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OF THE
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PHILOSOPHICAL SOCIETY, @, dye
4
VOLUME XV.
CAMBRIDGE:
PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS;
AND SOLD BY
DEIGHTON, BELL AND CO. AND MACMILLAN AND BOWES, CAMBRIDGE;
G. BELL AND SONS, LONDON.
M. DCCC. XCIV.
ADVERTISEMENT.
Tue Society as a body is not to be considered responsible for any
Jacts and opinions advanced in the several Papers, which must rest
entirely on the credit of thei respective Authors.
Tue Sociery takes this opportunity of expressing its grateful
acknowledgments to the Synptcs of the University Press for their
liberality in taking upon themselves the expense of printing this
Volume of the Transactions.
= Oe
613419
tei Soi
CONTENTS:
PAGE
I. A solution of the equations for the equilibrium of elastic solids having an aane of
material symmetry, and its application to rotating spheroids. By C. Curex, M.A.,
Hellow of ‘Kane's College: (Blates Te WUE) aiecses.2 seceeeee « on- seaessscessenes oncceseeecceeecee 1
MieVon-Luclidian Geometry. By PROFESSOR CAYLEY .........<cs202csccestocensesccnsea. seaceeseeaes 37
Til. On the full system of concomitants of three ternary quadrics. By H. F. Baxer, B.A.,
Hellow Ole Sty ohnis! Collemet nner. s<s8rs.cneccetetonec otat cos += as <1 SER oARCOR ER eee res sete ess 62
IV. On Str William Thomson's estimate of the Rigidity of the Earth. By A. E. H. Love, M.A.,
Hellowsorest ohn's Colleges ees. csac2s2 sess teeta eee one + s.c2h.ss0 s05 eRe eee nates eee 107
V. On Solution and Crystallization. No. Il. By G. D. Liverye, M.A., Professor of
Chemistry in the University of Cambridge. (Plate III.)...............c..c22ceescnecnsenees 119
VI. On Some Compound Vibrating Systems. By CO. Curr, M.A., Fellow of King’s College.
PESISTLERPRINV 5 eV is) tere oe ns ah tee ace cine ois ness cle ace OR EE MMM Tas oan s ~2 oa au asciec sa: CMaRE Re ER EA Toveaais 139
VIL. On Pascal's Hexagram. By H. W. Ricumonp, M.A., Fellow of King’s College ......... 267
VII. The Self-Induction of Two Parallel Conductors. By H. M. Macponarp, B.A., Fellow
ime NC ON CIC gece e mee cies cess ce Sen ae eRe ee nee e oc ins Wa c0 5 5 ORE RE EE Coras's = Scien 303
IX. On Changes in the Dimensions of Elastic Solids due to given Systems of Forces. By
C. Curez, M.A., Fellow of King’s College ................ Ree deo tciec0 20 dice. sae Gag pc eEERRN: 313
X. The Isotropic Elastic Sphere and Spherical Shell. By C. Curer, M.A., Fellow of King’s
iCal lees ers ereereee steele i satofs tics ois cS PRIS Me See ete swe ses sak alnsine oc atte ame ese Pam Mee meen eaees cone enseed 339
XI. On the Kinematics of a Plane, and in particular on Three-bar Motion: and on a Curve-
tracing Mechanism. By Proressor Caytry. (Plates VI. VIL.)................ccceceee eee 391
XII.
Examples of the application of Newton’s polygon to the theory of singular points of algebraic
Junctions. By H. F. Baxer, M.A., Fellow of St John’s College ...............0..00000- 403
I. A solution of the equations for the equilibrium of elastic solids having
an axis of material symmetry, and its application to rotating spheroids.
By C. Cures, M.A., Fellow of King’s College, Cambridge.
[Read Nov. 25, 1889.]
Selo edie t.. ¢ denote the stresses and wu, v, w the displacements in an elastic
solid of uniform density p, acted on by an external system of forces X, Y, Z, the
three internal equations are of the form
diz de dt. = du
da * dy * ic | caer
See ee ee ee ee i
If the axis of z be an axis of symmetry in the material, the stress-strain
relations are*
du dv gw dv dw
agen wt ie @=4a(z + Salt
dv ,dw dw du
=f § gt t+) F tas te =4(7 +9, “), Zi ea ee (2).
, (du dv dw du dv
t=@ (Gn tGy) to ae bey =t(3 + de)”
When the solid is in equilibrium in the absence of the bodily forces X, Y, Z,
substituting in (1) from (2) and arranging the terms we get
du a dw
f£V'ut+ (d— f) 9 =, pte+F) +(d+d’ —f- f) in > aa ntaleretelaisistatetelacieleve (3),
. d?v dw =
£V v+(d—- f) 7 iat +P) q, tata — —f-f') dyde.= OUR aaeemnaecee (4),
dw *
V*w +(c—2d—d’) ese) th U hanna see (5):
* Saint-Venant’s Théorie de VElasticité des Corps Solides de Clebsch, p. 77.
Vou. XV. Part I. ]
2 Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
where as usual
_du dv, dw .
= ie + dy + ae ain niu n/elaraleretataloi cetera’ sia?ule mia\elateteinteve’a ie a’eieleiatalelsrete (6),
a: rn
4 =ae Tap cde G cules tial getemiciesietesvateieis argiag seco eae i):
Differentiating (3) with respect to # and (4) with respect to y, then adding and
arranging the terms, we get
[ee +f) V?+(d—2F- £) za S= ce f—d—d’)V?+ (2d + d’—2f-f) | NG:
Differentiating (5) with respect to z we get
» @6 4 , a | dw _
(ata) a+ [av +(e-2a-a/) 7, wee secs sie sendencaecanemaeee (9).
Combining (8) and (9) we find for the equation from which 6 or = must be
derived
2
ja (2f+£) VU. V? + (26+ £’) (e—2d) —d@’ (24 4a’)} V? =
ee.
+ {ed —(2F +f) (ed) +d/(2a +4’)} al = 0.00010).
2 \dw
dz
In this equation it is obvious that 6 may be replaced by ata
§ 2. Confining our attention to solutions containing only integral powers of the
variables, it is obvious that (10) is satisfied by any term the sum of whose indices is
less than 4. For our immediate purpose we do not require to carry the expression for
6 above the terms of the second degree of the variables, and so the equations we
shall really have to do with at present are (8) and (9) not (10).
All possible terms not higher than the second degree are included in
é = 2. ai A, a+ Bly Sp ae tA,, (227 — a — y+ 3A,, (x* a) y) +F, (a? + y’)
+ 6B, wy + 3A, 22+ 3B, YyZ....c000 (11),
a = similax expression with dashed letters...............scceescees (12),
where A,,, A’, etc. are constants.
The first of the two suffixes attached to a letter indicates the dimensions of the
corresponding terms in the expressions for the displacements. A second suffix has not
been attached to F, and F, because these constants in consequence of (8) and (9) are
immediately connected with A,, and A’,, by the relations
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 3
(a—2f—f’) A,,+2(2f+f) F,=(2d4+'—2F-f) A’,, + 2(2f+f —d—d’) FY...(13),
(ed Ae eh (e =e yA eee oa OO) he ...5 02 deces- (14).
From (13) and (14) we could substitute at once for F, and F,, but it will be
more convenient to retain them at present.
Integrating (12) we find
w= A’ 2 a5 Ars ue ar By 5 FAL oe ats le (32° al we oie YZ)
+ 8A’,,2(a°— 9?) + 6B, ,wyz t+ $A’, oz? + 3B ys + Fiz (2? + y’)
2B) (Ga) aebciaos 200 nochcenosaqbLoonadeosétoascneesopeasneeecacc):Soeoc” sec dob ceneeeceBrORECeE (15);
where
$ (@, 9) = 4,0 + By + yaty + 6, (a —y') + 6 (2 +9)
+ n, (a — Bay") + 0, (y® — Byx*) +r, (a? + Bry") + w, (y* + Bya’)......(16).
Here a,, etc. are new constants; and all possible terms of less than the fourth
degree which can appear in the value of w are included.
On account of (5) we have the following relations between the constants occurring
in (11) and (15):
(deed) A+ (C= dd) AREA iG = 0) iineies was op segtasete (17),
(dicid))-A\. 4a (Gi idl —.dijetae nN, WOE. oem terrae es. (18),
(elec (Ge dd) ade — Ol. See scce.ckenecteeenens- (19).
If for shortness
A,=—(@4+f)A,,+(f+f —-d-d)4’,,,
A, = (bf) (A, =F 6A,,, A AC Ss dit i) (4% < 64’, i 2F), (20)
A,=6(£+f) (B,,,-B,,)-—6(4+4)B,,,
A, =3(f+ f’) (Als, Si A,,,) CAC tas qd’) Asp
then substituting from (11) and (12) in (3), we have to determine w from
Gu du CH= Se ae oe .
Seas a) $A Se = At Het Ay + Agrees (21).
A complete solution, so far as terms of not higher than the third degree are con-
cerned, is
wu=(A,+ Ae + A,y + 4,2) 2/24
+ a,0+ By + y,2+ 4, (a — y*) + Bey + yz + €,y2
+e + by" — £4 (n, + €,) 2 + 4, (x? — Bay’) + B, (y* — Bya*) + ysry2
+e,0°y + ny — fd (e, + 3n,) y2* + Gy'a + O,2° — FA™ (€, + 30,) we"
A Ng@?Z + pigyZ — FEA (Ag + fg) Devereccneecceenececencceenssesssseeeaeneseneeenens (22).
1—2
Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
Similarly if for shortness
B, = (f+ f') (B,,— B,,)-— (a+) B,,,
= =(f+f){A,,-4',,+ 6(4,,-4',,) -2(R-F,)} +(d+4)(4',, + 64’,, -2F)), (23),
B,=6 {(f+f) (B',,-— B,,) — (d+) B,,},
B, =3(f+f)(B,, -—B,,)-3 (d+) B,,,
we find from (11), (12) and (4),
v= (B+ By + Ba t+4Bz) 2/24
+4,'0+ By +42 +, (a — y') + Bay + y) a2 + €y2
+, 2° + Ey? — fd™ (n,' + &') 2 + @,' (x — 8ary*) + By (y® — Bye”) + y, xyz
+/a°y + yy? — fd" (e, + 3n,) yo? + 6, yx + O,2° — fa (6 + 30,') v2?
PALO 2+ ph, YS — FER (Ay Fay) 2 ve ceadowsceas techie deceitsenensitsnonwoseseanteeeem (24).
In consequence of the identity (6) the following relations subsist between the
constants in (15), (22) and (24):
Gi Bet Al eg — Ag gs cine so nov'os cnatiammee eee saps cand orcssacn= ace ee ey eee (25),
2a at By AY, F— Ay, = Ov ccc cnmemcmncerenectsheasy cerserseesaeannessteensecesenme (26),
Bow 2e 20" ASB eS Be ='0.4 Bye MnO ee cok ORed «cos; eee (27),
Vg HG A gg = Aig g HW wiesa ss one popcioae gp emapae ovbieep aegeis sage sisics es nelson hae penaee (28),
3a,+30, —3f,' +¢,'+4(4,,—A’,,—64,,, + 64’,, — 20, + BF, )SOvssiee. dese (29),
— 3a, +3, +388, + €+4(A,,-A’,,+64,,—64’,, — 2F, + 2F,') =0......000008 (30),
td (A, +B) — Al, + 4,5 —f4 7 (GBB Gy FON, HO... oe ec cscn-cevecrooneet (31),
€,— 3B, = Sa, 6) +83), — SB. S08 were ster eteett es censsabent oad eae (32),
Da + yy BAL, = BAe = Os see naive san tepeseeee cenouneee test oven - i See oon ee (33),
Yet Big, + BB, — SB By, = evn tevesbacaeteecet haste peetenseccsccheast.<eccttccsesageeaeee (34).
§ 3. Multiplying (31) by d/f, and adding it to the sum of (29) and (30), we
obtain an equation identical with (13). There thus exist between the constants of the
solution only 14 independent relations, viz. (14), (17), (18), (19), and (25) to (84). Since
65 constants occur in the solution this leaves 51 of them arbitrary, to be determined
by the surface conditions.
Certain of these constants fall into sets which seem fitted for application to different
problems. The constants of any one set are associated with one or more of the constants
occurring in the expression (11) for 6. The following table gives an analysis of the
constants :—
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS.
TABLE TI.
|
| Degree of
|. termsin | Associated Representative Connecti U iated
anleomment | Constants Constants ae ee Carseat
stants occur _
First | AN VAN a, [sk Zilog (25) a, a,’ B, B, N 1
A,, Ar, Ve €,, ra A, | (17), (28)
Second A,, A a, ish Ne A,, | (26) Ye Yo €,)) Ny &
B,, B,, a B, ea By, (27)
cA At Anya. Nie 4 in 4 (14), (29),
Ei ee a ea een ;
Third | A,, A’, 4%; As As A,, (18), (33) | 7, 7, 9, 9, r, Me
B,, By Ys Fs bs Bye (19), (34)
B,,. Bs; a,’ B, €; & By | (32)
There are thus 47 associated constants, of which however only 33 are independent,
and 18 unassociated constants. The associated constants all occur in the expressions for
strains causing a dilatation 8; while the unassociated constants answer to strains in which
the dilatation is zero.
§ 4. By applying the solution consisting of (15), (22) and (24) to the problem of
a straight cylinder of uniform elliptic section free from force on the curved surface, it
may be demonstrated* that Saint-Venant’s solution for an elliptic beam acted on only
by terminal forces is the only possible one when terms of the fourth degree of the
variables # and y, measured in the cross section, are neglected. The constants entering
into the solution are those associated with A,,, A,,, B,,, A,,, and B,,, and in addition
the unassociated constants ¢,, y, and y,. It can be shown explicitly that the conditions
on the curved surface require every other constant to be zero except certain of the
unassociated constants appearing in terms of the first degree in the displacements. The
terms however in which they appear merely represent rotations of the solid as a rigid
body about the rectangular axes, and so do not properly refer to the elastic problem,
For the same problem in the general case of any form of cross section the only
constants left after satisfying the conditions on the sides are those associated with
A,,, A,, and B,,. The solution agrees with Saint-Venant’s, which is thus proved to be
complete so far as it goes.
§ 5. The proof of the completeness of Saint-Venant’s solution is laborious, involving
some heavy algebraic calculations. As it merely confirms results that meet with general
acceptance,—based it is true on somewhat insufficient grounds,—it could hardly be
* The method of proof is the same as for an isotropic beam. Cf. Quarterly Journal, Vol. xxt1., 1887, p. 89, et seq.
6 Mr ©. CHREE, ON A SOLUTION OF THE EQUATIONS
expected to be found interesting. Accordingly the first application I shall make of the
previous solution is to the problem of a spheroid of uniform density rotating with uniform
angular velocity about its axis of figure, which is also an axis of symmetry of the
material. So far as I know, this problem has hitherto been solved only for the case
of an isotropic* material, and in the paper referred to it was hardly attempted to
deduce from the solution the true character of the phenomena. Thus the results obtained
here may possess an interest even for those who are not professed mathematicians.
§ 6. If @ denote the angular velocity and p the density of the spheroid it may
be regarded as at rest, but acted on by “centrifugal” forces whose components, per
unit volume, are
X =o'p2, Y=o'py, GEV
In place of (3) and (4) we get, reintroducing X and Y and slightly altering
the form
du du a
Qf+f) Tathagata Gg Tot (E+E) oo +444) OY 4 wpe =0.. (8 a),
du dv d*v
+f) Fay +8 +ereey Ti +a ts +(d+d) ai + apy =O.....n(4 8),
while (5) remains unchanged.
