) |= A.
—(uD,), —(vD;), D;—(wDs)
246 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
By adding the second and third columns to the first it is evident that
(tD,), —(vD,), — (wD,)
|A|=] (éD.), D.—(wDs), —(wD.) |.
(tD;), —(vD;), Ds— (wD)
By use of this form and consideration of the rules exemplified in the preceding
formulae it then becomes evident from (17 a) that
[5 (Ut) + 1G U,} + Ay (UiUc} | DiDDar| A\ wher
(tD,), —(vD,), = (wD,)
= D,'D,"D;"\ (tD.), D.— (wD), —(wD,) | {(wD,) — [uD,}} w. whorw*t.
(tD;), —(vD;), D;—(wDs)
Also from (17 }) and (17 c)
E (U.Ve} +5 (UV) +5 (0. v3} DEDmDy| A| uleurt
| (¢D,), — (wD,) — [wD,], — (wD,)
= D!D "D;"| (tD.), D.—(vD2) — [uDz), —(wD,) |(vD,) u. wh vrwtt.
(tDs), — (vD;) — [wD,], D; — (wD)
Again by interchange of v and w in (17 6) and (17 ¢) it is obvious that
E [U.W.} +5 (UW) — (Us) DADIMDj'| A | wf owt
(tD,), —(vD,), —(wD,) — [uD,]
= D|'Dy"D;"| (tD.), D,—(vDz), — (wD,) —[uD,] | (wD,)u. ufvrwtt.
| (tD;), —(vD,), D;—(wD;) —[uDs]
Similarly by interchange of v and ¢ in (17 6) and (17 c) it follows that
[5 (eee ae 4 (00s) DEDmD, | A\ uhrowrt
(tD,) + [uD], — (vD,), —(wD,)
= D,'D."D;"| (tD,) + [uD], D.—(vDs), —(wD,) | (tD,) wu. ufoow*t.
(tD;) + [wD,], —(vD,), D;—(wD;)
Now add these four equations together. The operator on the left will become
FU, a} +5 {Uy} +5 (0, 4
which it will be convenient to denote by {U, 1}.
INDEPENDENT VARIABLES. 247
On the right-hand side all the terms containing [wD,], [wD,] and [wD,] disappear.
For the coefficient of [wD,] in the operator is easily seen to be
(tD,), — (wD,), —(wD,)
=y DD,"D," (tD,), D, = (vD,), og (wD.)
(tD,), —(vD;), Ds- (wD)
(tD,), —(vD,), —(wD,)
+ D/D"D;"| (tD,),. D,—(vD,), —(wD,)
| (tDs), —(vD,), D,—(wDs)
=0,
The coefficient of [wD] is
(tD,), = (vD,), =a (wD)
D'D,"D;"| (tD,), — (wD,), —(wD,) |= 0.
(tD;), —(vD,), D;—(wDs)
The coefficient of [wD,] is
(tD,), —(vD,), —(wD,)
DD,"D;"\ (tD,), D,—(vD.), —(wD.) |=9.
(tD,), —(vD,), —(wD,)
Hence
{U, 1} D! DD," | A | ufo9wtt = DDD," | A | {(wD,) + (vD,) + (wD,) + (ED,)} wor w"t
= DUD EDA, ait eau re seatccs/snsesntceseueeltes fetta sacar os (18 a).
We note that {U, 1} may be written
{U, e}, {U, y}, (UY, 4
1 a”
== b, b’, b
ic y J / ”
6 e, c
If we take similarly
a, a, an ;
(V, =5 | (Va, yh (V4 |=FIBIV, 4B IV, y+ B(V, ah
C, Cc, c”
and
at, a, a” .
(W3=5] % 8, oY | FLOW, a} +0°(W, 9} +0" CM, af}
{W, a}, {Wey}, {(W, 2}
we shall find
{V, 2} DED ID," | A | wlotwt = D!D"D," | A| uloI wht .....eeeeeee es (18 b),
{W, 3} D2D.™D. | A| wort = DEDMD PY | A | ufo «2.0... ree eeeee (18 c).
248 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
We have therefore
DAA Det pe |A} ut vIw"t
{ U, 1} Di Dito) thi | A| ut—yow"t
{U, 1}/ DADmraApmtia | Al ywnt
={U, 1 {V, 2)9 (W, 3}* DAD Ds | A It.
But |A|t= D,D,D,t, so that the expression becomes
1G; 17 {V, .2)2{W, 3)". DED. Devs.
Therefore
sates 2 = as (— 174 DaADmt9-D nth | A ee
=S(-1)ftot ee BE ee at ie a DED D,rt
ap Maat: 00) ea Ane AOR rrp es (19 a),
since the operators are commutative. The fact that the operators are commutative has
been proved independently, but it is pretty obvious from the circumstance that the
D,, D., D; are commutative, so that the above reduction might have been effected in
different orders.
If it is desired to bring the independent operators {U, a}, {U, y}, &c. into prominence
the result may be written
A A’ A” B B BY
oltmt ng Si ap = = Soe of = = fen 2 SS, & = —{V, yr = —4{V,
Sarai 7x4 ten zi Ae Aires Ne ral aie ral hy 7 2}
C Cc’ (ola
-=—{W, x} -—{W, y} - —{W,z
eo TU EE og gh gg Pt oD are haael oe Ah tio teen (19 b).
: 1 gitmt nt
If we write Dinn = It m! n! duldu™ouw™?
ay ane a DED. mJ)”
we have Dis Se ON BSS ge ee eee (19 c)
lim! n!
LDN ED a De
Ul! m!n!
Since DD."D." is merely a linear transformation of D,'D,"D7, the operators required
for the purpose will be simply the ordinary operators of the theory of invariants.
These operators we shall define as follows :—
§ 10. It remains to express by means of operators acting upon djnn-
0
Dzy as {ET} — S (q + 11) Che. q+, ies ;
p=1, q=0, 7=0
0
Oye = {T's} a pS » P+ den, es. rag >
2
p=0, q=1, r=
Wg2 = [ET 3} = > tba,
Pp =
7)
rt1 a7 >
=1, q=0, r Od par
INDEPENDENT VARIABLES. 249
- 0
cde at ea aiid Pt don, arrag ’
=0,qg=0,rTr= por
“
o,={(nTji= REN ae he —
7 eae | ¢} oi ie di ) dp, q~1, ie a
a
a oe ee Ag¥t) d; q+ Od’
=0, q= = 7)
where 7’ is as in (2), and the operators are formed by expanding &7,, &c., and replacing
EPnity by = The upper limits of p, g, r in the summations are all infinite.
‘par
These operators w are identical with the operators © discussed in Elliott's Algebra
of Quantics, Chap. XVI. Their properties are there obtained by forming the alternants,
but as the formation is simplified by use of the symbolical method the process by
this method is given here.
We remark that the operators 0/0& and {£&7,} are independent and therefore, if 7;
denotes an algebraical expression,
(ET) Te= (0) 3p T = ap lET) T= 5p (ET)
Hence
Wx Oyx — Oyr@zy = {ET} {n Te} — {nT e} {ET}
a a)
= |nggEt,)} — {ES nt}
= (ap ala Ted pls ch caa cn aerntae cates sual tox emma *s soamaerane (20 a)
@ry@xz — Ox2Ory = {ET} {ET} — {ET¢} {ET}
Gy é )
= {Ese EM) - LES ED}
ace crocs ct ee en ee (20 b),
WryWzx — OzxOry = {ET} {ET} — {STs} {ET}
Cd) C)
= |Z) - fee ero}
sail ETT leigy oo ronan Sosa teatae slseig< rene v-sesiaccieaeons soaagecs (20 c)
Similarly
Dy Yaz Dy Drags — Tt Org ce sea tens eames vanpavenaeesane woes cise «5+ dcaceneas seen (20d),
Dry Orga Oey vari — | Ol aneenenceee ss seotea nea steasesnencke ices ase suas coer ssaaee (20 e).
Equation (20a) shows that if a function is annihilated both by @,, and @,, and
is isobaric in first suffixes it must also be isobaric in second suffixes and the partial
weights must be equal, and if the function is further annihilated by @,, and o@,, it
must also be isobaric in third suffixes and all the partial weights must be equal.
Equations (206) and (20e) show that any two o's are commutative if they have the
250 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
same letter for their first suffix, or the same letter for their second suffix. Equations
(20c) and (20d) show that the ’s are not independent but that any one of them can
be expressed as the alternant of two others, so that if a function is annihilated by
the three operators @,z, @zz, @zy, it is also annihilated by all the others.
When the case of more than three independent variables is considered there will
be pairs of o's which have no common letter in their suffixes, and reference to the
above proof of equations (20) shows that such pairs are commutative.
That the operators {U, x}, {V, a}, ... are not commutative with any of the o's is
easily seen by forming the alternants of typical pairs. Thus
{U, 2} Wy — Waxy {U, x} ={UTs,} {&.7,.} — {G73 {Us}
(ee) a)
a if On, ( urs) = {Dose er.)
= {&.U,,0r} — {U.T,,},
{U, 2} wy — @yx {U, x} = {UT} {nT} — (mTe} {UsTe}
a OMFG
Ns ae, (O.2.)} = ‘ces (nts)
— Uz T:},
(U, a} coye — coy (U, 2} = {U.Le} tn.Pe)} — fnPs} (ULe)
pee 3
=i ag, ( ust.) = \Uuag (03)
= (n.Uz,T:..