A particular solution of these equations is
(ff Se
wee)
8(2f+f) ’
ya we py(@e ty’) Pr sidtdd sows dsp wine bids lai» oie ebay eh afoln meee (35).
8 (2f+ f’) |
w=0
The general solution is contained of course in (22), (24) and (15). It would however
be a needlessly long process to substitute the whole of these terms in the surface
conditions. A comparatively small number of terms suffice to give a complete solution.
As by means of these the surface conditions are exactly satisfied, the solution is on an
entirely different footing from Saint-Venant’s solution for beams, and the neglecting of
the remaining terms requires no justification. The only terms required are those of the
first degree depending on A,, and its associated constants, and those of the third degree
depending on A,,, A,, and their associated constants. Further from the symmetry
around the axis of z we may at once assume
By = a,
Ns, = €,, — ee = (he Beemer cere een ere eee eesseesseesesess (36).
a,=P,=A,,= Aas =)
* Quarterly Journal of Pure and Applied Mathematics, Vol. xx11.1, 1889, p. 11.
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 7
Thus the solution we propose to use is in full, substituting 7° for «? +4’,
6=A,,+4A, | (22 —27) + For" — fot pr?/ (QE +14’) .....cccccesecen evens (37),
L Vv U 9 = / , / , ’ /
Ste ~=a,+ 6,7 + 32d” (f+) (A,,—A’,,,— 2F, + 2F,) +(d+d’)(A’,, — 2F,) — 8£9,}
Sp or (OR Iasi docs (38),
Wi ANN te eA. C2 (ee a) PD. 2EY ca cece sean os theteteseeies cone (39).
The constants appearing in this solution are connected, as shown in the table of
constants, by the relations (14), (18)—taken as more convenient than its equivalent
(31),—(25), (29) and (30). Owing however to the relations (36) the relation (25)
simplifies into
Gh CA ALL) Meters wectactehacias tic sete saccie asian (25 a);
while (29) and (30) both transform into the single equation
SONA Aeneas Ol sercsiaan-saicnalessiessienide sas (29 a).
§ 7. Let the equation to the spheroid, prolate or oblate, be
Gea Ce cs) ae Gd: al Log Mahe eno: LBRRO RS arco One ee PRE (40).
The direction-cosines of the normal at the point , y, z are in the ratio a*w : ay : 6.
Thus the conditions for a free surface are
Gal (Ghat Yb eth Zt eet Oe stents ssn cninceieoe sess oores eapseas ae (41),
a (7 Pe iT 8) ao 7 == Re (42),
Cie (Giese tia) tt mest — OS ae casncssesaeasiecee sso oieo Rinne (43).
The first two are however here identical as is obvious from the symmetry.
The relations between the strains and stresses are given in (2). Employing these
it will be seen that in the surface conditions the terms containing or the constants
associated with A,, and A,, are of the third degree in the variables a, y, z, while the
terms containing the constants associated with A,, are only of the first degree in the
variables. At the surface however the relation (40) holds; thus the terms in the surface
conditions containing the constants associated with A,, can be made of the third degree
by multiplymg them by a“r’+c%c* which is there identical with unity. Doing this,
and equating separately to zero the coefficients of wr? and wxz* in (41), we find
fi enh 2 7 } wpa’ (Bf + 2f’)
2(f+f')a,+d@'A’,+a (6f+ 4f') 0, + a’d’ (F, 44) = Ter somone ;
2(f+f)a,+d'A’, +o (f+f)d" (f+) (A,,—A’,,—2F, + 2F,)+(d+d’) (A’,, — 2F,') — 8£0,}
Fed Ata (@+f)4,,—4 0-22, 4 28,) 4d (Ay, 27) —8f0,} =0...... (45).
Treating the surface condition (43) similarly, and equating separately to zero the
coefficients of zr? and z*, we find
8 Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
2d’a,+cA’ +e {(f+f) (A,, — A’, — 2F,+2F,)+d' (A’,,, — 2F,) — 8£8,}
aw pad’
Fre ry 40)
2d’2,+ cA’, ,+cd’d" {(f+f) (A,,— A’, —2F,+ 27) + (d+ a’) (A, — 2F,) — 8£0,}
+ce A’, ,=0...... (47).
+a {e es 7x 4$A’,.) af 4d0,}
The equations (44)—(47) combined with (13), (14), (25a) and (29a) are obviously
sufficient, and no more than sufficient, ie determine without ambiguity the 8 constants
of the solution, viz. A,,, A’,,, A F,, F, and @,.
1,0? a, 3,0? as
§ 8. The actual determination of these constants is a somewhat laborious process,
and presents no novel features. Further a statement of the values of the individual
constants seems hardly likely to be of service in the solution of any other problem.
I shall thus not occupy space by recording here the values of the constants or the
algebraic steps by which they were obtained, but shall proceed at once to give the values
of the displacements. Their accuracy may be easily tested by reference to the equations
-which they require to satisfy, viz. (Ba) or (4a), (5), (41) or (42), and (48).
For shortness let
D=3e'f+ {ce (f+ f) —d”} 8e+4 2c°a*d™ {ce (2f + f’) —d’ (2d + d’)} + Sc'a* (2£ + f’)]...(48);
then the values of the displacements are as follows:
Diu’ D v D u,
wpr wp y ae
cf
==
—_) ; f’)-d = ;
rae ler ca a a Ee a*{e e+£)—a")|
— 32 [cd’d™ (2d—@’) + $c°d" (3f+4 2f) + 2c’a™ {c (BF + 2F’) — A”}]................--(49),
=tha'e? — 3h + 4c'ed™ fe (3£+ 2£’) — 2d”) + cia {c (8F + 2F) — A}
D ee (nll Tee ) did’ “ siti
= rale Ge _— x, 4 2
wot :| Jo'ea Prre ctr nate {fe (Bf + 2f) — 2d”} + 2c*a arse)
+r [oa + ca? \(c (f4")— 4 | +o(2e+f)}
+ 42° [1d {e (Bf + 26) (A + d’) + 24 (d — d))} + 20°C"! (OF + FY]. eseteeeees (50).
§ 9. The elastic constants occurring in the preceding solution are not those which
direct experiment would immediately lead to, and thus the application of the formulae
to a solid whose elastic properties had been determined by the usual methods might
be found laborious. It will thus be advantageous to transform the expressions into others
in which the elastic constants occurring are such as practical men may be expected to
become conversant with.
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 9
It is necessary of course to fix on five constants, and there is little doubt as to
what three of these should be. Suppose two straight bars of uniform rectangular section
cut out of the material, the axis of one of the bars coinciding with the axis of symmetry
of the material, while in the other this axis of symmetry is perpendicular to one of
the lateral faces. Let H and £’ denote the values of Young’s modulus for the respec-
tive bars under longitudinal tension, and 7, 7 the ratios of the lateral contraction to
the longitudinal expansion in the experiments determining # and J’, the direction in
which 7 is measured being perpendicular to the axis of symmetry; and finally let G@
denote the modulus of torsion for the first of the two bars twisted about its longitudinal
axis. Then the constants it is proposed to use here are #, EH’, y, 7 and G. The
notation is Saint-Venant’s, who has pointed out how the several constants may be found
by experiment.
Experimental methods at present in use ought to supply trustworthy values of
E, E’, and G with comparative ease. The determination of 7» and 7’ is by no means so
easy, and not improbably two more convenient constants might be selected. Still it
must be remembered that the strictures that have been so frequently passed on the
seemingly unsatisfactory determination of “ Poisson’s ratio” are really in the main directed
against experiments in which all substances, even hard drawn wires, are regarded as
isotropic bodies. There is no very obvious reason why satisfactory results should not be
obtained when observers take the trouble to find out what exactly are the quantities
whose magnitudes they determine with such extreme nicety.
§ 10. In Saint-Venant’s Clebsch, pp. 83, 84, are given the relations between the
several constants for the kind of material treated here. The following relations are in
part directly taken from this source, and in part deduced algebraically :
GG.
f= ,H/(1+7),
c= HW (1-7)/{E(L—7) — 2k'7,
d’/e = nh'/H (1-1),
(f+f)/e=tH/E( —7),
{e(f+f) —d*/e=f#/ (1-7),
ef/{c (f+ f) — d™} =(1—-7')/(1 +7’)
SS
—
Or
_
~
EB (1 ~1/) —2E'"\"
E*(1-7') §
§ 11. If now Da |
/ 2 yD E ‘ / ges Le
=4}(11 A a aa) \g- 20 +7’) + 4c°a 7p | ..(62),
the equations (49) and (50) transform into :—
Vout. XV. Parr I. 2
10 Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
ee =: =3(3 +7’) {= a } _ pag zo oe
—}r(1+7’) fat +ca” G + xa) + 2cta™* he oe =: aaa
ee; pe + Se +ca* alae Le +a ae AP OpAT CEE OC ReRAbeC HEC CHeaeE (49 a),
x w=— 2 =e len + Bay iE (1 — 9’) — 24"? +r} + Pt oe ae 9]
_ E (1 +1) +E 9) 5 {BL a1) (Bat) — AE + wee H
xf [BO +s ane 28 4 EE OD, gep20B B=B a9
From physical considerations alone we are led to treat D’ as essentially a positive
quantity. From (52) it is obviously positive when c/a is small, and if in any kind of
material it could change sign as c/a increased then a spheroid of this material could
be constructed such that all the displacements would become infinite however slow the
rotation.
These expressions it must be admitted appear somewhat formidable. It will be
found however that their length does not present an insuperable barrier to the drawing
of general conclusions. To permit the mind more easily to grasp these conclusions we
shall consider first some special cases of comparative simplicity.
§ 12. When terms in c* and 2 are neglected we get the following solution, applicable
to a very flat oblate spheroid,
v= OC? 3 4 af) at (1 +79",
E'(11 +1) (53)
es { eee ee ee | e .
2o'pnz '
w= EQi+7) {(3+7')a@—2(1+7)r |
This solution does not satisfy the equations (3 a), (4a) and (5), and there is no reason
to expect any approximate solution of the kind to do so; because while a term in wu
of the order x2* may be negligible when z is small, yet when operated on by = its
contribution to the equation (3a) is just as important as that of any other term in the
expressions for the displacements. It is thus impossible to test the accuracy of such
approximate solutions by means of the internal equations.
§ 13. It is well known that the distribution of electricity on a flat circular plate
has been deduced by a mathematical treatment which regards the plate as the limiting
form of a flat oblate spheroid. It would also appear that except near the rim there is -
a good agreement between theory and experiment. We are thus led to investigate whether
(53) may not satisfactorily be applied to the case of a rotating circular plate.
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. L1
The only way of testing the matter is by finding how exactly (53) may satisfy the
surface conditions for a right circular cylinder of radius @ and length 2c. These conditions
are the following :—
at, + yt. =0 when r=a, for all values of z between —c and +c, |
Tbirg +; Yty, = 0 ” ” ) BA
ty=t, =0 when z=+¢, for all values of r <a, | cao tie (Oe):
tee — 0 bbl ” >
Of these the first and the last, which answer to the vanishing of the normal stresses
on the curved surface and on the flat ends, are identically satisfied. This is not however
exactly the case with the other two, as the solution yields tangential forces of the
order za on the curved surface, and of the order cr on the flat ends. Thus while the
surface conditions are not all identically satisfied, they are approximately satisfied in a
thin plate, and the approximation becomes closer the thinner the plate.
It will be noticed however that if each term of the solution (53) were multiplied
by one and the same constant the resulting solution would satisfy the surface conditions
(54) to the same degree of approximation that (53) itself does. Thus all we are safely
entitled to assume is that (53), which gives very approximately the absolute magnitudes
of the displacements in a flat oblate spheroid, gives to a somewhat less close degree
of approximation the laws of variation of the several displacements and their relative
magnitudes in a thin circular plate. Considering that the volume of a flat spheroid is
less than that of the corresponding flat plate in the ratio 2:3, we should expect the
absolute magnitudes of the displacements to be decidedly larger in the plate.
§ 14. To derive its full interpretation from the solution (53) we require to know
something of the relative magnitudes of the elastic constants which appear in it. In
all ordinary elastic solids the constants ¢, f etc. can hardly fail to be positive quantities,
and the same is obviously true of EH, EH’ and G. It is conceivable that im some
exceptional substances » or 7 might be negative, though it seems a somewhat remote
possibility. If we assume here that all the constants are positive, then it follows from
the expressions in (51) for ¢ and d’/c that
Le)
E (1-1) > 22’
Thus in (53), u, must be everywhere positive and w everywhere negative. Consequently
every element of the flat spheroid, or of the thin circular plate, increases its distance
from the axis of rotation and approaches simultaneously the central or, as it may be
termed, “equatorial” plane.
Confining our attention at first to the flat spheroid, we notice that the centre of
an originally plane section perpendicular to the axis of rotation diminishes its distance
z from the equatorial plane by the amount
20'pa'zn (B+7'/)+ H(AL+ Bp) Neeeee a @ceis eer iteo deer “Meee (56) ;
12 Mr ©. CHREE, ON A SOLUTION OF THE EQUATIONS
and the section itself becomes very approximately a paraboloid of revolution, whose latus
rectum is
:
a EQ + 21) = Sea) pain (Uni) maces ccceeeseases deat oeseceacee +e (57).
The axis of the paraboloid is the axis of rotation, and the concavity is directed
away from the equatorial plane.
The curvature of the originally plane cross sections continually increases with their
distance from the equatorial plane, and for a given material and given angular velocity
is independent of the radius a—supposed of course great compared to the thickness 2c,
The diminution of the polar axis 2c is
Alexipa, cn (3 4 a") = ee (aD), ..-. nen en nes oeneeteeeeeeeenee (58).
It thus varies directly as the density, as the thickness and as the squares of the angular
velocity and the radius. It also varies directly as » and inversely as #. On the other
hand it is quite independent of Z’, and increases only about 20 per cent. as 7’
increases from 0 to 1.
The increase in the equatorial semi-axis, or radius, a is
Qen*pas (1 = 77) apes (Way) nds ae oyatew stent cee eee (59).
It thus varies directly as the density, as the square of the angular velocity, and
as the cube of the radius. It varies inversely as E’ and diminishes as 7’ decreases, but
is entirely independent of EF or of 7».
In the circular plate, as in the flat spheroid, every originally plane section per-
pendicular to the axis of rotation becomes very approximately a paraboloid of revolution
about that axis; and the latus rectum of the generating parabola varies inversely as
the original distance of the section from the central section, as the density, and as the
square of the angular velocity, while it is independent of the radius of the plate.
Owing to this change in its origimally plane surfaces the plate will present a bicon-
cave appearance. As the actual measurements of the displacements might be easier for
the plate than for the spheroid it may be as well to state explicitly the following
relations, the diminution in thickness being measured along the axis of rotation:
Increase in radius of plate a (l—7')#
— $$ —$__________—_ — _ — Tx... FN FW ceeeevesccscsessesss i
Diminution in thickness Ha 2c n(B+7) (60),
Curvature at centre of face of plate | 1 2(1+7’) (61)
Diminution in thickness a a ee , .
If the ratios on the left-hand sides of these equations could be experimentally
determined it is obvious that a great deal of light would be thrown on the nature of
the material.
§ 15. To arrive at a more complete knowledge of the effects of rotation, an
auvalysis of the strains is necessary. For our purpose the most convenient normal strain
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 13
components are
4 Sey CY
the longitudinal = Age
: du
, radial = ae p
» transverse =uwu,/r.
The first is directed parallel to the axis of rotation, the second along the perpendicular
on the axis of rotation directed outwards, and the third is perpendicular to the other
two.