§ 11. We can now express
DED2AD
Um! nt!
by means of these operators acting upon dimn.
We have, in the first place,
@z2A. mn => (n + 1) ay=n m, n+1)
+p)!
@z? dimn — we A a dip, m, N+p>
n+p)!(m+q)!
Ox! @z2? dimn = ee dp-9, m+q, n+p:
Therefore
DiDmDPt_ 1. (ADz+ A'Dy+ A”D\t 7
lim!n! Sarat eo), ) eae
abate Eee ib
J itmini~ pigil-—p—¢!
Al-14 A't A"? D714 Die Det
INDEPENDENT VARIABLES. 251
ee ote
Jt mini~ p!q!
i (A\is = : ae a) ( (ay Wy! Wzz" Chinn
rir hey ITH)
Al-?-9 A’a A"? (m +q)!(n Fp) Nidpop—o, m+q, np
\J/) ~ pig! \ 4,
1p Aen Awe
= (5) é sa Chins
since @,, and @,, are commutative.
To express D,'D,"D,"t in a similar manner we proceed thus. We have
JD, = AD, + A’D, + A”D,,
JD, = BD, + BD, + B’D, ;
whence AD, = BD, + c”D, — c'D,.
It may be noticed here that, if we were dealing with more than three independent
variables, ec’ and —c’ would be replaced by second minors of J.
We have therefore
DD." Dt A ee .
~ imint =D! (BD, + cD, Dy" D t
Am ry, " rD
= Fit © piigt rt BPD (CDs) (Dey Dt, where p+q+r=m
Aa-m " A\ts 2eetsies
Saal a an q(— dy Up) qntr)!(5) e Orbe aree
A! wry + Al wre
TANG COUN rare 1 /AB\P/ c\r “
=(5) Ga) © * peter) (=a) oe? 2 dn
Alaryt+A"wr: AB
c!
(5) ic carr pas ieieera ae d;
=| — = p + Gimn-
J, (4
Finally, since D,=a''D,+b’D, + ¢"D;, we have
maT tfm
DED: Ds t D, D, Cant (Dz = aD, = b’D,)"t
Umint Uimin!
i
c nr
1 n= p! ae (—a’)a(—b"y D7 Di" Dt, where p+q +r=n,
mi)
“ ” A! wry+ A" wre AB ¢
( a’’)4(— b yr A l+q ‘e ie SSS oye S wy:
AB ic:
@. =F Myx — 7 @
2 Jc” Ux ce’ uzZ>
a’ A bc”
ae iia Wa
{U, V, W} =U, +{V, 3+{W, 3}
=F[A (0, a} +AU, y} +4" (U, 2]
+ F(BIV, 2 + BV, yh + BV, 2]]
+ F[C{W, 2} +0 (W, y} +0" (W, 2}].
il gh t
hen Dinn = Im! n! duldv™dw”
Al-m cm
=i
Now if © denotes any linear operator which acts on two functions P and Q,
we have
EAU, Vs WH. eer’, GP | 6. Digan ses bdo se sneen (21).
e@P 62 Qe 6% PO)
where 9,, ©, are equivalent to but act respectively on P and Q alone. Therefore
eUPe2@ eon. PO
iP)
By repeated applications of this principle we find that
A 1+U—m—m' ec” m+m—n—n'
Jie
Dimn Demin = @ (UF, Wi ems Fees Sema dimn Ar m'n'-
And more generally if F(dimn, drmn’,-..) represents any function isobaric in each
set of suffixes being of weights p,, po, ps mm first, second and third suffixes,
APi-P2 ¢Pa-Ps
Te eG, V, WI meres I (damn, Armin'y «+-) «+++ (22).
TCD ree rata se)
§ 12. The asymmetry of that part of the operator which depends on the o’s is a
consequence of their non-commutative character. By arranging the work a little differ-
ently nine different forms of the result could have been obtained. In the case of two
independent variables the number of different forms will be four, and it will be
convenient for some of the subsequent applications to have these four forms set out
at length.
INDEPENDENT VARIABLES, 253
In modifying the work of § 11 for this case it is obvious by reference to the
argument that only two @’s will be required, viz.
0
@z, = {ET,} = pines (¢+1)dpa, gi Adpy’
; 0
Oyr = {n T's} = ee ED dy+1, q-1 Ad pq ;
and their effect on d;, is seen to be this :—
a oo —
Wry” din diy, m+p>
l !
yz? dyn =f +P) Airy, m—p-
Moreover, reference to the work shows that c” must be replaced by unity, and that
Therefore
eee eee ee eee
ipjm l m A comp AB
“r= (5) = ete din
We will next obtain the second form of the result. We have
DiD,."t = . (AD, + A'D,)! Dj t
=Fi25 2D,” Dt, where p+q=l.
Therefore
Dj! Dj"t _ =. 1 y(m aa APA dnp
l!m! m! p
ANNE 1 ANP
S (5) ai (=) Reps
t 4.
-(4 “) et Ant.
Now JD, = AD,+ A'D,,
JD, = BD, + B'Dy;
therefore, eliminating D,,,
A’D, = BD, — D;.
33—2
254 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
Hence
DiDnt = ana (B'D, — D,)" D,!t
= 5 ™ pe (_ 12 DH? Dest
A’™ pq!
= = a Be (—1)91 (1+ p)!q! (4) a den th S
Therefore
Bi (2 eae ae
willl (-oil ie 2 lly (Al Re ai
= G ( ) a) ( es
=(-1)" — La at oc’ a OR rete ope (23 b)
Interchanging B, B’ with A, A’ and J with m, and writing —J for J, we shall
find similarly
ae yG) &) Ran a ir tt ete Oe => (23)
= (Gy (Fe) Pare aect me Wa) cee Dee (23 d).
From these results it follows that, in the case of two independent variables, if F
is isobaric and of weights p,, p, in first and second suffixes,
F (Dig ee) = BOs CHOU HET dpdhon ace ieee eee (24 a),
where {UV} =F{A (UW, a+ A'(U, + BUY, a+ BV, yl] ovrrrerecceee (24 b),
and the quantity K and the operators @;, @, may have either of the two sets of values
APi-Pa AL AB
rae Ona a a> yy Oe atatale atelalaiateiajarcloielalttcrseielarere (24 ¢),
nce B AB
K= Jaan? 1 = Py Oye, @, = yi Ory sec c cere cece cncccccccs (24 d).
And, as another form,
FE (Digs jos Ce ene BL OGae cna)’ wast cate doses cc teen eee (25 a),
to
or
o
INDEPENDENT VARIABLES.
where K, @,, », may have either of the two sets of values
'Di~Pa d 4 oud
K= (- 1)? a » a= yz, O,=— - Day vscrececccsccseses (25 b),
K=(-1)" — So CS ate o— a ee Oust a atrcseadadacces (25 c)
Here A, A’, B, B’ are the first minors of J, and therefore A=b’, A’=—b, B=—-d’,
Bia.
§ 13. In the particular case when there is only one independent variable 2 which
is transformed to u, we have J=u,, A=1,
{U, V, W}={U, a} ={U,0;,} + {UT ,,} ;
there are no ’s, and we have, if F is an isobaric function of weight p,
f
and Di=——=—e
This form is not quite the same as that given by Mr Lendesdorf (Proc. Lond.
Math. Soc., Vol. Xvi.) and also established in my previous paper (7rans. Camb. Phil.
Soc., Vol. Xvi.), but one formula can be deduced from the other by the method of the
next section.
§ 14. Another form of the general result is often more useful than that stated in
equation (21). It is obtained by exhibiting separately the terms containing first
differential coefficients of ¢. For this purpose modified forms of the operators {U, 2}, ...
must be used; let [U, z] denote the result of suppressing all terms in {U, 2} which
contain dye, duo, don, 80 that [U, 2] may be formed in exactly the same way as {U, za},
except that in the process of formation the value of 7’ used is
dag? af oon? aR dof? + dy EN +...
instead of that given in (2). Let [V, «], [W, «], [U, y],... represent similar modifications
of {V, a}, {W, a}, {U, y},.... Therefore
{U, x} =[U, xv)+ Dio [U4],
{U, y} =[U, y) + doo [U4],
256 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
The twelve operators [U,], [V.], [W.], [U, z], [U, y], --
. are easily seen to be all
commutative with one another. For, by § 7,
{U, a} {V, y}-{V, y} {U, a}=0;
therefore
{[U, 2] + doo [Us}} {[V5 yl doo [Va]} — {[V; y] + dow [Vs]} ([U, 2] + dro [ Us]} = 0.
Hence, by selection of the coefficients of Gyo, dno, Arodoo, it follows that
[U, z](V, y]-LV, yw [Y, z]=9,
(ULV, yI-LV, y] (UJ =9,
[U, x] [V.]—-[V.] [U, «]=0,
[U,] [V4] -—[V.] [U.]=0;
and in similar fashion it may be proved that the alternants of all other pairs of the
operators are zero.
Now, by (19¢e),
-{U, vy, Ww} DED "Dr
Ui min! ~
Therefore if [U, V, W] is the modified form of {U, V, W}, so that
Dimn =e
(U, V, WI=F(ALU, 2]+4'(U, y] +4’, 2)
+5 (BV, o]+ B[V, y] +B’, 2)
+5(CLW, a] +0'[W, y]+0"(W, 2),
(u,v, -(4a+4'a'+4"a") (Ug —1 (Bat Ba +B"ANL va —-Lca+eva'+e"a") (Wa DD" Ds"
De i J ff
imn = @ oa me 6
Lim! n!
Now D'D,"D3% is a linear function of dyg,, Therefore the effect of [U,], [V4],
[W.] operating on DD."D,"t is to change dyg into Apgr, bygr, Cpgr and therefore to
produce
DiEDED Su, DEDED sy, DED Dw;
whilst repeated operations by [U,], [V.], [W.] produce zero results.
Hence
Dinn =" "I ALS. DID MD ot —(Ad + A’ + A’d’) DED "Dy
—(Bd + B’d’ + B’d’) DiD™Dv — (Cd + C’d’ + 0A”) Di DPD yw]
“1 ce, wy | DIDMDS, DEDDou, DiD"Dym, DLDMD Zw |
Si b c
INDEPENDENT VARIABLES, 257
§ 15. Up to the present there has been no restriction on l, m, n except that
they be not all zero; the last formula holds when 1+m+n=1 on the understanding,
assumed throughout, that D,!D,"D,"u, D,D"Ds"v, DD "Dw all vanish when l+m+n= 1.
But it is necessary to assume, in what follows, that 1+m+n>1.
Corresponding to the operators » of § 10 we introduce six operators 2 given by
the equations
Oxy = [6 Un.) + [EV 5) + [Es Wa] + [ET],
Oye = [Ue] + [mV e.] + [ns We,] + [Ze],
Ore = [FU] + [EV] + [6s W,) + [E.2e,],
Ore = [6:U¢,) + [SV e,] + [We] + (8.2),
Oy2 = [Ue] + [Ve] + [ns We) + [Tc],
Oey = [8105] + [SV] + [on] + [80],
where Oi a0 ons Vi=Oyee ees
Wa = Cams? + +. ; T's = dao E e+... ,
0 C)
m= 3p, On Vu = ap, Var ones
and after expansion of the expressions [£,V/,,], ..., £m!" is replaced by ae :
pyr
f) ri) a
EPIC" by aban , €PnC" by cee EPn ac’ by ll
par pyr pyr
The four components of each operator are independent of one another and therefore
commutative with one another; but as in § 10 the ’s are not all commutative.
In fact, applying the results of § 10 to corresponding pairs of the partial operators,
we find the alternants of various pairs of ’s to be
OgyQyz — QyzQey = — [6:Us,) — [82 Ve,) — [Es We] — [ELe.] + Pm U9.) + P02Vo.) + 92 Wo] + Pn],
NyyQzz — Az2Azy = 0,
Oe — De Ay = Azy,
Quy Qyz — AyzAzy = — Azz,
QQ — zy Ary = 0.
From these relations deductions can be made similar to those in § 10.
We next write
A’ A”
AB ec
Dares rate
eA b"c"
0, == =F Oe = Oy
258 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
DID" D St a Abm (om m—n 4 J
Hao agica, Sh.) op ek
D. ‘DoD ny Alum (oh m—n
rf m! = ae af Cree iam =:
Therefore
ee St ee
J d a b e |
is hea, | Shae (27)
qd’ a b” c"
Now write IFS |) ol b c =Ad+A'd'+ A’d’,
d’ we
da’ b” ec’
df= || @ d c = Bd+Bd + Bd’,
4 Gk c
a” ae c
d= || @ b d = Cd+ 0'd'+C"d’;
a’ b d’
a” bY da’
therefore
Alm mn
Da = yee el, y, WeM%e%e0s (Sdimn = aCe = Dian = EGran) Kjeteisiatelpiaiaia’wieisiate (28),
and as in § 11, if F denotes a homogeneous function of degree 7%, which is also
isobaric of partial weights p,, Ps, Ps;
APi?2¢''Pi Ps
F (Dim; ---)= yp, e- (U, Vs WleMe%e% F (Idan — Syn — Fabimn — FsCimns +++) «+-(29),
or using the operators [U,], [VJ], [W.] defined in § 14
APiP2¢"'P-Ps
Ti (Dry 50) = JP
ws
ai = a
e-[U, Vs MeMeMg%g— FEU 9 FV oT F(dinn, ---)
This last form does not require F to be homogeneous, though it must be isobaric.
§ 16. If in (27) we put t=a, y, 2 in succession we obtain formulae for the
interchange of the dependent and independent variables. Write
1 gitmtny,
Aimn= Tj m! n! duldv™dw”’
1 giimtny
Benn = Ll! m! n! duwdv™dw”’
if! glimtng
Cunn =
1! m! n! duldv™dw™
INDEPENDENT VARIABLES. 259
Then, provided 1+m+n>1,
A —nc!’m—n
Alin a ja —e-lU,¥, W) 76% 16%. e% (Adinn by Bhunn 4 Counn)
Ahm” //m—n | h
c -(U, ¥,W)] of a a Aimn imn Cimn
——— ya é merge eo. 67. Ee
a’ by Cima Wesead (31a),
ae ie cc’
A —me//m—n
— T y / ve
Bimn = Jin Cag at WleMe%eNs (A Gimn + Bbimn ar C Cimn)
eno ’m—n
mt ane SRA ee py, |! b c
yn e eo .e.e
Aimn loprers Clmn | ceeeesere (315),
a” b” cr
Ammen
= “4 o “u" ua ”
Comn = — - Jin e— LU, V, Wi eMigMe0s (A”dimn + B’Bimn + C”Cimn)
l—mp’m—n
ee Aes e-LU, V, Wle%gM%e0s b ¢
Jim , , ,
a b CB Wi eeececsc cess (31e).
Amn Dimn Ctmn
And if F(Dimn, Ain, Biman, Cimn; ---) 18 a function homogeneous of degree i in
Aimn; Bimn, Cimn, Din; -.. and isobaric of weights p,, p., p; in first, second and third
suffixes, we have
AM —Poc!/Ps—Ps
eae W1 @Mg%g05
1
F (Dimn, Amn; Beans Cuan = = i
. F(S,Qimn == Jobimn SF J sCimn = Jdimn; Adimn + Blinn ae Coumns
A’ Gimn sr Bbinn a OCs AD tisnn ar Born: + Co Cimas o° -) o ccccceeenas (32).
Since there are no d’s occurring in equations (3la), (31b), (8lc) the operators
occurring in these equations, but not im (32), may be simplified by the omission of
differential operators which affect only d’s. Thus [U,7;,] may be omitted from [U, 2],
fedeulietrom: O75 «-..-
It will be noted that Aim, Bimn, Cimn ave the coefticients of v'v™#" in the expansions
of &, », € when the series
= Gok SRose 4 GimnE'n™E” + ..5 5
V = bro& + +--+ OunnEé'n™l® +... ,
@ = Ci + 0. + CimnE'N”E" + ...,
ave reversed, and & 7, € expanded in powers of v, v, @.
Yo) oem. 0) Ol (ie oats oO I 34
260 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
§ 17. The formulae of § 15 may be adapted so as to give a symbolical form for
the differential coefficient of an implicit function. The method is applicable to any
number of variables, but for the sake of brevity the work will here be restricted to
the case when there is only one dependent and one independent variable.
As in § 16, let
u=F(x, y), v=G(#, y);
1 oy.
m! dum?
is obtained on the assumption that w is constant, so that if we take G@(«, y)=#, we
then formula (31b) shows how to determine B,,, or this difterential coefficient
™,
shall obtain a ou on the assumption that #, y are connected by the equation
F(a, y)=const. As in § 16 let ay, stand for
ORE ee
PP! gq) Caray?’
the b’s of § 16 are in this case all zero except b, which is equal to unity. Now
we have
J=
Gh Gi
i @
Ay dn
0)
=— yy
so that A=0, and therefore the forms of ©,, ©, used im (27) are not applicable.
Instead of these forms we may use forms similar to those given in (25a, 6) and obtain
S a
K=1,, O=0, O;—=—— 0.
01
We have therefore, if m>1,
B, + tw, vig-gtaw | % 1
mm Ga
Amo Ono
Now, in general, when dealing with special values of the letters, it is necessary
to carry out all the operations imdicated and then substitute the special values. But
in the present case, where all the b’s involved in the operators are zero, it is allowable
to suppress in the operators all terms which involve b's; for it is obvious from the
form of the operators that they never diminish the degree of any function in b's, though
they may increase the degree. It therefore follows that the terms which arise from the
b-parts of the operators will all be zero. We therefore have
[U, VI=-1U, y= SPF]
1 9
re [( Gop + Gu En + Aon? + AoE? +...) (AnE + 2m + dn & + ...)],
01
Oy = [EF] = [E (ané + 2an9 + anE+ «..)],
INDEPENDENT VARIABLES. 261
0
on the usual understanding that £m? is replaced by Demy Hence finally the value of
Co Pq *
diy ; ; y= Re
Jam 38 found from the equation F(a, y)=0 is given by
fr cs
Ldmy__ 1 o-Acrr) -“*(er,
Bom a: m! da” a.” am Namrata Mat ein tat lane adie Od aly © dpaceinie os vena e (33).