Referring to (53) we see that m a flat spheroid, or a thin circular plate, the
longitudinal strain is everywhere a compression, and the transverse everywhere an
extension, and that the numerical measures of both these strains are greater the nearer
the element considered to the axis of rotation. A cylinder whose axis is the axis of
rotation, and whose radius is
divides the volume into two portions in the inner of which the radial strain is an
extension while in the outer it is a compression. The expression (62) is necessarily less
than a so long as 7 does not vanish, so that except in this extreme case the radial
strain actually is a compression near the rim of the circular plate and in the super-
ficial equatorial regions of the flat spheroid.
§ 16. The next case that presents itself is that of a very elongated prolate spheroid
in which c/a is very large. Near the centre of its length the surface of such a spheroid
differs very little from that of a right circular cylinder of radius a. We are thus led
to expect that a solution obtained from (49a) and (50a) by making c/a infinite while
z/a vemains finite, being strictly applicable to the central portions of an indefinitely long
prolate spheroid, will apply very approximately to the case of a right circular cylinder,
provided the length of the cylinder be great compared to its radius and its terminal
portions be excluded from the solution. The solution in question is
ae opr : 25 Savi \ _ pee ' ae) 17722) |
u, 8E" (E — E’n) ul (1 7) (3+) 4E'n"} +7) {#1 7) 2h
: b...(68).
pa tee en
2E
Unlike (53) this solution, though deduced as an approximation from the general
solution, itself satisfies the internal equations. There can thus be no doubt that it gives
the absolute magnitudes of the displacements in any rotating solid whose boundary
conditions it may happen to satisfy. It will be found to satisfy identically the first
three surface conditions (54) for a right circular cylinder of finite length. The last of
equations (54) is not exactly satisfied, as from (63) we get for all values of z
t,, = o'p (a* — 2r*) En (1 +7) + 4(E — E’7’).
14 Mr ©. CHREE, ON A SOLUTION OF THE EQUATIONS
It will be noticed however that
-
"Onrt_dr =0,
Jo
and thus the sum of the normal forces over a terminal cross-section vanishes. Now
Saint-Venant’s solution for beams acted on by terminal forces only secures that the
integral of the stresses taken over the ends should have required values, and notwith-
standing it is regarded by the highest authorities as perfectly satisfactory provided the
length of the beam be great compared to its greatest transverse dimension. Thus (63),
which satisfies exactly 3 out of 4 surface conditions, and is as regards the remaining
condition in no respect less satisfactory than is Saint-Venant’s solution as regards the
terminal conditions in the ordinary beam problem, will doubtless be accepted by the
majority of elasticians as a very approximate solution for the case of a rotating circular
cylinder whose length is great compared to its diameter. The portions of the cylinder
immediately adjacent to its ends ought however to be excluded.
§ 17. Assuming 7/<1, and noticing that in accordance with (55) H-—EH’y* must
be positive, we see from (63) that each element of the long cylinder, as of the flat
plate, increases its distance from the axis of rotation and approaches the central plane
z=0. In the long cylinder, however, the longitudinal displacement varies only as the
distance from the central section, so that each cross-section remains plane.
The shortening in a length 2c of the cylinder amounts to
It thus bears to the shortening in the polar axis 2c of a flat oblate spheroid of
the same density and central section, rotating with the same’ angular velocity, the ratio
11+7/:4(3+7), which for uniconstant* isotropy is 45:52, and is for every material
less than 11: 12.
The increase in the radius of the long cylinder is
kL Ne arias aaas ene seo a ceint SeietucPate eae (65).
This bears to the increase in the equatorial semi-axis of the flat oblate spheroid
of the same density and central section, rotating with the same angular velocity, the
ratio 11+7':8, which is for every material a little less than the ratio, 3:2, of the
volumes of a cylinder and spheroid of the same axial thickness and central section.
We also see from (63) that throughout the long cylinder the longitudinal strain is
everywhere a compression, and the transverse strain an extension. Also the radial strain
is an extension inside and a compression outside of the coaxial cylinder
- L E (1-9) (8 +9')—4E 9 ]
3(1 +91) (E(L = 1) — 229"
* i.e. Isotropy in which Poisson’s ratio is 1/4, or in Thomson and Tait’s notation m=2n.
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 15
In order to apply to our problem this radius must not exceed a, which is the
case only when
En’ (1 =n!) > E’g (1 + 39/).
When this inequality becomes an equality the radial strain just vanishes over the
surface of the rotating cylinder, and if the inequality be reversed then the radial strain
is everywhere an extension. In the case of uniconstant isotropy the radial strain is a
compression throughout one-fifteenth of the area of the cross-section.
§ 18. The next case we proceed to consider is that of uniconstant isotropy. In
a material of this kind there is only one elastic constant. The one employed here is
Young’s modulus #, which is identical in Thomson and Tait’s notation with 5n/2 or
5m/4. The expressions for the displacements in this case are:
pr A She eee wee ee =
= seat as Df : 28 ie 2-2 20cta*
u, 60H (9 + 8a + léca) |L17a* + 195° + 280e%a ar (9 + 18c*a* + 20c*a*)
— 52° (51 + 56c’a)}......... (67),
= @ pz ( 5 2 ~ 2 22. = C = 20
a— 30E (9 rg Sea? p 16c‘a~) (39a°+130c°+ 60cta = 107 i (3 —- 19¢ 1 102 (5 + 27a “I. ..(68).
In considering the strains we shall also want the following expressions:
du w'p er ona. Heanor at 4 fs
a = © —— 28 one. 1: etl ( 1 —2 9 4,4
dr 60EF (9 + 8c°'a*+ 16c'a*) {117a? + 195c? + 280c'a 57° (9 + 18c?a* + 20c*a*)
— 52" (51 + 56c’a”)}..... ... (69),
dw —w'p OTS er Ee A Pe Nn ea.
_——_ os (997.1900 ?_10r?(3419¢a*)—302? (5+ 2c’a)}....(70),
dz ~ 30E (9 + 80a? + 16c'a) {39a°+130c* +60c%a 0r* (3+19¢*'a™) —302* (5+ 2c*a~)}...(70)
du, dw _ —w'prz (39 — 20c’a*)
det d= GEG + Bea? hea (71).
§ 19. Writing
v= o' pa’ (117 + 195c°*a* + 280c%a *)/60F (9 + 8c’a* + 16cta™)..... ee. (72),
@°= a? (117 + 195c’a* + 280c'a*)/5 (9 + 180°a? + 20cta>)........ noe eenene (73),
B =a; (UT =F W9bcaR +. 280cias)/Si(S 425607 aT).. ce ccce corer ceeeecec serena (74),
we get Td Pear (cea (eh x=—7f so) pe eontaconeeco-c oncOC as canae donee Are (75).
Thus as v, a, and £8,° are necessarily positive for all values of c/a, it follows that
wu, and u,/r are positive inside and negative outside the spheroid whose equatorial and
polar semi-axes are respectively a, and 8,. Obviously a,’ is more than twice a’, whatever
e/a may be. Treating a as constant and varying ¢, it is easily seen that 8, is greater
than ¢ so long as c/a is less than ,/39/20, but that for greater finite values of c/a the
value of 8, is less than ¢. The least value of 8,/e is very nearly ‘989, occurring when
c/a is approximately 2:08. Thus for all values of c/a exceeding ./39/20 the difference
between 8, and c is extremely small. They become equal when c/a becomes infinite.
16 Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
It follows that so long as c/a is less than J/39/20 every element of the spheroid
increases its distance from the axis of rotation, and the transverse strain is everywhere
an extension. When c/a exceeds /39/20 there is an extremely limited superficial volume
surrounding each extremity of the axis of rotation within which the elements diminish
their distances from the axis of rotation, and where the transverse strain is a com-
pression; elsewhere the distance of an element from the axis of rotation increases, and
the transverse strain is an extension.
When c/a equals /39/20, or when it becomes infinite, the volumes within which
the elements diminish in distance from the axis of rotation and the transverse strain
is a compression, become reduced to the extremities of the axis of rotation.
§ 20. Similarly from (68)
wy = 72 (La 2] Bo nt nn seschas teas ogee nee (76) ;
where T =o pa’ (39 + 130c*a~ + 60c'a *)/30 BF (9 + 8c°a* + 16c'a*)... eee (77),
a,” = a’ (39 + 130c’a™ + 60c'a™*)/10 (3 + 19c%a™)..... ees scenes seeeecneccenees (78),
BE=¢(89are 7 -F30-F 60cae) LOGE Oi) en nar aceseeeee s-seb see peer (79).
Thus 7, 2,° and 8,° being essentially positive, w is of the opposite sign to z inside
and of the same sign outside the spheroid whose equatorial and polar semi-axes are
respectively a, and 8,. It is easily proved that a, equals a when c/a has approximately
the values “43 and ‘90, and that it is only when c/a lies between these limits that a,
is less than a. The least value of a,/a is about ‘97, answering to c/4="65 approximately.
It is obvious that 8, considerably exceeds ¢ for all values of c/a.
It follows that when c/a lies between “43 and ‘90 there is a very limited superficial
volume close to the equator, the elements within which increase in distance from the
equatorial plane, while elsewhere the elements approach this plane. When c/a lies outside
these limits every element throughout the spheroid approaches the equatorial plane.
§ 21. From (69)
du,
dr
where vy is given by (72) and 8,’ by (74), while a,’ equals a,*/3 and so is known from (78).
= y(t TE 0) eee (80),
It is obvious from (73) that a, is always less than a. It may also easily be found
that as c/a increases from zero, a,/a commencing with the value /13/15 diminishes at
first, attaining a minimum value of about ‘908 when c/a is ‘65 approximately. It then
increases continually as c/a increases further, passing through its initial value /13/15
when c/a ‘equals ,/ 39/20, and finally reaches the value /14/15 when c/a becomes infinite.
It may be remarked as a somewhat curious fact that a,/a and a,/a attain their minimum
values for the identically same value of c/a. The variations in the value of 8, have
been already traced.
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. TZ
The conclusions from these data are as follows:—The radial strain is for all values
of c/a an extension throughout all but a small portion of the spheroid. There is always
however in the equator a superficial volume throughout which the radial strain is a
compression. As c/a increases from zero this superficial volume extends towards the
poles, and eventually reaches them when c/a=.J/39/20. For greater values of c/a this
volume forms a layer completely enclosing the rest of the spheroid. The thickness of
this layer in the equator continually diminishes from about 069a when c/a =,/39/20 to
about 034a when c/a=ax. At the poles the ratio of the thickness to ¢ attains a
maximum of about ‘01 when c/a=2°08 approximately, and then continually diminishes
and vanishes in the limit when c/a becomes infinite.
§ 22. From (70)
3 eo (Meese est) See ae (81),
where 7 is given by (77) and a4,* by (78), while 8,7=8,7/3 and so is known from (79).
Thus a is negative inside and positive outside the spheroid whose equatorial and
polar semi-axes are respectively a, and §,. The variation of a, with the value of c/a
has been already traced in § 20. As c/a increases from zero 8,/c diminishes from
infinity and becomes unity when c/a=/39/20. It attains a minimum value of about
‘986 when c/a=2:21 approximately, and then continually but slowly increasing becomes
unity when c/a becomes infinite.
The observed variations in the values of a, and 8, lead us to the following results :—
When c/a is less than “43, or when it lies between ‘90 and J/39/20, the longitudinal
strain is a compression throughout the entire spheroid. When c/a lies between -43 and
‘90 the longitudinal strain is an extension throughout a small superficial volume in the
equator, elsewhere it is a compression. When c/a has any finite value exceeding J/39/20
the longitudinal strain is an extension in a small superficial volume surrounding each
pole, being elsewhere a compression. Lastly when c/a becomes infinite the longitudinal
strain is everywhere a compression, except at the poles themselves where it vanishes.
§ 23. It will be observed that = he and = are the normal strains when for
the coordinate axes at each point we take the parallel to the axis of rotation, the
perpendicular on this axis produced outwards, and a third axis at right angles to the
du. dw
other two. The only remaining strain is the tangential or shearing strain Oi Fim
dz adr
the plane of 27.
From the expression (71) for the shearing strain it will be seen that it vanishes
along the whole of the polar axis and everywhere in the equatorial plane. On _ the
positive side of this plane it is everywhere of one sign, and this sign is negative or
WO SGV ART I g
Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
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FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 19
A clearer idea possibly of the general character of the phenomena may be obtained
from a study of the accompanying figures (see Plate I). Each figure is intended to
represent the state of some particular strain throughout a section of the spheroid by a
plane through the axis of rotation. The strain represented is the radial de when the
dr
: ; : 2 :
lines are straight and horizontal, the transverse = when the lines are curved, the
Be dw : ; é : ;
longitudinal rE when the lines are straight and vertical. When the lines are thin the
strain is an extension, when thick a compression. The boundary line is drawn thin or
thick according as the particular strain is an extension or compression in the surface at
the point considered.
The surface volumes in which the sign of a strain differs from that at the centre
are as a rule very considerably exaggerated in thickness. If drawn accurately to scale
some of them could hardly be seen without a microscope.
§ 24. The displacements whose experimental determination appears most feasible are
the increase uw, in the equatorial semi-axis, and the diminution —w, in the polar
semi-axis. The amounts of these quantities per unit of original length, ie. u,/a and
—w,/c, are given in the second and third columns of the following Table mt. The fourth
du,
dr ~
centre and, as will presently appear, see § 31, is the absolutely greatest strain existing
column gives the common maximum value v of u/r and This is found at the
anywhere in the spheroid. According to Saint-Venant’s theory of rupture if the angular
velocity be increased until v reaches a certain limit, determined by experiment, the
spheroid will rupture—or more correctly the material will cease to obey the laws of
perfect elasticity. The fifth column gives the maximum longitudinal compression, ie. 7
dw : 3
or the value of — de 3 the centre. The last column gives the maximum stress-difference
Z
at the centre—ie. the difference 4H (v+7)/5 between the algebraically greatest and least
of the principal stresses found there. On the maximum stress-difference theory of rupture
the absolutely greatest maximum stress-difference found in the solid supplies the place
taken on Saint-Venant’s theory by the greatest strain. In certain special cases the
absolutely greatest value of the maximum stress-difference unquestionably is found at the
centre, but I have not proved this universally true, so in general we are only entitled
to regard the value given in the last column of the table as an inferior limit to the
value of the absolutely greatest maximum stress-difference existing in the spheroid.
As a basis of comparison a@ may be regarded as remaining constant while c/a passes
through the values indicated in the first column. The displacements and strains are thus
all expressed in terms of w’pa’/#. This represents a numerical quantity whose value can
be easily calculated when the angular velocity, the equatorial diameter, the density, and
Young’s modulus for the material are known.
3—2
20 Mr ©. CHREE, ON A SOLUTION OF THE EQUATIONS
TABLE III.
Value of e/a Increase of Decrease of Greatest strain
equatorial diameter —_ polar diameter wpa
per unit length per unit length »/ 25
ve / Se aaa) ee
a E € E
infinitely small 13 ‘14 216
2 "1364 1507 "2234
+ 1456 1647 "2422
6 "1590 1748 2669
8 ol kya “1719 ‘2874
1-0 1803 160 298
12 1851 1472 3037
14 ‘1875 "1352 BOAT
16 ‘1886 255 3041
1s 1891 1180 3030
2 1892 112 °BO17
3 ‘1888 0968 "2972
+ 1883 “0910 2950
infinitely great 1875 083 2916
Greatest
longitudinal
compression
/ w*pa?
i / BE
‘14
‘1580
1913
2235
‘2367
231
‘2176
‘2029
1898
‘1791
1705
1469
‘1376
125,
Maximum
stress-difference
at centre
$E (v +7)/*pa*
28
“‘BOS1
3468
“B924
“4192
“42
“4171
“4060
3951
‘B857T
It will be understood of course that in the preceding as in the succeeding table
the entries do not as a rule give the exact values, but the last figure of each decimal is
chosen so as to make the result as correct as the number of figures retained will permit.