§ 18. The determination of the differential coefficients of implicit functions is equi-
valent to the solution of equations by series, so that the method of the last section leads
to a symbolical form for the solution of a set of equations of infinite degree. It will be
sufficient to illustrate the method by considering the case of a single equation,
O= F(x, Y) = Ayyt + Any + Aah? + Ay LY + Ayo y? + Ago? +... 5
it is required to determine that value of y which vanishes when # vanishes. The
solution is
y=Byct+ Beet...
a
where B,=— a
o1
, and B,, is given by (33). Now let P denote the terms of F(z, y)
which are independent of y; then the required solution of the equation F(z, y)=0 may
be written
1 + [rr] 2" (Fy)
ag er A Mn Oa Guts ChRE te-ceaddenancdcetneesssetse: serine (34),
ol
where the operators [FF,], [EF,] are the same as in the last section. For an equation
of finite degree n it is necessary to suppose all the operations carried out, and then all
the coefficients ap, for which p+q>n must be made zero.
If f denotes any rational integral function
%o 5
, Ene a af EN
TY =C@ & LFF] ~@ A Leal Ff (- rE) aia ereleiciciciele viaiviciee sin win'eien ainie (34a).
§19. As an illustration of the general methods established, we will employ them to
effect the change when the variables are linearly transformed. Let the scheme of transfor-
mation be
N= an + aly + a’g+ at,
V =f0+By+6’+p"t
Z=yat yt et yt
T.= da+ Syt+ 82 +8",
and 7 being regarded as the dependent variables. Let
1 Qitment
U! m! n! datoy™ez”’
i giminT
lim! n! ex'eY™oZ”
d inn =
Dian =
34—2
262 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
It is required to express Dimn in terms of dimn, dymn')---- In the formulae T, X, Y, Z are
to be written for ¢, u,v, w so that a,b, c,... will denote first differential coefficients of TY
not ¢, and therefore
a=at+a't,, b=B+B'tz,...
Now equation (27) shows that by writing X, Y, Z, T for u, v, w, t the value of Dimn
can be found in terms of differential coefficients of X, Y, Z, 7 with respect to a, y, z
and therefore expressed in terms of fyg,,---- But the operators [U, «],... which produce
the expression can be replaced by others involving differential operators Maer
For the operator [U, x] or [X, x] may be written
a Q a
ZA, PAA? (XXz) aXset ES AzPA {AY (Xz) Ven + SAPA SAS (XZ) a7 —
+ EA, PA, 007 (Xs) a
pqr
where p+q+r>2.
Now since all first differential coefficients are removed after operation with A’s it
is obvious that in A,?A,2A,7(XX,),... X, Y, Z, T ‘may be replaced by at, B'”'t, yt, 8”.
Moreover, if p+q+r¢ 2,
X pg = O"bpar » Yyqr = Boar» Spar = boar » Lyge = 8" boar 5
and therefore, for operations on a function Of eXpors Lipa 2oers Lear ees
) mt 0 uy 7) ut 0 ir 0
Ot par 4% OX por at OV yar tae OZ par ve OL par j
Also SA PAYAL (XXz)=a' 27D APA, AZ (tz),
TAPA MAS (XVz) =a"B” SAzPA,IAS (tte), &e.
Therefore [U, #] becomes
0 0 0 é
wey Pp q rT wn Ua we nt
a >A, A, Az (ttz) E ake =F B ave Of Aine 5 |
=a" SAPAIAY (ite) =.
‘par
Now denote the operators
0
Ot por
0
>> A,PA {Al (ttz) > > A,PA,IA7 a > DA PAYA, (tt, Oh:
‘par ‘par
by Vi, Vo, V;. When working with d
‘nq ©nstead of pe it will be more convenient to
form the operators V by writing
Vi=trd, Velen), Ve=lrrd
where T = ogy E? + don? + ---
and after the algebraical multiplications £?n7£" is replaced by aa
par
INDEPENDENT VARIABLES.
We shall then find that [U, w), [U,y], [U, 2], [V, a],... become
263
ol’ V, F al” Ve. al” V;,
Bovawens and it finally appears that [U, V, W] becomes
Aa” + BB” + Cr” 7 Ala!” + BBM! + Oy” Alege: BY BY + Cl” =
i Vi+ 7 -_ Y, —— V,=V, say,
so that
Piz seat a yen
d/ a a a’ al’
b b’ b” [si
c c cc’ ff”
wil V, Vz V; 0 |
di) at Cates a + at, a” + at, a” |
B+Btz, B+ Bt, BY’ + Bt, BY
yty"t of tet ? Ye +t, yf” i
Vitae eos, O
Jf a a’ a’ al”
: ” P}, ll (oa To dUodoodeNOnbg cba HcOOAESaECuRAROCceCE: (35).
Pee hae es
Y y yy” yf”
When there are n independent variables the corresponding formula for V is
Teh ee is | Wor V2, os Wo ? 0
“ ee A) 5 a™
DC a cyte ee een ee ee (36).
Bees = ar), Be
Vey eye oo po aay
eee Seo ae ey eo
The formula (26) then gives
D Almem—n pe DIDO D YT. dD! D DX, Di! ‘Dm dD. Ve DED" Ds LRA
Mm —.
JH ie ie Ve Zz (37)
1, x, y, Z, meeeec(an):
Te XxX, Ve Zz.
Now, 1+ m+n being greater than unity, the determinant becomes
Oe al” (sy yf” D, 1 D, m Dj"
é+ Oy at alt, B+B't: yt ote
OFF Ok a’ + Cat B+ Bue of +7ty
8" ef Ot a’ te ae Be 4 Bitte yf” + 9y'"tz
264 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
= al!’ BY” mw Di Dr Dt
B
OL a “Ria
Of Gy Beenie
=ja B vy 8 |DID™D
aay,
a) BR shy lity Ou
ee ee ote 2)
= ME DED D i asays
so that M is the modulus of the linear transformation.
To transform D,!D."D,"t we use the operators
Oxy =[Et,], zz = [Ere],
@yz=[nTe], — @y2=[T¢),
@2=[Ste], ay = [Sr],
where T= do &+dyen?+..., and the operators are formed in the usual way by replacing
Bene with 5
investigated in § 10, and just as in previous cases we find that if we write
The properties of these @’s are precisely similar to those of the o’s
A’ PAu
Oa, Way + Ai xz;
AB c
O, ae eu Dyx — Za Dyz;
a’A bc”
Ory Oe tae
we shall have finally
Almne!/m—n
Duan = M — 6 Vetiaias Dem ntuccet. ase eee reece (38).
And if F(dimn, drmn’,..-) is a pure homogeneous function of degree « and isobaric
of weights p,, ps, ps; in first, second and third suffixes,
F (Danan; Dem’ pee =) — =e APi-?Pi¢’P2Pae- VeM% eg iM (dimn; Adymn’ os 3) pecoud (39).
The value of J is
at a’’d, a +ad, a” + ald”
B+A"d, pte", 8" +—"0"
fy a Sy Ry ee oy enya
INDEPENDENT VARIABLES, 265
and A, B, C,... are the minors of this determinant. It will be noticed that products
of d, d’, d’ do not appear in J, which is therefore a linear function of these quantities.
In fact another form is
J= a, a, a”, a”
Bs §8 ee RY
% Hs Co
| OT (41D);
M'* Br: B A’B’
Le ae Or BR or oO;= Ti Dry eee ce scescccnscne (41¢).
And, equally well,
TE (ID 9, Sg) LO EMG cog) aadocpensonedccnesccoschene (41d),
where V is as before, and K, 0, ©, may have either of the two sets of values
MiA'?:?: A A’B'
K=(- 1 ey ee me ; =a Myx, 0, =— a Dyy serererereeeees (4]e):
M* BP B AB
K=(- tame oe » Q= Bow Og= => (SF) Soecnpndodar act (41f).
For example, suppose it is required to change the variables cyclically so that
t, z, y are changed to w, y, ¢ and w is the new dependent variable. Let «,, stand
1 Qumy
T Tm! ayoe™
The scheme of transformation is
X=0.e+1.y¥+0.¢,
Y=0.%+0.y4+1.4,
F=1.¢2+0.y+0.t
266 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
Therefore = | © 1 | =—& M=1;
5 aa
pee a =
Jl o ee OMe
0 0 1 |
Here B’=0 and therefore the form (41c) is not applicable, but the form (41b) gives
fyjPx—Ps pas AE
=Car? te ty is O82 TE (dumesccs). asecce eee ace (42a);
and the forms (41e), (41) give identical results, viz.,
EN (Gens eee
(- ty) Pie ou
=e
(Gms -- )= DD" Cpe ty) 2 @ tt, HI 1IK(Gbns aac)
ele ee
=(- 1) pe te te PP (dm, ---) vdee nana Sauapanesoeets (42D).
If it is required to make the second cyclical change from ¢, 2, y to y, t, @ so that
l+m
y is the dependent variable, let ym = =. Then the scheme of transformation is
X=0.2+0.y¥+1.2,
Y=1.2+0.y+0.4,
T =0.¢+1.y4+0.t.
Therefore Jf =|te ty |=—ty,
a) 0
Vi ae
see Ac | 2s
i 0 ty”
1 0 0
Here A=0, and the form (41b) is not applicable, but the other forms give
t,)pP. Vn _ ty te
F (Ym, 500) = & year? ty @ tz” ty HEM (Cams ea) arotete aislayetelsravelovereinetalotere (43a)
¥.
and
(-ayerm Ta tay,
FY im; +) = (= ae Pee we ty ” F (dma; «-+)
Ve
=(- i) a HOU ty 5 Fig, <=)
_V, ho
Sei: ne tier Ge (deaies Vek (43b).
INDEPENDENT VARIABLES, 267
§21. The alternants of the operators » with,one another have been already examined
in § 10; it remains to examine the alternants of the operators V combined with @’s.
These alternants have, in the case of two independent variables, been given by Prof.
Elliott (Proc. Lond. Math. Soc. Vol. x1x. p. 9); but the proof is much simplified by
making full use of the symbolical form for the operators. Only typical cases sufficient
to establish the general results will be considered. We have
Y= [TT], Vey = i Dyx = [nT], Oyz >= [nT],
where 7’= dy &+..., and after multiplication £?n7f" is replaced by xy ;
par
Therefore V,V.—V.V,=[TT;](7T,] —[TT7,](TT:]
‘yyy yy fj 0 L4 i- yy 0
= ht, se an| = [PEt + Te (2,)|
Very — Oxy Vi = [TT:] [er = [é7,,] [TT]
= E on) iw lente sé) |
SPRATT RE a ee coe 2k (44),
Vioys — yz V,= (PT nF) — [nT] [PTA
= | nge Ot) |- | ott + Tn]
Vi@ye — @yzV1 = [TT] [nPe] — [nq] [LT]
= |nz¢ n)| = [nsf + rs (ot)|
Similarly ee Vp asa ane dee to Me (44e),
and generally V, or [77;] is commutative with all w’s except those which have z for the
first suffix, whilst all the V’s are commutative with one another.
§ 22. The applications to the theory of pure cyclicants are easily made. A cyclicant
is defined as a function of differential coetficients which is unaltered when the dependent
and independent variables are interchanged in any way whatever, save for the introduction
of a factor which involves first differential coefficients only. The cyclicant is pure if it
involves no first differential coefticients.
In the case of three independent variables, and the method will be perfectly
general for any number, we shall show that if the function, supposed pure, is invariable
Vou. XVII. Parr ITI. 35
268 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
save for the factor mentioned above, when ¢ is interchanged with x and y, z are
unaltered, and also unaltered when ¢ is interchanged with y and z, # are unaltered,
and also when ¢ is interchanged with z and a, y are unaltered, then the function is
invariable when any interchange whatever is made, and moreover the function is
invariable when the general linear transformation is applied. We shall find that the
necessary and sufficient conditions for the mvariance of a homogeneous and isobaric
function are that it be annihilated by the three operators V,, V., V; and the six
operators @zy, @zz, Wyz, @yz:, @zz, @2y. These conditions, though necessary and _ sufficient,
are not independent. For it is evident, as in § 10, that annihilation by three o’s
such aS @yz, @zz, @z, Will ensure annihilation by the remaining o's, and it is proved
in § 21, that annihilation by the @’s and JV, will ensure annihilation by V, and J.
Now annihilation by the o’s implies that the function is invariable when the independent
variables only are linearly transformed, so that a pure function will be a cyclicant
if it is unaltered by linear transformation of the independent variables and unaltered
also by the interchange of the dependent and one independent variable.
In consequence of annihilation by the o’s any pure cyclicant will be an invariant
of the system of quantics in & , §&
Aso = aE dean? a5 opal? ate di EN aF din EG + dunf,
Ag +... + dyn En + ... + dnEn§,
SSS ee i)
and conversely any invariant of these quantics which is annihilated by V, will be a
pure cyclicant.
When the number of independent variables is n, there will be n operators of the
V type and n(n—1) operators of the w type.
§ 23. To make the transformation by interchange of ¢ and x the scheme will be
X=0.2+0.y+0.24+1.4,
Y=0.2+1.7+0.2+0.¢,
Z=0.e2+0.y4+1.2+0.¢,
T=1.£+0.y¥+0.2+0.¢,
: . f D 1 b il gitming
Se ne CEE eee Se caper aa atoy™oz” ©
Here M=-—1,
Tae Ue ee
0 1 0
0 0 1
A=4, A=—0, AV=0) B=—f ce —0,7 a —t.,)0 =O) cr
INDEPENDENT VARIABLES, 269
Therefore O,=0, O,== m Oye, O=— 4 rip
z x
and V= | V;, Ve Ke Ons Ve
| 0 0 0 1
| 0 1 0
| O 0 1 0
Therefore if F (dinn, dymn,...) is a pure homogeneous function of degree i and
isobaric of weights p,, po, p; in first, second and third suffixes,
; Vi t t
(- 1) Sat SS tes
EF. (Dimns Dyin’; j= titPr te ty ae ty f= Bl (yaa drm'n’; vee)e
x
- : : 1 4 :
The right-hand side can be arranged in powers of 8 7, z, and since these are
x Zz xz
independent quantities it is obvious by observing the coefficients of their lowest powers
that, in order that 2 (dim,,...) may be invariable, save for a factor, it is necessary
that the function be annihilated by V,, ow, and o,,. These conditions are obviously
sufficient, and therefore if the conditions are satisfied we have
aa\e
FD sce Decem = Fe Fg 2
Similarly the necessary and sufficient conditions that F may be invariable when t
is interchanged with y, and wz, z are unaltered, are that F be annihilated by V2, ox,
and ,,; and, when ¢ is interchanged with z and «, y are unaltered, the necessary
and sufficient conditions for the permanence of F' are that it be annihilated by Vs,
yz and @y,.
If F is annihilated by all the operators V and o, equation (39) shows that F will
be permanent in form, save for a factor, when any interchanges of variables are made or
when both dependent and independent variables are changed by any linear transformation.
Since the annihilation of a function isobaric in first, second and third suffixes by
the w’s implies that the three weights are equal, equation (39) shows that if F be a
pure cyclicant the effect of the general linear transformation upon it is to transform
it into
Mw
Jee
where 7 is the degree of F and p the weight in either set of suffixes.
In order that a function may remain permanent in form when the variables are
changed by the general linear transformation it is therefore necessary that it be
homogeneous and isobaric in each set of suffixes throughout.
35—2
270 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
§ 24. As another illustration we will give a proof of a theorem established by
Prof. Elliott in a paper “On Pure Ternary Reciprocants, and Functions allied to them”
(Proc. Lond. Math. Soc. Vol. x1x.). In this paper he considers two independent variables
x, y with a dependent variable z. His operators V, and V, are the same as those
considered in $19, 20 and 21, with z written instead of t: his Q, is w,, and his ©,
1 QPt%@z
p! q! dar dy?”
(Pe, Pastas eect ean) a nO),
is @z,; and he writes 2,, for He considers a “reciprocantive covariant”
where wu, v are any quantities and Py, P,,...Pm are functions of 29,... where p+q¢ 2,
such that P, is a homogeneous and isobaric function annihilated by V,, V2 and @,,, and
@x,Po=MP;, OnyL. = (i —1) Passe. Onl — ms Oay Lan — Os
In consequence of these conditions the function is a covariant of the emanants
(dsp, dy, du Qu, v), (dso, din, Can, dys Ou, v)§,
Therefore, if w,, w. are the partial weights of P,, m=w,—w, and the function is
only altered by the factor (—1)" when w, v are interchanged. Hence if in P, each
quantity 2,s 1s replaced by 2s, the result is equal to (—1)”P,,_,, and the quantities
P therefore satisfy the conditions
@yzPm=MP ma, SyzPma=(m—1) 1 epace Opin ey Oya)
Prof. Elliott shows also that all the P’s are annihilated by V, and V,; this property
following from the relations
Oya Vi — Viedyz = 0, Oxy Vo — Vex, = 0,
Ory Vy — Vi@zy = Vs, @yzV2— Voeye = Vi.
See § 21.
Now let the variables be cyclically transformed from z, x, y to «, y, z so that « is
; 1 92t2e
the new dependent variable. Let «,, denote Alli ByPOe4 ’ and let P,(x) denote the result
of substituting 2,,,... for 2)4,-.. in Py.
Similarly when the variables are transformed from z, 2, y to y, 2, x so that y is the
: OP+4y
new dependent variable, let y,, denote allel ae and let P,,(y) denote the result of
substituting y,,,-.. for Zp... i Pp. Then Prof. Elliott’s theorem states that*
P, (2) iG)
(— 1)it Th mF =(- yarat = = (Ce 1%. IES, tee ie) (— 21; Zn
where 7 is the degree of Py.
* Prof. Elliott gives different powers of (—1) in his statement of the theorem, but there is a slight error
in his work which accounts for the difference.
A a ate
INDEPENDENT VARIABLES, 271
The theorem follows at once from the results of § 20. Using equation (42a), we
have
EF ad ato Vy he FF at is
Py (a yy a 2, #30 ty "6%, ” P,(z).