§ 25. The approximate positions and values of the maxima of the several quantities,
supposing @, p, EZ and a to be constants, can be obtained from the preceding table. The
following more exact results were obtained by direct calculation from the formulae :—
TABLE IV.
Value of c/a supplying maximum | Maximum
Quantity
2 2
Ug | @ pa
a E
w, | wpa’
c/ E
ow? pa?
es
7
wpa’
E
teo+n/ wp
2:06
658
./39/20 = 1:396
‘826
956
1892
| “L749
3047
‘2367
“4.246
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 21
§ 26. The most notable results in the two preceding tables are the extremely small
change in the increase per unit length of the equatorial diameter or in the value of the
greatest strain as c/a increases from 1 to », and the fact that the absolutely largest
value of the greatest strain—and so according to Saint-Venant the greatest tendency to
rupture—occurs in the critical spheroid.
It is important to bear in mind that the above maxima are calculated on the
hypothesis that the length of the equatorial diameter is the same in all the spheroids.
If this be varied and some other quantity kept constant different results of course will
be obtained. If for instance ¢ and a both vary while the volume remains constant, a
biquadratic equation in c*/a? is obtained whose roots determine for what forms of spheroid
the greatest strain v—or Saint-Venant’s tendency to rupture—has its greatest and least
values. All the terms of this equation are however of the same sign, and so no true
maximum or minimum can exist. The correct interpretation is that when the mass of
the spheroid is constant Saint-Venant’s tendency to rupture continually diminishes as the
polar axis 2c increases from 0 to ©. The same conclusion also follows if the constant
quantity be the moment of inertia about the axis of rotation.
§ 27. Taking the axes specially for each point considered, as in the case of the strains,
we get for the stresses in the case of uniconstant isotropy the following expressions :—
sn Ue i, apy
Fie (3 a oe a) Des Ea ac tem cedar see (82)
9 du, U,. dw ee cseeee
ed Faas? oP
R,=3E( du, _
2 dz dr
The first three are normal stresses directed respectively parallel to the axis of rotation,
along the perpendicular on this axis directed outwards, and along the perpendicular to
these two directions. The last is a tangential or shearing stress in the meridian plane, or
plane containing z and r.
From (67)—(71) we obtain the following convenient expressions for the stresses :—
pa eeeaoamee) fh Gy
Ot 8ea?+léca le ee meter Bets perce neat Rar ao be
@ ¢
2 a2 2 m3) a
R ue {e9 ~ 20c%a"*) (1 “ =) +45 (18 +25¢%a) (1 =e a) Ae (840,
~ 15 (9+ 8c'a* + 16c'a*) GG
@ pa” ie 5) (ou aoe yr? 2
a ~ 15 (9 + 8a + 16c'a*) {(s9 ate) (1 ~ ae 4 a me nzoed”) (1 Gee 3)
+(18 + 36c*'a? + 40c*a™) a sa ahisaagsiant (85),
_ —o'p (39 — 20c*a) rz
= 15 (9 + 8e'a + 1l6c‘a*)
22 Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
§ 28. There are at every point, as is well known, three principal stresses parallel
to three rectangular axes, whose directions are such that the tangential stresses vanish
ever the elements whose normals are these axes. ©® is one of these principal stresses,
and the corresponding strain u,/r is everywhere one of the three principal strains. The
two other principal stresses lie in the plane 27, but coincide with Z and R only when
R. vanishes, and so in general only along the polar axis and in the equatorial plane.
These principal stresses are the two values of
Liat (GR= Zs AR tec. crete nota okt (87).
If we suppose the square root always to represent a positive quantity, then the
algebraically greatest principal stress in the meridian plane answers to the upper sign,
and the angle a which its direction makes with the perpendicular on the axis of rotation
directed outwards is given by
2 2 3
sores ee ee
As this expression concerns us practically only when R. is not zero, we may say that
tana is everywhere of the same sign as R,, It is thus by (86) negative or positive
for z positive according as c/a is less or greater than the critical value /39/20. It
follows that the angle which the direction of the algebraically greater principal stress in
the meridian plane makes with the perpendicular on the axis of rotation directed outwards
is oblique or acute according as c/a is less or greater than the critical value.
§ 29. On the surface of the spheroid 1—7r/a?—2*/c? vanishes, and it is very simply
proved from the expressions (83)—(86) that the two principal stresses in the meridian plane
are there directed along the tangent and the normal. Also, from above, the principal
stress along the tangent is the algebraically greater or the algebraically less according as
c/a is less or greater than the critical value. Further the principal stress directed along the
normal is zero, this being in fact a consequence of the surface conditions, Thus the
tangential meridional stress is a tension or a pressure according as c/a is less or greater
than the critical value. The algebraical expression for this stress may easily be found to be
w'p (39 — 20c*a*) a*c
ACES era a mar ee ede
where p is the perpendicular from the centre of the spheroid on the tangent plane at the
point considered. Comment on the applications of this remarkably simple result seems
unnecessary.
The complete change that takes place in the character of the meridional surface stress
as c/a passes through the value J/39/20 seems an ample justification of our designation
of it as the critical value. There also appears for this value of c/a an important change
in the character of the surface value of ® the stress perpendicular to the meridian plane.
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 23
For from (85) we find for the surface value of ® the expression
o*= wpa
* 15 (94 8c?a* + 16c*
_ (39 — 20cta) Z 4 (18+ 36c'a? + 40c'a*) t ...... (90).
ay ty a’)
So long as c/a is less than the critical value it is obvious that ®, is positive for all
values of r/z and so all over the surface. When c/a attains the critical value ®, is still
everywhere positive but just vanishes at the poles. For all greater values of c/a, ®, is
negative within a small area surrounding each pole, being elsewhere positive. Thus for
all values of c/a below the critical the surface stress perpendicular to the meridian plane
is everywhere a tension. But for all values of c/a above the critical there is a small
area round each pole within which this stress is a pressure.
It may also be easily proved that the surface tension at right angles to the meridian
has its greatest value at the poles or on the equator according as c/a is less or greater
than °55 approximately.
§ 30. In the critical spheroid the state of stress is extremely simple as the only
stresses which do not vanish are R and ®, and these are everywhere principal stresses.
Of these R vanishes all over the surface and elsewhere is positive, while ® vanishes
only at the poles being elsewhere positive. Excepting at the poles ® is everywhere greater
than R; and so, as both are positive and the third principal stress is zero, ® is everywhere
a correct measure of the maximum stress-difference. Its greatest value obviously occurs
at the centre. Thus the critical spheroid is one of the special forms in which it is actually
proved that the tendency to rupture on the maximum stress-difference theory, as well as
on the greatest strain theory, occurs at the centre. It will be noticed that over the surface
of the critical spheroid ® varies as the square of the perpendicular on the axis of rotation.
§ 31. For values of c/a other than the critical the determination of the algebraically
greatest principal stresses is a matter of some little difficulty. It is however worthy of
notice as it leads at once to the greatest principal strain, which is required im applying
Saint-Venant’s theory of rupture,
Let P and Q denote the algebraically greater and less of the two principal stresses
in the meridian plane. Then the algebraically greatest principal stress is either ® or P.
From the formulae for ® and P we easily find
b=P according as
2)
2u, du, dw>(/du,_ dw\’ , (du, , dw)’
“LUE 1( - =) (Git iz) “eer eee (91).
5 ee Qu. du, dw . ae ;
Thus ® is the greatest principal stress when Nae —qz_ 38 Positive, and when its
du, dw\? /du, _ dw\’. : : faa
square exceeds (Ge- 1) + ee + i) ; otherwise P is the greatest principal stress.
* Here and in what follows surface values are distinguished by the suffix s.
24 Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
Substituting the expressions for the strains from (67), (69), (70), (71), I find by a
straightforward and not very laborious calculation on the above lines that so long as c/a
is below the critical value, ® is everywhere—excepting the axis of rotation where it equals
R which is there a principal stress—the algebraically greatest principal stress. Thus for
all values of c/a below the critical u,/r is at every point in the spheroid the greatest
strain, and so is the correct measure of Saint-Venant’s tendency to rupture. A glance at
(67) will show that its greatest value is found at the centre. This is given in Table MI.
under the heading ».
When c/a exceeds the critical value there is a small superficial volume round each pole
within which ® is not the algebraically greatest stress, though elsewhere it continues to be
so. Within these small volumes, however, the values of the maximum stress-difference and
of the greatest strain are for finite values of c/a much less than are the corresponding
values found at the centre of the spheroid. Thus so far as the question of rupture is
concerned, the fact that when c/a exceeds the critical value small regions exist around the
poles in which ® is not the greatest principal stress nor u,/r the greatest strain is of
no material consequence, though of course a point well worthy of notice on its own account.
This leaves the value of y given in Table II, a correct measure of the tendency to rupture
on Saint-Venant’s theory even when c/a exceeds the critical value.
§ 32. The determination of the maximum stress-difference throughout the whole of
the spheroid would be a laborious process which seems hardly worth the trouble. The
value at the centre is given in the last column of Table m1. In the critical spheroid it
was shown above that this is the absolutely greatest value of the maximum stress-difference,
and in a previous paper* it was proved that the same was true for a sphere of any
isotropic material.
If the values m and n of the elastic constants in the general case of isotropy be
substituted in the general expression (53) for a flat rotating spheroid, it can easily be
proved that the stress Z everywhere vanishes, and that consequently, excluding the surface
where all meridian stresses are of order z at least, the principal stresses in the meridian
plane are respectively R and zero, when terms in 2 are neglected. Further the value of
R is nowhere negative. The third principal stress is © along the perpendicular to the
meridian plane. ©® is everywhere not less than R—it is equal to R along the axis of
rotation —and its greatest value exists in the axis, where it is constant so long at least
as terms in z* are neglected. Thus the greatest value of the maximum stress-difference
is correctly given by the value of at the centre of the flat spheroid.
The expressions obtained from (63) for a very elongated prolate spheroid of isotropic
material, whether uniconstant or not, are even more simply treated. The stresses Z, R and
® are everywhere the principal stresses, and &—Z is everywhere a correct measure of the
maximum stress-difference. It is easily proved that its greatest value occurs in the axis of
rotation, at every point of which the value is the same.
* See the Society’s Transactions, Vol, x1v., pp. 292—294.
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 25
We are thus certain that in the cases of the flat oblate spheroid, the sphere, the
critical spheroid, and the elongated prolate spheroid, the numbers in the last column of
Table ur. give the greatest value of the maximum stress-difference occurring anywhere,
and there seems to me every probability that such is in general the case. I thus believe
this column to give in each instance the true measure of the tendency to rupture on
the stress-difference theory; but except in the four special cases just mentioned, we are
strictly speaking only warranted in regarding the results as supplying minima for the
correct measures of the tendency to rupture.
§ 33. After our examination of these special cases it will be unnecessary to enter
into great detail in discussing the general case, for which the displacements are given by
the expressions (49 a) and (50 a).
Assuming the original elastic constants c, f etc, as well as 7, 7’ etc. all positive,
we have as already explained the relations (55). From the latter of these it follows that
B> Pe is
Ha Sees ee ).
Bearing in mind these relations, we see from (49 a) and (50 a) that :—
ajr= vA—rfa?—27/8),
=—72(1—1'/a’?— 2*/8',’),
d i Breage ‘
= Sy a oy: _ Z3/Q! 2), b vceeseeseeneereneenecesceneenes (93),
dw aati ’ ee 2 y 2
ae (Ea cei /5), 9) |
where v’, 7’, @,”, 8’, 47, BY”, v7 =,"/3, and f’,’=8',/3 are all positive constants depending
on the values of c/a and on the elastic constants. For the special case of uniconstant
isotropy these reduce to the corresponding undashed constants »v, 7, etc.
There is thus for each displacement, or normal strain, a determining spheroidal surface
, ; : : du ea, Se
over which the displacement, or strain, vanishes. Also u,/r and aa are positive inside and
negative outside their determining spheroids, while the reverse is true of w and a When
z
a determining spheroidal surface lies wholly outside of the material rotating spheroid the
corresponding displacement or strain is, if u,, w,/r, or ap , everywhere positive, but if w
or ay everywhere negative throughout the solid.
dz
The only remaining strain is the shearing strain in the meridian plane, whose value
is given by the simple expression
Dy (ii 4 2) Paes aie ene)
wp\dz dr/ G\| 4 a)
Vou. XV. Part I. 4
26 Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
Thus it vanishes everywhere along the polar axis and in the equatorial plane, and throughout
the rest of the spheroid changes sign only with 2 The sign is — or + for z positive
according as c/a is less or greater than the critical value
E (1-7) cae il
4E'n (1 +7’)
In the critical spheroid whose axes possess this ratio the shearing strain is everywhere zero.
§ 34 The expressions for v’, 2? ete. are somewhat complicated, and a consideration
of the magnitudes of the semi-axes of the determining spheroids does not so easily lead to
the desired results as does the following method.
The signs of the displacements and strains at the centre of the spheroid are already
known. Thus if we determine their signs at the surface of the material spheroid we can
tell whether any portion of the solid lies outside of the determining spheroids. To get
the sign of any displacement or strain at the surface, it is simplest to make the expression
for it homogeneous by substituting @*r*+¢%2* for unity. There are then in each expression
only two coefficients whose signs have to be considered. Employing this method we find
over the surface
(DBGEN Te es 2(1—7’) pale 7 (1+ Fa Nee 19
eS Shey ees +a a= aa t+ = “(E-E *)|
1,2) 2-2 ( -71) (3+7/) _ 4n (1 +7')
+4 ne i EB (96):
Employing the last of equations (55) it is easily proved that for all values of c/a,
however large G/E may be, the coefficient of 7* is positive. The coefficient of 2 is
obviously positive or negative according as c/a is less or greater than the critical value.
It follows that for all materials of the class here considered, so long as c/a is less
than the critical value, every element of the rotating spheroid increases its distance from
the axis of rotation and the transverse strain is everywhere an extension. When, however,
c/a exceeds the critical value there is in all such materials a superficial region surrounding
each pole wherein the distance of each element from the axis of rotation is diminished
and the transverse strain is a compression.
§ 35. The expression for the ‘surface value of w is not quite so manageable. It is
the following :—
Fw), =92|- 74 ea” Sn(s+n)- 2E'n? — we |
*p 8
QF” 4h} G E* (1-7)
2-2 1(3 +77) 1 ! 2H’ (5 +3) E'n (8 +7)
+2|-¢ Wy ~ gEaayy (A-) B +9) E 7E GE
2 2 L'n (E — E'7")
fuk wee ae
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 27
By means of the second of equations (55) it is not very difficult to prove that, whatever
be the value of G/E, the coefficient of z* is negative for all materials of the kind here
considered. The coefficient of r°z is certainly negative if c/a be either very small or very
large, but in general it will be positive when c/a les between certain limits depending
on the material, the superior of which is decidedly less than the critical value. C.f. § 20.
It follows that if c/a be either very small or very large every element diminishes
its distance from the equatorial plane. In most if not all materials, however, of the kind
treated here,—certainly in all isotropic materials,—there is between certain limiting values
of c/a depending on the material a superficial equatorial region within which the elements
increase in distance from the equatorial plane.
§ 36. For the surface value of -- we get
Mapes 9S 7 A=) 4. s(n ond ta} Cir ; . is :
=e | ee 3 ee E "EF G—7) En (1 —1') — E'y 1 +3%)} |
Pee OC s2)) ee
+42 {ore : _ ys “ | pile lation 2A) (98).