But @,, annihilates P,(z), and V, annihilates not only P,(z) but ‘every function of the
form @x,’P,(z); therefore
LE
a ay” Pris)
Le (2) = (- zg am
= zy" 2x ll /e3\2.. =
= Ee Zz ee E — 2, Oxy + aie) Oxy — a 12 (z)
1 _ m(m—1
= apyinm [Poo mPa t+ BONED Paste t— a + (Pate
(-1
= Fra (Pas Pay Pas ve P)( 2p 60)"
_C 1 jit.
eae i+,
(PoP iees 0m) (= Zas 2)
Again, the first and second partial weights of P,, are w, and w, respectively; there-
fore by equation (43 a)
_Va iy
Zyl —2 _2
Pr(y)= es Guwenmene 7 Ps (4).
Z yet
But @,, annihilates P,,, and V, annihilates not only P,, but every function of the
form @yz”P,; therefore
x
ae. fer 2y LU eee Sup *y\" m
=Cz,)Fe E = 5 tie 31 € yx? — ..e + mal 2 ge | tlre
ae [P. (— Zy)™ + MP, (— 2y)™ 1 22 + gee), P,(— 2)" 7277 +... + Paes
~ (—2y)r ite 22
1
> (az yirte, (Po, Pas P, mee Pin) (— 2y, Zn).
¥,
The two results establish the theorem.
§ 25. It has been seen that when the variables are linearly transformed from
t, a, y, 2 to T, X, Y, Z a pure function of differential coefficients will be unaltered
in form, save for a factor, provided it is annihilated by the operators Vis Vacs and
eg Ond; <6.) Lb, willbe convenient temporarily to denote V,, V2, Vs by Vz, ee
It is evident that this permanence of form would be ensured if the transformed
function expressed in terms of differential coefficients of 7 with respect to X, Y, Z
272 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
were annihilated by operators Vy, Vy, Vz, wxy, wxz,... formed with the quantities
Dyor, Apgr, --- In the same way as Vz, Vy, Vz, wzy, @zz,--. are formed with yor, Gor) ----
It must be possible therefore to express the effect of these operators Vy, Vy, Vz,
@xy, ®xz,-.. on the transformed function by means of the original operators acting on
the original function. It is now proposed to examine into the manner in which these
operators Vy, wxy,... can be so expressed.
The scheme of transformation is, as before,
X=ar +ay +a"z +a't,
Y =Br+ B’'y+p'2+B"'t,
Zaye toy +y'2 + "4,
T =6a + &y +82 + 8",
and ¢, 7’ are regarded as the respective dependent variables.
Suppose now that a, y, 2 receive increments & 7, € and let the consequent
increments in ¢, X, Y, Z, T ber, &, 7’, ©, 7’, so that
T=AywE +doon +dmf + dong? + Aoogn? +... ,
T= Dyook’ + Doon’ + DenS’ + Doyo&’? + Dyson’? + -- «
Then VA = Vz = (T = DyooE >= don = dof) (Tz i, yoo).
Bry = S (tT, = doo);
Oye =] (Te a Ay), &e.,
where after expansion £72" is replaced by a :
Py
Therefore also
' f , f 0
Vy = (7 a Dayo = Dion = Donk ) & = Din) ;
oxy =& & a Dan) >
, (Or
@®yx = (se - Pw) , &.,
when &?n 2%" is replaced after expansion by a :
OD par
Now let F(&, 7’, &) be the symbolical expression for an operator obtained by
expanding F’(£’, 7’, €’) in powers of &, 7’, & and replacing &?n'9f’" by = it being
par
understood that # contains no term for which p+qg+r<2. Then Prof. Elliott has
INDEPENDENT VARIABLES. 273
, , . . seRO TU. ‘ P
proved* that the expression for this operator in terms of aq... 8 obtained by expanding
Oo
par
a certain expression in powers of & », € and replacing £?n%f" by - - This expres-
Od nar
sion is
i ; Or OT Or’
ay F U ’ TTA A or Ae a Sw BAN ” zit
HP, 1, 6) (A"-aF wears),
where M=| a, a’, (4 ce
B, [sit Bas Bee
| |
| Yr ta is, aan
oe iene ae
and A, A’, A”, A’” are the minors of 6, 8’, 8”, &” in MV.
The application of this rule to the operators considered here is simplified by use
of formula (47) which we now proceed to establish. The rule itself may also be deduced
from this formula, but Prof. Elliott adopts a different mode of proof.
We have, in consequence of the scheme of transformation,
EB =aE+ an + a’O4+ a's,
n =BE+ B+ B'S+ B's,
CayEtyntyo +77,
7 = 6& + 8 +06 + 87.
The simplest way of finding Dio, Doo, Do. is to determine them as the coefficients
of &', 7’, ¢’ when 7’ is expressed in terms of these quantities.
Now, neglecting higher powers of & », € than the first, the last set of equations
may be written
EB’ = (at adiy) E+ (a +.a”dow) 9 + (a” + @don) §
1 =(B +B dro) E+ (B +B" doo) 7 + (B" + B' den) &
C= (9 + 9dr) E+ (oy +. dowo) 9 + (y" +9" don) &
7 = (8 + 8d) E + (8 + 8’ dao) 9 + (8” + 8’ doar) &
Therefore, eliminating &, , € we have
la+ abs: a ap a” duos al” if al” doer
B+PB''diw; i Rae B'+ Bo den, 7
ms , m ” m” , — 0 see eeeeeeees (46).
y+ 9"diw, y +9douo, y+ "den 4
§ + 6 di, 8 + 8douo, 6" + 8"'dun, T
* «The Transformation of Linear Partial Differential Operators by Extended Linear Continuous Groups.”
Proc. Lond. Math. Soc., Vol. xx1x.
274 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
Write this equation in the form
7 = Diy E’ + Doon’ + Donk’,
and Dy, Doo, Do: are immediately determined.
As previously, write
J= | ata’ dy, a +a’ doo, a + a"don | _0(X, Y, Z)
B+B"dm, B+8du, B"+B"dm | 2 ¥ 2)”
y+ yy rw, yy 35 yd ’ y a5 door
The increments £, 7, € 7 have only been temporarily assumed small for the purpose
of finding Dy, Dao, Do. Let them now be regarded as finite; then the determinant
on the left of equation (46) will be the value of
J (7 — DywE’ — Doron’ — Dont’).
Multiply the first three columns by —& —y, —@ and add to the last. We then
find, by means of equations (45),
J (7' — DywoE’ — Down’ — Df’)
= (7 — Ayo — Anon —AonS) | & + roo, a’ + a dy, a” + a dun, a” |
B ote B dro; Isy SF B' duo; [x ae Bidens Be
yt" dw, +7 "da, oY +7",
6 + 6'di, +0 du; 8 +8'"du, &”
Multiply the last column by do, dno, @o and subtract from the first, second and
third columns respectively. We then obtain the important equation
, ; ; M
T = DyoE — Daon — Doo = 7 (7 — dio& — dow — dons) soanuasadcacesd («!i/)).
This theorem is the generalization of a statement by Prof. Elliott (Proc. Lond.
Math. Soc., Vol. xv, p. 147) made with reference to two independent variables when
the linear transformation consists of a cyclical interchange.
§ 26. Now
/ f ’ va 0 4
Vy=(7 F Dioo& — Doon — Donk ) & 7 Dw) .
Therefore the transformed expression for Vy is obtained from
1 , / , / Or’ uw Or , OT uw Or
vad — Dyk —Daon — Do f’) (ae — Dae) (a —A Aare A on A =
by expressing it in terms of & », € The first factor is transformed by equation (47)
which gives
/ f , lA M
LD — DywE — Dawn’ — Din =F (r- Ayo — Ano — don).
Again, from this last equation
pom T (Palit aga)
INDEPENDENT VARIABLES. 275
where 5 a e are to be obtained from equations (45). These equations give
‘a m OT\ 0& nm OT\ On Te Te OTN GG
1=(a+a se )ae + + (a! +a a) ae t (« +4 at) ae”
/ we Or)\ 0& 7 yy 1 OT ag
0=(8+,'"" ( fe =) ae +( ;
8 B ae) ae’ + +( 8’ +8" in) OE \8 + p" 5t) é ae
= ye gm 2) BE (pg on BVO (rg aw Or) Of
0=(y+y a) oe tv +9” mee +(y +9" 56) ag
Now let
i — a + on OT a’ +. m OT al’ ae i ot
a | oF’ en’ 0g
Me OT nor ss ite!
+e" =, + p's
at Se ee a
m” Or , m” Or mw Or
ere: ae? bia Soret of +y¥ at
iy On, Are OT,
Hee T an SOE?
and let @, A’, A” denote the minors of the first row of J; therefore
pete es Ob ees
ee! ae re Ani ae’ eine
Hence
Or’ M rn iy
I (je—Dw) = 7 fi \a (eo dw) +B (=- day) +2 (3 og - den)
M r) a
mk =F th Dig ae don
| vet OT wi ae r 0
+8", A+R", BRS
wt Or (jie mt za: se uw Or |
M | or Or Che
aay ae ti: Bn a0 ae do |
B+B' dm, B +B" du, B’ +B’ dun
yt dw, +9" du, ¥ +7'"dun |
M ”
=7 {4 (F-dw) +4’ (3 (je— dns) +4 (F-an)| pers (48)
where A, A’, A”,...
Wits SAVANE
are the minors of J.
Parr III.
36
276 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
The Page, expression for G is therefore
, (OT n(@
a a dro & — doom — eon g) \4 @ = din) +A (i — da) +A Ga a dn.)
= wid V,+AV,+ A’ V,),
or, in the previous notation,
2 (A Vik Ve ACV ee, Arcee te anes (49).
Similarly the transformations of Vy and Vz are
= (BV, + B'V,+ B’V;), = (CV. HO, Clie ee (49a).
; , (Or
Again Oxy=& & = Daw) >
and the transformation of xy is therefore
me &- Dayo) (A Aa — a" 5)
= aC +a'nta’f+a’r) \B (F — dh) +B (= = day) + +B” (= ag tha)
as in equation (48),
1 wn , aw a mr wt
= {(a +a!” do) E+ (a +a doo) 9 + (a + don) § a" (7 = dio E — dnon — don £)}
(n(n) +7 da) 3 Gaal
Now let G,, G, G; be operators defined by the equations
Or a
G=€ (ae = de) = pd por Od par’
0 )
é 0
G,= iS (Fe — dae) = Srey ae
5 p
where in each case p+q+r>1.
Then the transformation of w yy is
1 mn ’ mr / a” wm A
7 {(a + a” diy) BG, + (a! + a” do) B’G, + (a" + a” dy) B’ Gs}
1 wae A , mn" Wa ” mt
ty (a+ dyo)(B ory +B’ wxz) + (a + 2”do) (Bey + B’@yz) + (a" +0” don) (Boz +Box)}
al” *
+5 (BV, + BV s+ BYVs) coeeeessssssee sonsssseeseessnessnecsssesseceansnnncscensanneasse (50).
INDEPENDENT VARIABLES. 277
If the function F' operated on is isobaric, and of weights w,, ™,, w, in first, second
and third suffixes, we have
GF=wu,F, G.F=wF, GF =w,F.
If further w, = w,=w,, since
B (a+ a” dy) + Bi (a! + a” do) + BY (a" + a” don) = 0,
we find for the transformation of wyy, a linear function of the operators V and wo, viz.
5 {(a-+.0'"dyy,) (B’ wny + B’ tye) + (a! +.0”doyo) (Beye + BY toyz) +(a” + dy) (Boze + B’w,,))
“ur *
+ - EN FY BPO arose gh tv at net ies a IS SD srs ca le (51).
The transformations of the remaining operators yy, wyz,... can be written down at
once from this last expression.
>
§ 27. For example, suppose a cyclical interchange is made from z, a, y with 2
for dependent variable to «, y, z with a for dependent variable. Using the symbolical
notation, let
Vi = (= 20& — 200) (S¢ — 210) 5 Vo = (€— 20& — 2019) (5 — 201) 5
Way = E (F, — 2) 5 Oyxz = 1 (& — 2);
Vy =(& — Xn — nb) (E, — U0) ; Vi! = (€ — aon — nb) (Eg — 2m) ;
@yz =) (& — fin); Oz, = €(E, — 2p).
The scheme of transformation is
X=0.2+1.y7+0.z2,
Y=0.2+0.y+1.z,
4=1.¢2+0.y¥+0.z,
so that M=1, and
df= || (0 1 | =—2.
The transformations of the operators are therefore respectively, if for simplicity we
suppose them to act on pure isobaric functions with equal partial weights,
ie, 2 s
I Ber Wy — ea a) se nel e oe See anaes aenrecs coed Saigeape sie wa ran vad=~ <~aNMawenaps (52a),
210
es aa
Vig = Aa Vi Binlelele(eleie'e:wia-v.0.u,siniu(dinie’em/ela'a'aieie elueisWin'a'neieinia wala bala\viow « ntle'a'a a\plein 6/c\a'n' erute ela s’a setgine ae (52b),
, 1 9
Myz ee Wyz TOT errT ere ee ree eee eee eee eee eee eee ee rr (52c),
, 1 1 )
On, = — Zu (20 Vi- ZV) = Zao {210 (— 20) Wyy + 2)» Zo, » Myx}
2 uot ; 52d
=— a (2 Vi — ZV 2) + e EROS e ir) ) Bacdnes Jodngnace cece epireeeeaGal scr ene (52d).
36—2
278 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
§ 28. As an application of the last results we may employ them to prove two
formulae which are fundamental in Prof. Elliott’s paper “On the Linear Partial
Differential Equations satisfied by Ternary Reciprocants,” Proc. Lond. Math. Soc., Vol. Xvi.
Let F(Zpq,-.-) be a pure function of the differential coefficients, and let w be any
number. It is required to evaluate
é F fa) ins
alee ee Eb
where in the differentiation with respect to aw, it is assumed that a, a», @,,... are
regarded as constant, and, in the differentiation with respect to a), @ , ©», @y, ++. are
constant. We shall, for simplicity, assume F to be homogeneous of degree 7 and isobaric
with equal partial weights w.
Now the change from a, yz to z, wy is the second cyclical interchange from a, yz;
therefore by making suitable interchanges of letters in equation (43b) we have
Vy Zio
1
” 1 be ——w'y
F (2p9; = (— 1)7 tre ° TE Xo LT Gina S06
where V,’ and @,,’ are the same operators as in the preceding section. It may be
remarked that V,’ and @,,/ are commutative by §21. Therefore since fo=—
ol
0 F pot w 1 1 Do , - Shee a. ==).
a = = | ab va ie Wy: (= 1)§ vat i—we me TM F (Gin; =)
La Toy 01
by (52b) and (52c).
Similarly
0 /(F 1 1 ; a ea
aig (Aut) 7 age (Tag) OH OD garam HH Oa F Gea)
1
= — Fp Oval pas mee)
by (52c).
These are Prof. Elliott’s formulae.
§ 29. The theory of cyclicants is a generalization of the theory of ordinary
reciprocants; in the case when there are two independent variables it plays a part
which has the same reference to the theory of surfaces that the ordinary reciprocant
has to the theory of plane curves. But the ordinary reciprocant may be looked at
from another point of view. Regarding y as a function of «, let us suppose & 7 to
INDEPENDENT VARIABLES, 279
: : a 1 dy — 1 ae
be corresponding increments of « and y; then writing a, for — —” and A, for — ;
n! da ni dy"
we have
N= GE + a€?+...,
E=A,.n+ Asay’ +....
The second series is that obtained by reversion of the first, and a reciprocant may
be looked upon as a function of the coefficients of a series which is unaltered in value,
save for a factor involving a,, when the series is reversed. From this point of view
the generalized reciprocant may be defined in the following manner. Using the notation
tt "i . OPt+ItTy
of § 16 let u, v, w be functions of a, y, z, and let Gpgr Aenote ——— —_____
Pp: 4:7! dxPoytoz
1 opta+ry
pi qi r! durevidw"
and let A, , denote Then F'(Gpor, Bpgr, Cpgr, ---) Will be a reciprocant if
F (Agger, Bygr, Cpgr, +++) = PF (Gpqr, Bpars Cpars -++)s
where w is a function of first differential coefficients only. The function will be called
an n-ary reciprocant if there are mn independent variables involved, and F will be a
pure function if it is free from first differential coefficients. This kind of reciprocant
may also be regarded as a function of the coefficients of series which is unaltered, save
for a factor, when the series are reversed and the coefficients of the reversed series are
substituted for those of the original series.
Sufficient conditions to ensure the permanence of such functions, when pure, are easily
obtained from the results of § 16.
Reciprocants of the kind here considered have been discussed by Prof. Elliott* for
the case of two independent variables. The conditions here obtained for n variables
agree with those obtained by Prof. Elliott, who however does not examine into the
question of the independence of his conditions.
Suppose F (dimn, Bima, Cimn,---) to be a homogeneous function of degree i and
isobaric with equal partial weights w. Then
ci);
F (Aina; Berns Ch ce . -) = Jive e7[U, Vs W) eM eM es
F (Adimn aF Blinn 5F Commas A’dimn sf BOimn 3° CCimn ’ A Cian 3 B’bimnn 30 ClCmaa . .) at (53).
The function will therefore be permanent in form if it is an invariant of the system
of emanants
Aso E? + Aya? + oon F* + Gro En + ..-, (Sey a eee ts)
Doo E? +... Osteo se was "a. \P. sae eee (54),
Cana E* + -: |
* «On the Reversion of Partial Differential Expressions with two Independent and two Dependent Variables.”
Proc. Lond. Math. Soc., Vol. xx1r. pp. 79—104.
280 Mr GALLOP, ON THE CHANGE OF A SYSTEM OF
which remains an invariant when Au+pv+vw, Nutpety'w, Mut+p'vt+v’w are sub-
stituted for u, v, w, and which is further annihilated by the various operators [U, <],
[V, 2], [W, 2], (U, y],.... The operators as defined in § 14 contain terms with
differential operators involving ¢; such terms will of course be omitted here.