The coefficient of z is positive or negative according as c/w is less or greater than the
critical value. The coefficient of 7° is negative for all values of c/a for all materials
in which
JH ig ead dary (Uk 5877))) aneehonsobeck. -.acpocpopacecnencsense (99).
This includes all isotropic materials in which m < 3n.
For other materials however, including isotropic materials in which m>3n if such
exist, the coefficient of 7* becomes positive when c/a is sufficiently increased above the
critical value.
We conclude that while c/a is below the critical value the radial strain is everywhere
an extension, except in a superficial volume about the equator where it is a compression.
As c/a increases the superficial volume approaches the poles and eventually reaches them
when c/a attains the critical value. In materials whose elastic constants satisfy the relation
(99) there is for all values of c/a above the critical a superficial layer completely
surrounding the spheroid wherein the radial strain is a compression, while elsewhere it is
an extension. In materials whose elastic constants do not satisfy (99),—including isotropic
materials for which m>3n,—when c/a exceeds a certain value, greater considerably than
the critical value, the superficial volume in which the radial strain is a compression splits
up into two volumes one surrounding each pole, and as c/a further increases these polar
volumes continually contract. The materials in which this splitting up of the superficial
layer into two polar volumes may naturally be expected are those in which Young’s
modulus for the direction parallel to the axis of rotation is small compared to that for
the perpendicular directions.
4—2
lo
io 2)
Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
: SLOU IoD iy
§ 37. The surface value of dz 8 given by
z
D (dw) _ s[_ 20-9), C0" (igs ny omy MBE) ong Hale ~E'y)
@'p z).7 ? |- On =< 45° eS + n) 2H n G i “ca ES (l —7)
2g? 2 ta) , 2H at (L+0))
+2 1 we aE TW ee (100).
The coefficient of 7° is the same as that of r*z in (97), and its sign has been already
treated of in considering that expression. The coefficient of 2° is negative or positive
according as c/a is less or greater than the critical value. The conclusions these data
lead to are as follows :—
For small values of c/a the longitudinal strain is in all materials everywhere a
compression. In most if not in all materials,—certainly in all isotropic materials—there
exists within certain limiting values of c/a, the superior of which is decidedly below the
critical value, a superficial region about the equator wherein the longitudinal strain is an
extension; elsewhere it remains a compression. Between this superior limit of c/a and
the critical value the longitudinal strain is everywhere a compression. Finally when c/a
exceeds the critical value there exists in all materials a superficial region round each pole
wherein the longitudinal strain is an extension; elsewhere it is a compression.
§ 38. It will be observed that on the whole the variations of the strains and
displacements in the general case follow very closely the variations which occur in the
special ease of uniconstant isotropy. In fact, with one exception presently to be noticed,
when a2, ete. are replaced by a, etc., /39/20 by the “critical value” (95), and 43 and ‘90
by the two positive values of c/a obtained by equating the coefficient of r* in (100) to
zero, Table m1. in § 23 may be applied to all but certain exceptional materials whose
existence is somewhat problematical.
The single exception is that of materials in which the relation (99) does not hold. In
such materials, as already explained, the superficial volume wherein the radial strain is a
compression becomes for large values of c/a limited to circumpolar regions. This is a rather
noticeable departure from the phenomena described in uniconstant isotropy, and is worthy of
special attention because the relation it requires between the values of the elastic constants
seems likely to be by no means uncommon in materials in which Young’s modulus in
the direction of the axis of symmetry is small compared to that in the perpendicular
directions.
§ 39. The expressions for the stresses in the general case are on the whole wonderfully
simple. The tangential or shearing stress in the meridian plane = ( x (corresponding shearing
strain), and so is the product of the right-hand side of (94) into w’pG/D'. Its fluctuations
in sign have been already noticed in treating the shearing strain. It will be noticed that
the surfaces over which this shearing strain and stress have constant values are generated
by the revolution about the axis of rotation of rectangular hyperbolas whose asymptotes
are the axis of rotation and an equatorial diameter.
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 29
The expressions for the normal stresses, referred as previously to the fundamental
directions at each separate point for axes, are as follows :—
Oe pany, ' 2-27 le (te ER
Bm gpg ray py EA = 1) 8+) — AaB (1 + MILA (1 SS )h acces (101),
wpa” N\ fe , 2.-25Y \) r
R= gay) p | HA-WB+H)—se0 B'y(1+n)3 (1-4)
Se ee i a. VAGUE ; _ 2
+cat sq Gta) +0430) +e (26+n)—22'%')} (1-5 - 4)]...102)
és wp E'7* 1—1/ ian il oe _, E (1-1) -2E'n?
@=—R 2D’ | Via ae (A+ <a) ap AG — Pan eleleletalslotataietetotefetaleterctalayalsreys (103).
§ 40. From (101) it appears that for all values of c/a, whatever be the character
of the material, the longitudinal stress vanishes over the surface of the spheroid whose
equatorial and polar semi-axes are respectively a//2 and c. It is a pressure inside and
a tension outside this surface when c/a is less than the critical value, a tension inside
and a pressure outside when c/a is greater than the critical value. The volume throughout
which it is a tension is thus under all circumstances equal to that throughout which it
is a pressure. In the critical spheroid itself the longitudinal stress everywhere vanishes.
Over the surface of the material spheroid for all values of c/a the longitudinal stress varies
as the square of the perpendicular on the axis of rotation.
In (102) it will be noticed that the coefticient of (1 —71°/a*—*/c*) is essentially positive
for all materials of the kind considered here, and that the coefficient of (1—7*/a’) is positive
or negative according as c/a is less or greater than the critical value.
Thus so long as c/a is less than the critical value the radial stress is everywhere a
tension, but when c/a exceeds the critical value it becomes a pressure in a superficial
volume, whose thickness is greatest at the poles and zero in the equator. Over the
surface of the spheroid, whatever be the value of c/a or the character of the material, the
radial stress varies as the square of the perpendicular on the equatorial plane. The radial
stress thus vanishes where the equatorial plane cuts the surface and in general nowhere
else. In the critical spheroid however it vanishes at every point of the surface.
The stress ® at right angles to the meridian plane is equal to the radial stress at
every point on the axis of rotation and everywhere else is algebraically greater than it. It
is everywhere a tension so long as c/a is less than the critical value, but when c/a exceeds
the critical value it becomes a pressure in a superficial volume around each pole.
The remarks made on the position of the principal axes in the case of uniconstant
isotropy, cf. § 28, apply verbatim to the general case. The stress ® perpendicular to
the meridian plane is everywhere a principal stress. Along the polar axis and in the
equatorial plane the longitudinal and radial stresses Z and KR are principal stresses, and
this is also the case at every point of the critical spheroid, which has thus one of its
30 Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
principal stresses everywhere zero. With these exceptions however the principal stresses
in the meridian plane do not act along the fundamental directions, and the angle which
the algebraically greater of them makes with the perpendicular on the axis of rotation
produced outwards is everywhere obtuse or acute according as c/a is less or greater than
the critical value.
On the surface the only stress in the meridian plane is along the tangent, and it is a
tension or a pressure according as c/a is less or greater than the critical value. Over
the surface of any given spheroid it varies inversely as the square of the perpendicular
from the centre on the tangent plane.
§ 41. In the general case it seems scarcely worth while constructing tables for the
values of the changes in the lengths of the equatorial and polar diameters and for the
strains at the centre of the spheroid. To be practically useful such tables would have
to assign numerical values to 7, 7’, G/E and #'/E. It is doubtful if satisfactory ex-
perimental determinations of these quantities exist for materials of the class here con-
sidered, and a large amount of time would be required to make the arithmetical
calculations necessary if all values theoretically possible were to be included.
Further, materials of this class can doubtless support a greater strain in some
directions than in others, so that the value of the greatest positive strain, or the
greatest value of the maximum stress-difference, cannot on any possible theory immediately
determine the tendency of the body to pass beyond the limits of perfect elasticity or
to approach rupture. Saint-Venant it is true has applied his theory of rupture in a
generalized form to such materials, but it seems on the whole advisable to postpone
consideration of the question until a reasonable expectation exists that the theory cor-
responds to the facts.
§ 42. In the case of uniconstant isotropy the variation of the more important
strains and displacements with the value of c/a have been already shown in Table m1.
Since however in this country the biconstant theory of isotropy is almost universally
accepted, I have calculated the values of the several quantities of that table for the
values 0, 2, 4, ‘6 and 1 of the ratio of the elastic constants n : m. These answer
respectively to the values ‘5, “4, ‘3, ‘2 and 0 of Poisson’s ratio. Every solid probably
that has the least claim to be regarded as isotropic will be admitted to have positive
values for Poisson’s ratio and for the rigidity, so that 0 and 1 are respectively the least
and greatest values which can be attached to n/m. The results are thus of the utmost
generality so far as isotropic materials are concerned. They are given in the following
tables, v.—1x. The corresponding results for intermediate values of n/m could in general
be obtained to a close degree of approximation by interpolation from the tables.
§ 43. The quantity treated in Table v. is the total increase in the equatorial dia-
meter divided by its whole length. It is for shortness spoken of as the increase per
unit length, but it must be clearly understood that the radial strain varies from point to
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 31
point of a diameter, so that the change in any particular unit of length varies with
the distance from the centre. In this as in the following three tables the numbers in
the table must be multiplied by w’pa’/E to get the absolute values. This factor is an
arithmetical quantity, and as such independent of the particular system of units employed.
The value of # must of course be determined by experiment and expressed in terms
of the same system of units as the other quantities.
In comparing the results answering to a given value of n/m the equatorial semi-
diameter a must be regarded as constant, so that the variations in the value of c/a
must be treated as proceeding from variations in c alone. Thus what Table v., for
instance, immediately shows is how the increase in the equatorial diameter of a spheroid
of given equatorial diameter, formed of given material and rotating with a given angular
velocity, depends on the ratio of the polar to the equatorial diameter.
Table vi. gives the total diminution of the polar diameter divided by its whole
length. The actual longitudinal strain of course along the polar diameter is not in general
constant but varies with the distance from the centre.
Table vu. gives the algebraically greatest principal strain at the centre. It might equally
d : :
correctly have been represented by ( a) , because the radial and transverse strains are
i) °
there the same. In certain cases—e.g. for the values 0, 1, ~ of c/a—this has already
been proved to be the algebraically greatest strain occurrmg anywhere in the spheroid,
and is then known to be the exact measure of the tendency to rupture on Saint-Venant’s
theory. It may further be shown, as in the corresponding case in uniconstant isotropy,
that this quantity is in general the correct measure of Saint-Venant’s tendency to rupture.
Table vill. gives the numerical value of the third principal strain at the centre.
It is a negative quantity and so is a compression, and its direction is the polar diameter.
It does not in itself supply a measure of the tendency to rupture on any theory and
so is of less importance than the greatest strain. Its variations have been deemed worthy
of tabulation because the centre is in itself the most important poimt in the spheroid,
and because the value of any given normal strain throughout the spheroid is as a rule
small or great according as its value at the centre is small or great.
The quantity tabulated in Table 1x. is the maximum stress-difference at the centre.
For the values 0, 1, © of c/a it measures exactly on the stress-difference theory the
tendency of the spheroid to rupture. For other values of c/a it can be regarded only
as an inferior limit to the true tendency to rupture, as the existence of greater values
elsewhere has not been formally disproved.
Being of the nature of a stress it is measured in terms of w’pa’, and is thus given
in absolute measure in terms of the system of units of length, time and mass which
may have been adopted.
Table x. is of a totally different character from the previous five. It gives the
value of c/a in the critical spheroid answering to the assigned values of n/m. The
32 Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
importance of the critical spheroid has been pointed out and most of its properties have
been noticed in treating of the general case or of uniconstant isotropy. In the latter case
it was stated in § 26 that the absolutely largest value of the greatest strain, for a given
material and given equatorial diameter, occurs in the critical spheroid. This is not how-
ever a peculiarity of uniconstant isotropy but, as may easily be proved from the expression
for the greatest strain, is true of the general case of biconstant isotropy. We can thus
lay down as a general law that :—
In a rotating spheroid of given equatorial diameter formed of an isotropic medium,
the absolutely largest “greatest strain” at the centre, and so the greatest “tendency to rupture”
on Saint-Venant’s theory, invariably occurs in the critical spheroid.
TABLE V.
Increase in equatorial diameter per unit length.
Ma, wipa’
an
| y )
| Val f Val f 4
dapptlStigalip= Pd inal 2 4 2 :
0 087 105 124 143 is
2 091 109 ‘127 146 183
4 102 119 ‘137 154 191
eS 126 144 163 181 217
eek) 132 151 171 190 229
| 0 131 154 ‘178 2009 2475
40 127 1514 1760 2006 2498
x 125 ‘15 175 2 25
TABLE VI. 7
Diminution of polar diameter per unit length.
=. _, w'pat
cn eae
> | :
| oe of | ro of 0 2 “4 6 1
0 304 ‘239 175 114 0
2 308 244, 181 121 007
se 314 255 194 136 025
8 287 241 195 149 0565
1-0 263 2 181 140 0571
20 197 163 129 095 027
40 174 141 "108 O74 0076
wo | 16 13 1 06 0 |
FOR THE EQUILIBRIUM OF
TABLE VII.
Greatest strain at centre.
ELASTIC SOLIDS,
Val f Value of is
a 2 ; ‘
0 152 179 204 229 27
2 164 ‘188 212 ‘234 276
“4 190 213 233 "252 ‘286
8 218 "252 277 296 325
10 210 255 286 31 343
2:0 160 231 282 319 371
40 135 215 22 3150 3747
00 125 208 268 3125 ‘BT5
TasLeE VIII.
Longitudinal compression at centre.
zs | a
(a ;
Val f Val f %
ae “ ales . 0 2, 4 6 1
0 304 239 175 ‘114 0
2 327 256 190 N27 ‘O11
4 380 298 225 158 ‘038
8 437 B47 271 204 0847
10 421 337 264 2 ‘0857
2-0 319 ‘256 198 143 040
40 269 "215 163 ‘112 ‘O11
oa 25 2 15 “1 0
Win, OWS 1a IE
34 Mr C. CHREE, ON A SOLUTION OF THE EQUATIONS
TABLE LX.
Maximum stress-difference at centre.
2H U, dw\) _ 2
3 —n/m {( i + (a), Sones
V BA of V ai of 0 2 4 6 1
0 304 298 "292 286 27
2 327 318 B09 301 286
4 380 365 352 342 324
8 437 428 422 ‘417 410
1-0 421 422 “4.24 “4.25 429
2:0 319 348 B69 356 411
40 269 307 335 356 386
co 25 292 321 344 375
TABLE X.
Value of c/a in the critical spheroid.
| n[m= 0 ai
2 4 6 8 9 1
|
| sc aaa ‘764 853 954 1-217 1633 2518 3715 oo
§ 44. The calculations on which these tables are based proceeded to 4 places of
decimals. The last of these however has been retained only in a few cases where the
variation of the quantity considered with the value of c/a is exceptionally slow. When
less than 3 places of decimals are shown the value given in the table is the exact value
of the quantity.
The results of the Tables v.—1x. are also shown graphically in the accompanying
figures 1—5, Plate IL, as they seem peculiarly well adapted for this form of treatment.
In all the figures the abscissae of the curves answer to the values of c/a, a special curve
being drawn for each value of n/m. In the first four figures the curves for the value
n/m="5, answering to uniconstant isotropy, are also drawn. In the last figure this curve
is omitted as in its earlier portion it could hardly be shown distinctly between the curves
answering to the values “4 and ‘6 of n/m.
FOR THE EQUILIBRIUM OF ELASTIC SOLIDS. 35
In the first four figures the ordinates give the numerical value of the coefficient of
w pa?/E, which is thus treated as the unit quantity. In the last figure the unit quantity
is wpa’ simply. In the first four figures and the corresponding tables when a direct
comparison is instituted, for a given value of c/a, between the values of the quantities
which answer to the various values of n/m, the materials compared must be supposed to
have the same Young’s modulus and density.
§ 45. From fig. 1, or Table v., it is seen at a glance that the way in which the
increase in the equatorial diameter varies with the value of c/a is very similar for all
possible values of n/m. As c/a increases from 0 to 1 the increase in the equatorial diameter
rises continually in every case, though somewhat slowly. As c/a increases further the
variations in the quantity considered are remarkably small, so that the increase in the
equatorial diameter is practically nearly independent of the eccentricity in all prolate
spheroids. When n/m=1 the curve continually approaches an asymptotic value as a superior
limit. In the other curves the ordinates show true maxima for finite values of c/a, all
greater than unity and so denoting prolate spheroids, and the value of c/a answering to
the maximum continually diminishes as n/m diminishes, i.e. as Poisson’s ratio increases.
Also it is obvious that for a given value of Young’s modulus and a given density, the
increase in the equatorial diameter invariably increases as Poisson’s ratio diminishes, whatever
be the value of c/a.
§ 46. The ordinates of all the curves of fig. 2 show distinct maxima which answer
to values of c/a less than 1, so that for a given material and a given equatorial diameter
the diminution per unit length in the polar diameter is greatest in some form of oblate
spheroid. It is also obvious from the figures that the spheroid in which the quantity
is a maximum becomes more and more oblate as n/m diminishes, te. as Poisson’s ratio
increases,
The dependence of the diminution of the polar diameter on the value of Poisson’s
ratio is very marked. When Poisson’s ratio becomes zero, the diminution of the polar
diameter totally disappears in the limiting forms of the oblate and prolate spheroids
answering to the values 0 and o of c/a, and is extremely small in all spheroids which
differ much from the spherical form.
A comparison of figures 1 and 2 shows very strikingly how the class of isotropic
materials in which the increase in the equatorial diameter is most marked is precisely
the class in which the diminution in the polar diameter is least conspicuous.
§ 47. The curves of fig. 3 resemble pretty closely those of fig. 1. Except in the
case of n/m=1, the ordinates show true maxima for finite values of c/a, and the value
of c/a at which the maximum appears continually diminishes as Poisson’s ratio increases.
The exact positions of the maxima are, as already explained, given by Table x. Except in
the case of n/m=0, the dependence of the greatest strain on the eccentricity is decidedly
more conspicuous in oblate thap in prolate spheroids.
5—2
ie)
for)
Mr C. CHREE, ON EQUILIBRIUM OF ELASTIC SOLIDS.
§ 48. The curves of fig. 4 show a general resemblance to those of fig. 2. Their
ordinates however exhibit much more pronounced maxima. The spheroids in which these
maxima occur are all oblate, and the oblateness increases but only to a very small extent
as Poisson's ratio increases. It will also be noticed that for a given magnitude of spheroid
the longitudinal compression at the centre diminishes rapidly as Poisson’s ratio diminishes,
and absolutely vanishes along with Poisson’s ratio in the limiting oblate and_ prolate
spheroids answering to the values 0 and o of c/a. In fact the isotropic materials in
which the greatest strain at the centre is largest are precisely those in which the
longitudinal compression is least and conversely.
§ 49. In fig. 5 the closeness of the curves for all values of c/a less than unity
seems very remarkable. This would indicate that on the stress-difference theory the
numerical measure of the tendency to rupture at the centre in all oblate spheroids of
isotropic material is nearly independent of the values of the elastic constants. This would
not of course imply that the angular velocities causing rupture in oblate spheroids of the
same size and shape are nearly the same for all isotropic materials of the same density,
because one such material might stand a very much greater stress-difference than another.
There is also a critical value of c/a, lying in every case between ‘9 and 1, at which the
value of the maximum _ stress-difference regarded as a function only of n/m becomes
stationary. In all oblate spheroids in which c/a is less than ‘9 the maximum stress-difference
continually increases, though only to a small extent, as Poisson’s ratio increases; whereas
in all prolate spheroids the maximum stress-difference continually diminishes as Poisson’s
ratio increases. In oblate spheroids in which c/a lies between ‘9 and 1 the maximum
stress-difference is practically independent of the values of the elastic constants.
In all the stress-difference curves the ordinates possess distinct maxima. When n/m=1
this maximum appears when c/a is nearly 1:2. In each of the other curves the maximum
appears when c/a is less than unity, i.e. in an oblate spheroid, and the oblateness of this
spheroid continually increases as n/m diminishes, ie. as Poisson’s ratio increases. In no case,
however, does the spheroid in which the maximum occurs differ very much from the
spherical form.
Il. Non-Euchidian Geometry. By Proressor Cay ey.
[Read January 27, 1890.]
I cONSIDER ordinary three-dimensional space, and use the words point, line, plane, &c.
in their ordinary acceptations; only the notion of distance is altered, viz. instead of taking
the Absolute to be the circle at infinity, I take it to be a quadric surface: in the
analytical developments this is taken to be the imaginary surface a*+y’+2*+w*=0, and
the formule arrived at are those belonging to the so-called Elliptic Space. The object
of the Memoir is to set out, in a somewhat more systematic form than has been hitherto
done, the general theory; and in particular to further develope the analytical formule in
regard to the perpendiculars of two given lines. It is to be remarked that not only all
purely descriptive theorems of Euclidian geometry hold good in the new theory; but that
this is the case also (only we in nowise attend to them) with theorems relating to
parallelism and perpendicularity, in the Euclidian sense of the words. In Euclidian
geometry, infinity is a special plane, the plane of the circle at infinity, and we consider
(for instance) parallel lines, that is lines which meet in a point of this plane: in the
new theory infinity is a plane in nowise distinguishable from any other plane, and there
is no occasion to consider (although they exist) lines meeting in a point of this plane,
that is parallel lines in the Euclidian sense. So again, given any two lines, there exists
always, in the Euclidian sense, a single line perpendicular to each of the given lines,
but this is not in the new sense a perpendicular line; there is nothing to distinguish
it from any other line cutting the two given lines, and consequently no occasion to
consider it: we do consider the lines—there are in fact two such lines—which in the
new sense of the word are perpendicular to each of the given lines.
It should be observed that the term distance is used to include inclination: we have,
say, a linear distance between two points; an angular distance between two lines which
meet; and a dihedral distance between two planes. But all these are distances of the
same kind, having a common unit, the quadrant, represented by 47; and in fact any
distance may be considered indifferently as a linear, an angular, or a dihedral distance:
the word, perpendicular, usually represented by 1, refers of course to a distance=}7.
We have moreover the distance of a point from a plane, that of a point from a line,
and that of a plane from a line. Two lines which do not meet may be 1, and in
particular they may be reciprocal: in general they have two distances; and they have
also a “moment” and “comoment”, the values of which serve to express those of the
38 Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
two distances. Lines may be, in several distinct senses, as will be explained, parallel;
and for this reason the word parallel is never used simpliciter; the notion of parallelism
does not apply to planes, nor to points.
Elliptic space has been considered and the theory developed in connexion with the
imaginaries called by Clifford biquaternions, and as applied to Mechanics: I refer to the
names, Ball, Buchheim, Clifford, Cox, Gravelius, Heath, Klein, and Lindemann: in particular
much of the purely geometrical theory is due to Clifford. Memoirs by Buchheim and
Heath are referred to further on.
Geometrical Notions. Nos. 1 to 16.
1. The Absolute is a general quadric surface: it has therefore lines of two kinds,
which it is convenient to distinguish as directrices and generatrices: through each point
of the surface there is a directrix and a generatrix, and the plane through these two
lines is the tangent plane at the point. A line meets the surface in two points, say
A, C; the generatrix at A meets the directrix at C; and the directrix at A meets the
Fig. 1.
A
generatrix at C; and we have thus on the surface two new points B, D; joining these
we have a line BD, which is the reciprocal of AC; viz. BD is the intersection of the
planes BAD, BCD which are the tangent planes at A, C respectively, and similarly AC
is the intersection of the planes ABC, ADC which are the tangent planes at B, D
respectively.
According to what follows, reciprocal lines are 1, but 1 lines are not in general
reciprocal; thus the two epithets are not convertible, and there will be occasion throughout
to speak of reciprocal lines.
2. Two points may be harmonic; that is the two points and the intersections of
their line of junction with the Absolute may form a harmonic range: the two points
are in this case said to be 4
Two planes may be harmonic: that is the two planes and the tangent planes of
the Absolute through their line of intersection may form a harmonic plane-pencil: the
two planes are said to be 1.
Pror, CAYLEY, ON NON-EUCLIDIAN GEOMETRY. 39
Two lines which meet may be harmonic: that is the two lines and the tangents
from their point of intersection to the section of the Absolute by their common plane
may form a harmonic pencil: the two lines are said to be 1.
The locus of all the points 4 to a given point is a plane, the reciprocal or polar
plane of the given point; and similarly the envelope of all the planes 1 to a given
plane is a point, the pole of the given plane: a point and plane reciprocal to each other,
or say a pole and polar plane, are said to be 1.
3. If a point is situate anywhere in a given line, the 1 plane passes always through
the reciprocal line: each point of the reciprocal line is thus a point of the 1 plane
Le. it is 1 to the given point: that is, considering two reciprocal lines, any point on
the one line and any point on the other line are 1. Similarly any plane through the
one line and any plane through the other line are 1.
A line and plane may be harmonic; that is they may be reciprocal in regard to
the cone, vertex their point of intersection, circumscribed to the Absolute; the line and
plane are said to be 1. The 14 plane passes through the reciprocal line, and conversely
every plane through the reciprocal line is a 1 plane. It may be added that the line
passes through the 1 poiut of the plane; and conversely, that every line through the
1 point of a plane is 1 to the plane. Moreover if a line and plane be 1, the line
is 1 to every line in the plane and through the point of intersection,
A line and point may be harmonic; that is they may be reciprocal in regard to the
section of the Absolute by their common plane: the line and point are said to be 1.
The 1 point lies in the reciprocal line, and conversely every point of the reciprocal line
is a 1 point. It may be added that the line lies in the 1 plane of the point: and
conversely that every line in the 1 plane of a point is 1 to the point. Moreover if a
line and point be 1, the line is i to every line through the point and in the plane of
junction.
4. We may have a triangle ABC composed of three lines BU, CA, AB in the same
plane: the six parts hereof are the linear distances B, C; C, A; A, B of the angular
points, and the angular distances of the sides CA, AB; AB, BC; BC, CA. Similarly
we may have a trihedral composed of three lines meeting in a point, say the planes
through the several pairs of lines are A, B, C respectively: the six parts hereof are
the angular distances CA, AB; AB, BC; BC, CA of the three lines, and the dihedral
distances B, C; C, A; A, B of the three planes. According to the definitions of distance
hereinafter adopted, the relation of the six parts is that of the sides and angles of a
spherical triangle: in particular, if two sides are each =47, then the opposite angles are
each=47, and the included angle and the opposite side have a common value; and so
also if two angles are each=47, then the opposite sides are each =47, and the included
side and the opposite angle have a common value,
5. Let A, C be points on a line, and B, D points on the reciprocal line; by what
precedes, each of the lines AB, AD, CB, CD is = 37: also each of the angles ACD, ACB,
40 Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
CAB, CAD is =}x. The line AC is 1 to the plane BCD and to the lines BC, CD,
in that plane; it is also 1 to the plane BAD and to the lines BA, AD in that plane;
and similarly for the line BD. From the trihedral of the planes which meet in C, distance
of planes ACB, ACD = distance of lines BC, CD, viz. the dihedral distance of two planes
through the line AC is equal to the angular distance of their intersections with the 1 plane
BCD; and it is therefore equal also to the linear distance of their intersections with the
Fig. 2.
B
other 1 plane BAD: and so from the triangle BCD, where BC, CD are each =47, the
angular distance BCD is equal to the linear distance BD; that is the distance of the
planes ACB, ACD, that of the lines BC, CD that of the lines BA, AD and that of
the points B, D are all of them equal; say the value of each of them is =@. And
in like manner the distance of the planes ABD, CBD, that of the lines AB, BC, that
of the lines AD, DC and that of the points A, C are all of them equal: say the value
of each of them is =6.
The theorem may be stated as follows: all the planes 1 to a given line intersect
in the reciprocal line: and if we have through the given line any two planes, the distance
of these two planes, the distance between their lines of intersection with any one of
the + planes, and the distance between their points of intersection with the reciprocal
line are all of them equal.
And it thus appears also that a distance may be represented indifferently as a linear
distance, an angular distance, or a dihedral distance.
6. Consider a point and a plane: we may through the point draw a line + to the
plane, and intersecting it in a point called the ‘foot’: the distance of the point and plane
is then (as a definition) taken to be equal to that of the point and foot. It may be
added that the 4 line is in fact the line joining the point with the 1 point of the
plane; and that the distance of the point and plane is equal to the complement of the
distance of the point and the + point. Or again, we may in the plane draw a line 1 to
the point, and determining with it a plane called the roof: and then (as an equivalent
definition) the distance of the plane and point is equal to the distance of the plane and
roof. It may be added that the + line is in fact the intersection of the plane with
the + plane of the point, and that the distance of the point and plane is also equal
to the complement of the distance of the plane and the + plane of the point.
Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY. 41
7. Consider a point and line: we have through the point a line 1 to the line and
cutting it in a point called the foot; the distance of the point and line is then (as a
definition) equal to the distance of the point and foot. It may be added that the foot
is the intersection with the line of a plane 1 thereto through the point.
Again consider a plane and line: we have in the plane a line 1 to the line and
determining with it a plane called the roof: the distance of the plane and line is then
as a definition equal to the distance of the plane and roof. It may be added that the
roof is the plane determined by the line and a point 1 thereto in the plane.
8. If two lines intersect, then their reciprocals also intersect. Say the intersecting
lines are X, Y; and their reciprocals X’, Y’ respectively; then K, the point of intersection
of X, Y, has for its reciprocal the plane of the lines X’, Y’; and similarly XK’, the point
of intersection of the lines X’, Y’, has for its reciprocal the plane of the lines X, Y:
hence KK’ has for its reciprocal the line of intersection of the planes XY and X’Y’;
say this is the line A, meeting X, Y, X’, Y’, in the points a, 8, a, 8’ respectively. Since
Fig. 3.
K, K’ are points in the reciprocal lines X, X’ (or in the reciprocal lines Y, Y’) the
distance KK’ is =}; and since the plane XY passes through the line A which is the
reciprocal of KK’, the line KK’ is 1 to the plane XY and also to each of the lines
X, Y (it is also 1 to the plane X’Y’ and to each of the lines X’, Y’). Again since
the lines KK’ and A are reciprocal, each of the distances Kz, K8 is =47; that is the
line A is + to each of the lines X and Y (and similarly it is 1 to each of the lines
X’ and Y’). Moreover the angle at K or distance of the lines X and Y (which is equal
to the distance of the planes K KX and K’KY) is equal to the distance af of the
intersections of A with the lines X and Y respectively. We have thus for the two
intersecting lines X and Y, the two lines KK’ and A each of them 1 to the two lines:
where observe that KK’ is the line of junction of the point of intersection of the two
given lines with the point of intersection of the reciprocal lines; and that A is the line of
intersection of the plane of the two given lines with the plane of the reciprocal lines.