It is obvious that functions which satisfy the conditions just laid down will be
unchanged or at most changed only in sign when wu, v, w are interchanged; and that
they will be unchanged or changed only in sign if first, second and third suffixes are
interchanged. Such functions therefore, if homogeneous, will be of equal partial degrees
IN) pers 'o=*> Opors ==> Gpgrs =-- °
When linear functions Av+pu+vw,... are substituted for uw, v, w im a combinant
the function is multiplied by the 7th power of
A, Bb Vi;
uy ’ ‘¢
| Ne) a ee.
Ie ise yp’ |
where 7 is equal to any one of the equal partial degrees of the combinant.
In the case of the reciprocants here considered the determinant is
| ae Be Cae
Wee Peers Bere?
| A”, Ba (64
which is equal to J*; and the determinant is equal to J” when there are n inde-
pendent variables.
Reference to equation (32) then shows that the factor for a reciprocant of equal
partial degrees 7 and equal partial weights w is
(- 1 ye Ja) i K (- 1 ynt
Jnirw tru
so that
(- 10}
7 (pers Opgrs Cpgrs +++):
F (Aper; Bears Ohren no0)) =
§ 30. One example of such reciprocants is easily seen to be the eliminant of the
quadratic emanants just written down. For this is an invariant of the required type,
and since it involves no differential coefficients a,,. for which p+q+r> 2, it is obviously
annihilated by [U, ], [V, a],....
This example for the case of two independent variables is given by Prof. Elliott.
The eliminant in this case is
(Gao, — boa.) (@irDo2 — Oe) — (GaDo2 — boy on)*-
INDEPENDENT VARIABLES. 281
The partial degree i=2, and the partial weight w=4; therefore by the last result of
§ 29 this expression is equal to
J* {(AwBy — By An) (AnBu — Buda) —(AnBu— ByA o)*};
where J =ab'—ab.
The invariant character of the function just considered corresponds to a simple
theorem in the theory of the reversion of series. Let
UV = MoE + Ayn + Ay E? + AnEn + dyn? +... ,
V = DdyoE + dan + byE* + buEn + bun? +... ,
and suppose that from these equations are deduced the series
E= A,ut Ayv + Anu? + Ayuy + Ayy?+...,
n= By + By + Bov? + Byvy + Boy? +....
The theorem then states that, if the quadratic terms in the original series have a
common factor linear in &, », the quadratic terms of the series obtained by reversion
will have a quadratic factor linear in v, ».
The theorem is easily proved independently. The property referred to is one un-
altered by a linear transformation, and therefore we may take & for the common factor.
The method of successive approximation then shows at once that the quadratic
terms in the last two series must have a common factor.
$31. The conditions for pure reciprocants laid down in § 27 although sufficient
are not independent. This statement can be proved by forming the alternants of various
operators. If we assume F to be an invariant of the system of emanants (54) which
remains unaltered save for a factor when Au+pv+vw, Nut+pouty'w, Nutpv+ vw
are substituted for uw, v, w, then it can be shown that annihilation by one of the
operators [U,«#] will ensure annihilation by all the others. In fact since F is invariant
when Au+uv+vw is substituted for u, therefore # must be annihilated by the operators
which in the usual symbolical notation will be denoted by [V,] and [W,], so that
a
or dla
[Vi] = [Dak + ...] = 6 =P ecc'5
[W,] = [Caaf + ---] = Caw 5 Pal
200
Similarly F must be annihilated by [U,], [W.], [U3], [Vs].
Now, in the present case, we have
[U, #] =[U,U¢,] + (U.V:,] + (Us We);
therefore
(U, «)[Vi]—[VA](U, 2] =[02V%,] * [Vi] — [Vi] * [U, Ue,] — [Vi] * [02%] — [Vi] * [U,We,).
282 Mr GALLOP, CHANGE OF A SYSTEM OF VARIABLES.
7
And [Varese a, [Gi Uy = 72 1) v=o.
Therefore
[U, #)[¥)- (VALU, #]=(0,Ve)-[0e,%a + U,Ve.1-[%%e]—-1V2 Me]
=—[V,U;,]—[V.V:,]—[VsW,]
= = [Vy Gl) esacesctes bisa ctissinetow ios soot sets dasoeese eoee doseee (55a).
Similarly
LU el Wolter | Weta oa eon es (555);
and other equations can be written down with y, z in place of #, and with U, V, W
interchanged.
Again Oy = [EU y+ [EV 9,1 + [Es Wag] ;
therefore
Oz, [U, 2] -[U, ©] Ory = (&,U;,] * [UU] + [& Uy, * [UV] + [60,1 * [Us We]
+ [&Vn,]* (V2V2,] + [&W,,] * [Us We]
—[U,U¢,] * [&U,,] — [U2Ve,] * [&V5,] -—[Us We,] * [EW ,]
= | 0,,Us, + Usp GUy)| +1EUe, Ved (660, Wel + | Uaze EM) |
+[0.26"_)|-[62 Cr)|-[b9 Gr» |-[ae cmp
=([U,U,,]+[U2V,,]+ [Us W,,]
8 1/0 I ea aN MR ne Sn hee (550).
Similarly
OF Oe 23 — NO a] Ore [kU alle oce ace sistent eee eee reece (55d);
and other equations can also be obtained with V, W in place of U, or with a, y, z
interchanged.
Equations (55a), (55b) show that any fuaction annihilated by [U, 2], [Vi] and [W,]
will also be annihilated by [V, z] and [W, x]; and then equations (55c) and (55d),
with similar equations in which V, W are written for U, show that the function will
also be annihilated by [U, y], [V, y], [W, y]. [U, 2]. [V, 2], [W. 2].
Hence defining a combinant of the emanants (54) as an invariant which remains
invariant when wu, v, w are replaced by linear functions of wu, v, w, we see that any
combinant of the emanants which is annihilated by any one of the operators [U, 2],
[U, y],... will be annihilated by all the others and will therefore be a reciprocant in
the sense defined in § 29.
IX. On Divergent Hypergeometric Series. By Prof. W. M°F. Orr, M.A.,
Royal College of Science, Dublin.
Addition*. [Received 3 April 1899.]
13. WE have obtained the complete solution of equation (3) in divergent series
only in the case in which m=n+1. It has been shown by Stokes (Camb. Phil. Soc.
Proc. Vol. vi.) that in any case in which m 00
yi? | OPT CaUll ale (Q) ints cats vates nessa teMemee vac nee naanese (70),
Lee
where (v) when v is small is of the order of a power of v, and when »v is great is
approximately equal to evi", provided the argument of v lies between — 3/2 and
+37/2, and ¢ is an indefinitely small positive quantity. It should be noted that the
limitations placed on the argument of « in the integrals which have been expressed
by divergent series were only imposed in order to make those series arithmetically
intelligible in the sense of equation (28), but that while the integral forms are retained
no such limitations are necessary. We may accordingly suppose that in (70) the limits
of the argument of y are still further extended to —27 and +27; for in evaluating
Fig. 6.
the integral when the argument of y lies between —37/2 and —27 we may change
the lower limit to a point whose argument is — 37/2 without altering its value, and
so have all along the path of integration y(v)=e-"v*@-»), As regards the path of
37—2
286 Pror. ORR, ON DIVERGENT HYPERGEOMETRIC SERIES.
v a consideration of (37) of which it is a limiting form, and of Fig. 3, shows that if
the argument of y lies between 0 and 27 the path must be such as ABC, or A’BC’,
(Fig. 6), while if the argument lies between 0 and —2z7 the path must be such as
the image of this with respect to the axis of real quantities.
We can take y so great that wW(v) is as nearly equal as we please to evi“ °?
for values of v for which e~”’” is as small as we please, and accordingly so that the value
of (70) is as nearly as we please equal to that obtained by replacing y(v) by this
approximation. We would then have to consider an integral of the type
[erred nce a my
ey
This is a particular case of another with which we will have to deal, viz.:—
hea)
| GME OG IAL weet £1 NG Lis Ee eee (72),
ey
s being a positive integer, and e an indefinitely small positive quantity, the argument
of v at the infinite lmit being zero, and the argument of y lying between —(s+1):/s
and +(s+1)7/s, the path of integration thus bemg permitted to make round the
origin a number of revolutions determined by the initial argument. See Fig. 7, in
Fig. 7.
which ABCDEFG represents a case in which the argument of y is positive, as we will
at first suppose.
The value of e*’-""™ is stationary for values of v given by the equation
G8 UPR 0 cine waters sansfenie'dieceiseeree ore onenenoeetes (73);
let v, be that root whose argument is s/(s+1) times the argument of y, and thus lies
between 0 and +7. It may be noted that if the point corresponding to any other
solution of (73) lies in the region traversed by the path of integration in (72), the
real part of v at any such point is very much greater algebraically than the real part
of v,, and therefore the modulus of e*’-*"™“ very much less than that of e~-wn,
We will now suppose the path of integration to pass through »v,, and consider separately
the two portions of the path from v, to » and from ey to v,. Considering the former,
let part of it be a straight line starting from », in a direction whose argument is
Prop. ORR, ON DIVERGENT HYPERGEOMETRIC SERIES. 287
half that of »,, a direction which makes an acute angle with the positive part of the
axis of real quantities and an obtuse angle with the line joining », to the origin.
Expanding v~* in powers of v—v, we have for points on this line
sv + yeu * =(s +1) 0, +8(8+1)(v —,)/20, + R,
where mod. R |