The linear distance along AK’ between the two lines is =0; the dihedral distance between
the planes which KK’ determines with the two lines respectively is equal to the angular
distance between the two lines. The linear distance along A is equal to the angular
Wore OV. PART IL. 6
42 Pror,. CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
distance between the two lines; the dihedral distance between the two planes which A
determines with the two lines respectively is = 0.
9. If two lines are such that the first of them intersects the reciprocal of the
second of them, then also the second will intersect the reciprocal of the first; the two
lines are in this case said to be contrasecting lines; or more simply, to contrasect: and
contrasecting lines are said to be 1. Supposing that the two lines are X, Y and
their reciprocals X’, Y’ respectively, we have here X, Y’ intersecting in a point XK, and
X’, Y intersecting in a point XK’: and the planes XY’, X'Y intersect in a line A
which meets the lines X, Y, X’, Y’ in the points a, 8, a’, 8’ respectively. As before
the lines AK’ and A are reciprocal: the distance KK’ is =}7; and KK' is 1 to
the plane XY’, that is to each of the lines X, Y’; and also to the plane X’Y, that
is to each of the lines X’, Y; it is thus 1 to each of the lines X and Y. Again
each of the angles at a, 8, a, @’ is =47; that is the line A is + to each of the lines
Fig. 4.
X, Y’, X’, Y, or say to each of the lines X and Y. Moreover the angle at K or say
the angular distance of the intersecting lines X and Y’ is equal to the distance #8’; and
similarly the angle at K’ or say the angular distance of the intersecting lines X’ and
Y is equal to the distance a’8: but the distances aa’, BS’ are each equal to $7; and
hence the distances af’, a@@ are equal to each other and each of them is equal to the
complement of the distance a8. Thus in the case of two contrasecting lines we have the
lines KK’ and A each of them 1 to the two given lines; where observe that KK’ is
the line joining the point of intersection of X with the reciprocal of Y and the point
of intersection of Y with the reciprocal of X; and that A is the line of intersection of
the plane through XY and the reciprocal of Y with the plane through Y and the reciprocal
of X. The linear distance KK’ between the two lines along the first of these lines is
thus =4}7.
10. We have KK’ and A reciprocal lines; on the first of these we have the points
K, K’ which are 1 points: hence also the planes AK and AK’ are 1; but the plane
‘AK is the plane AXY’ or say the plane AX, and the plane AX’ is the plane AX’Y
or say the plane AY; hence the planes AX and AY are 1. Similarly the line A cuts
the two lines in the points a, 8; and the line KK’ determines with these two points
Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY, 43
respectively the plane AKK'a, that is KK’X, and KK’, that is KK’Y; and thus the
linear distance between the two points a, 8 is equal to the dihedral distance between
the two planes KK'X and KK’'Y. Thus the 1 line A cuts the two lines in two points
a, 8 the linear distance of which is, say 6: and it determines with them two planes
the dihedral distance of which is =}. And the other 1 line KK’ cuts the two lines
in the points K, K’ the linear distance of which is =47, and it determines with them
two planes the dihedral distance of which is =6.
11. Consider a line X and its reciprocal X’: a line intersecting each of these also
contrasects each of them and is thus 1 to each of them: and similarly if Y be any
other line and Y’ its reciprocal, a line intersecting Y and Y’ also contrasects each of them
and is thus 1 to each of them. Hence a line which meets each of the four lines
X, X’, Y, Y’ is also 1 to each of them, or attending only to the lines X, Y, say it
is a 1 of these lines: there are two 1s; and clearly these are reciprocal to each other,
for if a line meets X, Y, X’, Y’ then its reciprocal meets X’, Y’, X, Y, that is the
same four lines. Looking back to figure 2 we may take AB, OD for the given lines, and
AC, BD for the two 1s; as just remarked these are reciprocal to each other. The 1 AC
cuts the two lines respectively in the two points A and C the linear distance of which
is say =6; and it determines with them two planes ACB, ACD, the dihedral distance of
which is say =9@. Similarly the other 1 BD meets the two lines respectively in the two
points B and D the linear distance of which is =@, and it determines with them two planes
BDA, BDC the dihedral distance of which is =6. In the plane triangles which are the
faces of the tetrahedron ABCD, there is in each triangle an angle opposite to AC or BD
and which, or say the angular distance of the two including sides, is thus =6 or 6. Except
as aforesaid the sides, angles, and dihedral angles, or say the linear, angular, and
dihedral distances of the tetrahedron are each of them =47.
12. Considering the lines X and Y as given, the distances 6 and @ depend upon
two functions called the Moment and the Comoment: viz. moment=0 is the condition in
order that the two lines may intersect (or, what is the same thing, in order that their
reciprocals may intersect): comoment=0 is the condition in order that the two lines may
contrasect, that is each line meet the reciprocal of the other one. It may be convenient
to mention here that the actual relations are
sin 6 sin @ = Moment, cos 6 cos = Comoment.
In particular if moment=0, then the lines intersect; we have, say 6=0, and therefore
cos @=comoment; if comoment=0, then the lines contrasect, that is they are 1: we have,
say 0=47, that is siné=moment. These are the two particular cases which have been
considered above.
13. Consider as above the two lines, X, Y met by the 1 6 in the two points A
and C respectively. Consider at A a line J 1 to the lines X, 6; and take I the plane
of the lines (X, 8) and © the plane of the lines (X, J). Similarly consider at C a line
K « to the lines Y, 8, and take II, the plane of the lines (Y, 6) and Q, the plane of the
6—2
44 Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
lines (Y, K): we have thus through A two planes II, Q meeting in the line X; and
through C@ two planes II,, ©, meeting in the line Y. It requires only a little reflection
to see that the distances of these planes are
(I, 1)=6 (9,9) =
(1, 2) =47, (I, 0,)=$7; (I, O,)=42, (I, 2)=97.
Fig. 5.
In fact II, Il, are the before mentioned planes ACB, ACD the distance of which was
=6@: , Q, are planes having the common 1 AC, which is the line through the poles of
these planes, and such that the distance AC is equal to the distance of the two poles,
that is the distance of the two planes. Moreover from the definitions the distances (II, Q)
and (II,, 2,) are each =}: the plane II passes through the 1 at C to the plane Q,
that is (II, Q,)=47; and similarly the plane II, passes through the 1 at A to the plane
O, that is (II,, ®)=}7; and we have thus the relations in question,
The consideration of these planes leads, (see post 31 and 32), to the before mentioned
equation, cos 8cos@=comoment; if instead of one of the lines, say Y, we consider the
reciprocal line Y’, then the angles 6, @ are changed each of them into its complement, and
we deduce immediately the other equation, sin 6 sin @ = Moment.
14. It may happen that instead of the determinate number 2, we have a singly infinite
system of 1s: viz. this will be so if the lines X, X’, Y, Y’, are generating lines (of the
same kind) of a hyperboloid. They will be so if the lines X and Y each of them meet
the same two lines (of the same kind) of the Absolute, say if X, Y each meet two directrices
D,, D,, or two generatrices G,, G,; but it seems less easy to prove conversely that the
lines XY and Y must satisfy one of these two conditions. Suppose first that X, Y each
meet the two directrices D,, D,; say X meets them in a,, a, and Y in B,, B, respectively.
We have at a, a generatrix which meets D,, suppose in a,’ and at @,, a generatrix which
meets D,, suppose in @,'; joining a,’, «,, we have the line X’ which is the reciprocal of A;
viz. X’ meets each of the lines D,, D,: similarly the generatrices at 8,, 8, meet D,, D,
in the points 8,, 8,’ respectively, and joining these we have the line Y’ which is the
reciprocal of Y: thus Y’ meets each of the lines D, and D,: the line D, meets the four
geueratrices in the points 4,, 4,’, 8,, 8,’ respectively, and the line D, meets the same four
Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY, 45
generatrices in the points a,, @,, B,', 8,: thus AH (4,, 4,8, 8,)=AH (a,,4,, BY, 8,), AH
denoting anharmonic ratio as usual. But AH (a’,, a,, 8,', 8,) = AH (a,, a,', B,, 8,') and thus
the equation may be written AH (a,,a,', B,, B,) =AH (4,, ,', B,, B,') viz. the lines X, X’, Y, Y’,
cut D,, D, homographically; and there is thus a singly infinite system of lines cutting
D,, D, homographically: that is X, X’, Y, Y’, are lines (of the same kind) of a hyperboloid.
And similarly if X, Y each cut the same two generating lines G,, G,, then will X’, Y’
also cut these lines and X, X’, Y, Y’ will cut them homographically, that is X, X’, Y, Y’
will be lines (of the same kind) of a hyperboloid.
Fig. 6.
The condition may be otherwise stated; if the lines X, VY have for 1s any two
directrices D,, D, or any two generatrices G,, G, of the Absolute, then in either case
there will be a singly infinite series of 1s: the 1 distances are all of them equal; say
we have 6=6, and therefore sin*6=moment, cos*6=comoment; and therefore moment
+comoment=1; or as the equation is more properly written, + moment + comoment = 1.
15. Two lines X, Y each of them meeting the same two directrices D,, D, are said
to be “right parallels”; and similarly two lines XY, Y each meeting the same two generatrices
G,, G, are said to be “left parallels”: the selection as to which set of lines of the Absolute
shall be called directrices and which shall be called generatrices will be made further on,
(see post 35). We have just seen that if two lines are right parallels, or are left parallels,
then in either case there is a singly infinite series of 1s. It may be remarked that
reciprocal lines are at once right parallels and left parallels; and that in this case there
is a doubly infinite series of 1s, viz. every line cutting the two lines is a 1.
Observe that right parallels do not meet, and left parallels do not meet: their doing
so would imply in the one case the meeting of two directrices, and in the other case
the meeting of two generatrices.
16. If instead of the foregoing definitions by means of two directrices or two generatrices,
we consider a directrix and a generatrix of the Absolute, and define parallel lines by
reference thereto, then it is at once seen that there are 3 chief forms, and several sub-
forms; the directrix and generatrix meet in a point, or say an ineunt, of the Absolute,
and lie in a plane which is a tangent plane of the Absolute: we may have two lines
X, Y which
46 Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
1°. Each pass through the ineunt, neither of them lying in the tangent plane.
2°. Each lie in the tangent plane, neither of them passing through the ineunt.
3°. One passes through the ineunt, but does not he in the tangent plane: the other
lies in the tangent plane, but does not pass through the ineunt.
Observe that in the cases 1° and 2° the lines X and Y intersect, but in the case
3° they do not intersect. The lines in the case 3° are I believe what Buchheim has
termed §8-parallels, his a-parallels being the foregoing right or left parallels*. The subforms
arise by omitting in 1°, 2°, or 3°, as the case may be, the negative condition in regard
to the two lines or to one of them; as the question is not here further pursued I do not
attempt to give names to these several kinds of parallel lines.
Point-, line-, and plane- coordinates: General formule. Art. Nos. 17 to 20.
17. We consider point-coordinates (#, y, z, w): line-coordinates (a, b, ¢, f, g, h), where
af +bg +ch=0, and plane-coordinates (£, , §& @); if we have a line which is at once
through two points and in two planes, then the line-coordinates are given by
a : b : c ; fa ; g : h
= Yy20— Yo, > F lg — 2%, > UYo— UY, > Y%
wz E,o, = E,0, = @o > 1g, > 53% — 5,0, : 7,65 a 726, : Ga F oe, : En, 7 Em.
Similarly if a plane be determined by three points thereof, then the coordinates of
the plane are given by
W,— UW, 2 YW, — YW, > ZW, — ZW,
Eesenwe, Co2x@i=i|9 1 [ices 1 | 1 ii sl! ales
a, Yo 2, WwW, | | v,, Yp> 41> WwW, | vs Y»> 2 wW, v5 Yyp> 2 Ww,
Ty, Yor 22 We | Tey Yor 21 Wy | | Lor Yor Zar We Xe, Yor 2, Wy
Le, Yor 2, Ws | sy Yo, 2g, Ws | | Bs, Yor 23. Ws | Us, Yar 2, Ws
and if a point be given as the intersection of three planes, then the coordinates of the
point are
2 slgises ew=,| * 1 | 1 : Wife
| t3 I> G, @, Es My» Sy @, | eh, p> os a; E., My> Six @,
| £4 Ne» ee @, Be Nor Syr My é., No» Sor Wo E,5 Ne» o o,
| ea Ng> Gs @, | ES Ns» oS ®, Es, Ng, Sgr Os | E,, 3) an @s |
18. The conditions in order that a point (z, y, z, w) may be situate on a line
(a, b, c, f, g, h) are
hy —gz+aw=0,
—-he . +fy+bw =0,
ge—fy . +cew=0,
—axr—by-—cz . =0,
viz. these constitute a twofold relation.
* See Buchheim, A Memoir on Biquaternions. Amer. Math. Jour. t. 7 (1885), pp. 293—326.
Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY. 47
Similarly the conditions in order that the plane (& 7, § ») may contain the line
(a, b, ¢, f, g, h) are
en — b6 + fo = 0,
—c& . +af+go=0,
bE—an . +ho=0,
—f&-—gn-ht . =0,
viz. these constitute a twofold relation.
19. The condition in order that two lines (a, b, ¢, f, g, h), (A, B, C, F, G H) may
meet is
Af+ Bg+Ch+ Fa+Gb+ Hc =0.
Supposing that the two lines meet, we have at the point of intersection
hy —gz+aw=0, . HBy—Gz+Aw=0,
—he . +fz+bw=0, —-Hxe . +F2z+Bw=0,
gxe—fy . +cew=0, Ge-Fy . +Cw=0,
—ax—by—cz . =0, —Ax-—By-Cz . =0;
and from these equations we can find the coordinates , y, z, w of the point of inter-
section in a fourfold form, viz. we may write
a:y:2:w=fA+bG+cH: gA-aG : hA-aH : hG —gH
= fB-bF :gB+cH+aF: hB-bH : fH-hF
= fC-cF : gC-cG@ :hC+aF+bG: gF-fG
a) OSE 2) cA al! 2 YaBB tA SsfAPg BERG.
There is no real advantage in any one over any other of these forms, but it is con-
venient to work with the last of them
eiy:2:w= bC-cB : cA- : aB—bA :fA+gB+h.
20. In like manner if two lines intersect the plane which contains each of them
is given by
Ein:€:o=aF +gB+hC: oF-fB : cF—fC : cB—bC
= aG—gA :bG+hC+fA: cG@-glC +: aC—cA
SS eA: vin, bE — RB, = CH EAs GR: 17bA eb
= gan \hB-fE we fesgor : oF +bG+cH;
or say we have
E:n:€:0= gH-hG : hF-fH : fG-gF :aF+bG@+cH.
48 Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
The Absolute. Nos. 21 to 27.
21. The equation is
in point coordinates a*+ yt2t+w'=0,
in plane coordinates &+7°+°+o*=0,
in line coordinates a? +b'+e°+f*+g?+h'=0.
Hence 1 of plane (&, 7, € @) is point (&, 9, & o),
1 of point (x, y, 2, w) is plane (a, y, 2, w).
Reciprocal of line (a, b, c, f, g, h) is line (f, g, 4, a, 6, ¢);
Points (a, y, z, 2), (#’, y', 7, w) are + if wa’ + yy'+22'+ww'=0;
Planes (£, 7, § ©), (&, 7, &, w’) are 1 if EE’ +n’ + &’+aw'=0.
22. A line (a, b, c, f, g, h) and plane (£, 7, § w) are 1 when the line passes through
the 1 point of the plane, that is the point (& 7, §& ): the conditions (equivalent to
two equations) are
hn —9€ +a =0,
he . +f€+bo=0,
gE—jfn . +co=0,
—a&—bln-—c& . =0.
A line (a, b, ¢, f, g, h) and point (a, y, z, w) are 1 when the line lies in the 1 plane
of the point, that is in the plane (a, y, z, w): the conditions (equivalent to two equa-
tions) are
cy — bz — fw =0,
—ce . +az+gw=0,
be —ay . +hw=0,
—fe-—gy—hz . =0.
Two lines (a, b, ¢, f, g, h), (v, Uc, f’, g’, WV) which meet, that is for which
af’ +b +ch'+af+b'g+ch=0, are + if
ad + bb’ +-ce' + ff’ +99 + hh’ =0.
23. There will be occasion to consider the pair of tangent planes drawn through
the line (a, b, ¢, f, g, h) to the Absolute. Writing for shortness
P= . hy—gz+aw,
Q=—-he . +fze+bu,
R= gaz—fy . +cu,
S=—-aaz—by—cz . ,
Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY. 49
it may be shown that the equation of the pair of planes is
P+Qt+h+S'=0,
In fact writing for a moment (&, 7, & @) and (&, 7’, &’, @’) to denote the coefficients of
(x, y, Z, w) in P and Q respectively, so that (&, 7, 6 ) =(0, h, —g, a), (&, 7’, €, w’) =(—h, 0, f, 6),
then equation of the planes is
(EP — EQ)’ + (n'P —0Q)* + (&P — £Q)’ + (w'P — wo)’ = 0,
that is (E° +4" + 6° +”) P— 2 (EE + 97’ + &' + ww’) PQ + (E+ 9° + 67 +’) Q’=0,
viz. this equation is
(+h? +f") P?+2(fg—ab)PQ+(@7 +9 +h’) Y=.
But P, Q, R, S are connected by the identical equations
cQ —bR+ fS =0,
—-cP . +ak+ gS =0,
bP-aQ . +hS=0,
—fP—gQ-hR . =0,
and using these equations to express R, S in terms of P, Q, viz. writing
R=—7(fP+9Q, S=-7 (bP -aQ),
we see that the last preceding equation is equivalent to P?+@Q’+ R?+S°=0.
24. Similarly if P=) cyber,
Q,=-ce . +az+ gu,
R,= be-ay . +hu,
S,=-fe-gy-hz . ,
functions which are connected by the identical relations
hQ,—gk, + aS, =0,
—hP, . +fR,+bS,=0,
gP,—fQ, . +cS,=0,
—aP,—bQ,-cR, . =0;
then in like manner we have P2+Q'+R?7 +8? =0,
for the equation of the pair of tangent planes from the reciprocal line (f, g, h, a, b, c)
to the Absolute. And we may remark the identity
(P+ Q+ +S) + (Pit QP + Rit 8) =V+P+ e+ f+ ge +h’) a+ y+ e+’).
Vou. XV. Parr I. 7
50 Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
We in fact have
xv y z Ww
P+ Q+R4+S8=2 v+gth’, a—fy ac—hf cg—bh |;
y, ab—fg |B+h+f*| be-—gh ah —of
|
z| ac-hf be-gh &+f*+g’| bf—ag
w | eg — bh ah —of bf-ag | @+0?+e
and in like manner
x y z w
P2+Q24R24+S8i=2\/F+E+f? | —ab+fy | —acthf | —ceg+bh
cS
|
y
a4
Ran
C+04+9 | —be+gh | —ah+ef
|
—bf+ag
—bftag | fi+g +h
z| —acthf —-be+gh a+b'+h?
—ah+ef
25. For the distance of two points (a, y, z, w) and (a’, y’, 2’, w’) we have
ax + yy +22 + ww"
cos 6 =
Veit PtetfrtgGth’
ety tetw Va +y2+272 +0?"
whence also sin 6 =
where in the numerator (a, b, c, f, g, h) stand for the coordinates of the line of junction
of the two points, taken to be equal to y2’—y'z, za’ —2'x, ay'—ay, aw’—a'w, yw'—y'u,
zw’ —z'w respectively.
Similarly for the distance of two planes (&, 7, & ©) and (£’, 7’, ¢, w’) we have
a EE + nn’ + £6 + wo" ;
cos 6 = $y
VE 474+ 0+ 0° VE? + 97467 + 0”
2 2 2 2 2 2
whence also sin 6 = Vat +bF+e sol ima i aL
VPait lta Wty tobe
where in the numerator (a, b, c, f, g, h) stand for the coordinates of the line of inter-
section of the two planes, taken to be equal to &w'—£’w, nw’—1'0, bw’ — Cw, nf’ —7F,
cE’ —C'E, En’ — &’y respectively.
Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY. ol
The distance of a point (a, y, z, w) and plane (£, 7’, ¢, w’) is the complement of
the distance of the point (#, y, z, w) and the point (&, 7’, &, w’) which is the 1 point
of the plane; viz. we have
: vb’ + yn! + 26' + wo’
Veitytetui VEP 4 +E? 4+?
V, 2 2 2 P2 2 2
oe G+0+EO+fPr+gth _
Veit pt ett VER +g? + C4 a
where in the numerator (a, b, c, f, g, h) stand for the coordinates of the line of junction
of the two points. Of course the same result might have been equally well derived
from the formula for the distance of two planes.
26. If we now consider a plane triangle ABC, and write
(2, %, 2, W,) for the coordinates of A,
(Dryas ea, 0) * y Bb,
‘s (25, Y3> 43) Ws) ” ” C,
then the coordinates
a, b, Bs fs . g h
of the line BC will be
Ysera Yo%q1 2g ZyBqs VYy—UeYo, VWy— LW, YsWy—YoWy, ZW, — ZW;
and similarly for the coordinates of the lines BC, CA; the equations
a, f, + O,9,+ 6h, + af, +b.g, +¢,h, =0, &e.,
which express that these lines meet in pairs in the points A, B, C respectively are of
course satisfied identically; and we then have for the sides and angles (linear and
angular distances) of the triangle
TL, + YY, + 2.2, + WW,
cos ¢) ——— ————— = = == i
Va, ar Yo° aa ate W,. va3 ar Ys ar zy ar Ws.
sina = Va, at OF at Grasiie + a alg h,*
V Ee +ye +2 +, Veet yet Zz, +w,
cos A = U0, BE bp, AF C05 tiled ar IoIs ar hh, &e.
Va,’ =F sy a5 Cy. ya + Os at h* Va," ar b, ats Cs. iar ate Is. + h,
and this being so, if with the values of cosa, cosb, cosc, we form the expression for
cosa — cosb cose, then reducing to a common denominator, the expression for the numerator
is at once found to be
= 4,0, ai bib, als CLC, AN UF InDs aie Nhs,
cos @ — cos b cose
sin b sinc
and thence easily cos A
viz. the expressions for the angles in terms of the sides are those of ordinary spherical
trigonometry.
7—2
52 Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
27. Hence also
: /1 —cos*a — cos*b — cos?c + 2 cosa cosb cos¢
sin A = ———— : :
sin b> sine
whence sin A : sinB: snC=sina: sinb : sine,
cos a(1 — cos*a — cos*b — cos*c + 2 cos a cos b cos c)
and cos A + cos B cos C= ENTE Saar Ee ~ ;
sin’« sin b sinc
cos A + cos B cos C
sin B sin C .
and consequently COs & =
which completes the system of formule.
And similarly for a trihedral, that is if we have three planes A, B, C (meeting of
course in a point, 0) then the dihedral distances BC, CA, AB and the angular distances
CA, AB; AB, BC; BC, CA are related to each other in the same way as the angles
and sides of an ordinary spherical triangle.
Distance of a point and line. Nos. 28, 29.
28. The point is taken to be (z,, y,, 2,, w,), the line (a, b, c, f, g, h). Drawing
through the point a 1 plane, say (&, 7, & w) meeting the line in the foot, and taking
the coordinates hereof to be (z,, y,, 2,, w,), then &a,+ny,+ €,+wo,=0 and
hn—gf+aw=0, giving say, F= . cy,—bz,+/fu,,
—h—E . +f€+bo=0, n=-Ch, . +402,+ 9u,,
g§—-fn . +cw=0, f= ba,—ay, . +hw,,
—a&—bn—cf . =0, wo =— fa, —gy,—hz,
We have here P47 t+ C4 a= (++?) «74+ &c.,
where (b’+c*+/*) w+ &c. denotes the before mentioned quadric function of (a,, y,, 2,, ,),
which equated to zero, and regarding therein (#,. y,, 2,, w,) as current coordinates re-
presents the pair of tangent-planes from the reciprocal line (f, g, h, a, b, c) to the
Absolute.
Resuming the question in hand we have then
Ex, te NY + oz, =F ow, = 0,
which with . hy,—gz,+aw,=0, gives say —2,= . cn—b&+ fa,
—ha, . +fz,+bw,=0, —y,=-cE . +af+ go,
9£.—fy, - +cw,=0, —z,=—-bE-an . +h,
—az,—by,—cz, . =0, —w,=—ft—gn—-he
Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY. 53
that is “,=(P+eP+f*)x,+ (—ab+fg)y,+ (-ac+hf)z,+ (- cg + bh) w,,
Y= (—abt+fg)x+(P+at+y')y,+ (—be+gh)2z,+ (—ah+cf)w,,
2,= (-cathf)a,+ (—be+gh)y,+(@++h’)2z,4+ (—bf+ag)w,,
wW,= (—cg+bh)x,+ (-ah+cf)y,+ (—bf+ag)z,+(fP+g +h) w,.
We have therefore
LL, + YY, + 2,2, + uw, = (+e +f") x? + &e.,
and wy +Y, +2) tu) =(@4R 44+ Pte +h’) {(C+e+f*) x2+ ke},
where (b*+¢?+/*)a,?+&c. denotes in each case the above-mentioned quadric function of
(G1, Yu» 2) W,)-
In verification of the expression for «,’+y,’+z2,’+w,? it is to be remarked that we
have identically
Ps pt Oto? + (aftbgt chy (af + y+ 2, + w,)
H=(V7 4B 44+ PtP +h) (V+ +f 22+ ke};
here on the left-hand side the whole coefficient of «,’ is
(BP +c°+f*) + (ab—fg)’ + (ca—hf y+ (cg — bh)’ + (af + bg + ch)’,
where the last four terms are together = (b°+¢?+/*) (a+ 9°+h’), and thus the whole
coefficient is (as it should be) =(b'+e4+/*)@+P+e+f? +g? +h’): and similarly for
the coefficients of the remaining terms.
29. Writing then 6 for the required distance we have
LL, + YY + 2,2, + WW,
Vai + yl +2, +, Vag +y2t+22 +02 ;
VB +e +f%) «2+ &e.
Ver tye +z tue VitP tet ft yg th’
cos 6 =
that is cos 6 =
where (b?+¢?+/")«,’+&e. is the above-mentioned quadric function
a, ¥y 1 x, UW,
“|b? +e'+f? | —ab+fg | —acthf | —cg+bh
Yy,| —ab+fg o+atg! —be+ gh SaneneF |
z,| —ac+hf | —be+gh |a?+0°+h?| —Uftag
w =cg+bh | —ah-+of —bf+ag | f*t+g+h’
Distance of a plane and line. No. 30.
30. This may be deduced from the last preceding result: the formula as written
down gives the distance of the 1 plane (#,, y,, 2,, w,) from the reciprocal line
(f, g, h, a, b, c): hence writing (&, 7, & w) for (#, y,, 2,, w,) and (a, B, «, Jang, 2) for
54 Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
(fA. ga, h, a 6, c) we have for the distance of plane (&, », & w) and line (a, b, ¢, fi g, h)
the expression
Viaeit+g +h) & + Ke.
cos § = —— a —
VF 4p + Oto Vit P ++ Pte th
where (a°+ 9° +h*) &+&c. denotes the quadrice function
é ” g @
é | at+g+h®| ab—fg ac —hf cg — bh
x i
| ab-—fg |B+h? +f?) be-—gh ah — of
n
£; ac—hf be—gh |C+f°+g?| bf-ag
| a eee
w cg—bh ah —of bf-ag | @+b+e°
The theory of two lines. Nos. 31 to 38.
31. Considering any two lines X, Y it has been seen that these have two 1s,
viz. each 1 is a line cutting as well the two lines X, Y as the reciprocal lines X’, Y’,
say that one of them cuts the lines X, Y in the points A, C respectively, and the other
of them cuts the two lines in the points B, D respectively: and take as before the
distances AC and BD to be = 6 and @ respectively.
The coordinates of the lines X, Y are
(a, b, c, fg, h) and (a,, b,, ¢, fi, 9, h,) respectively;
and if we consider as before the planes II, ©, H,, ©, the coordinates of which are
(l, m,n, p), (4 # ¥% @), (U,, M,, %, Py), (Ay, My My. @,) Tespectively, then X is the inter-
section of the planes [I], 2, and we have
a : b : Ce ie as fig. 8 mht
=la—A\p:mo—pp : ne—vp: my— np: nu—lv : lw—mMA,
and similarly Y is the intersection of the planes II,, ©, and we have
a, : b, : o : uh : Gh : h,
=la,— 4p, 2 M,5,—-/,P, 2 4F,—Y,P, ? My,—Nw, : MA,—Ly, : Lu,—m,)r,.
Also the planes (II, Q), (I,, ©,), (II, ©,), (II,, ©) being mutually 1, we have
IX +m +1 +poa =0,
LD, + mp, + My, + Pye, = 0,
Ir, + mp, + nv, +poa, =9,
LA +m +ny +pyo =0;
Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY. 55)
and for the inclinations to each other of the planes (II, II,) and (Q, Q,)
Cos 5S ea Ee,
Vv + &e. VAP + Ke.
Ul, + mm,+ nn, + pp,
VE+ ke VIP &e, |
cos 0 =
32. The expressions for the coordinates of the two lines give
ad, +bb,+cc,+ff,+99,+hh,= (Ul,+mm, + nn, + pp,) (A, + wy, + vv, + wa,)
— (Dd, + mp, + nv, + po,) (LA+ m+ ny + p,z)
= (ll,+mm,+nn,+ pp,) (AA, + wpe, + vv, + oa,)
= VP+&e. V1?+ &e. Vr? + &e. VA? + ke. cos 8 cos 6.
But we have
C4P oe +f? tg +h? =(C+ m+n’ +p’) (4p 4? 4+ 0°) — (A+ mpetne + po)?
= (l? + &.) (A* + &e.);
and similarly
a; =f be f Be +f? +9, + h? = (i? + m, ae ny +p,) (xe 4 pe + v? a a) wa. (LA, + mp, + n,y, +p,s,)
= (1? + &.) (A,? + &e.).
Hence the last result gives
aa, + bb, + ce, alah +99, + hh,
fe ee = cos 6 cos @;
Va? + &e. Va,’ + &e. ‘
or calling the expression on the left-hand side the comoment of the two lines, and
denoting it by M,, the equation just obtained is
cos § cos = comoment, = M,.
And if for either of the lines we substitute its reciprocal, then for 6, @ we have
4a —6, 47 — 6 respectively, and consequently
af, + bg, + ch, + af+bg+ c,h
i ae = sin 6 sin 6;
Va + &e. Va, + &e. :
or calling the expression on the left-hand side the moment of the two lines and
denoting it by M, the equation is
sin 8 sin é = moment, = WM,
where observe that M=0 is the condition for the intersection of the two lines, M/,=0
the condition for their contrasection*,
* The foregoing demonstration of the fundamental Rigid Body in Elliptic Space,’ Phil. Trans. t. 175 (for
formule cos 5 cos @=M,, sindsin@=WM, is in effect that 1884), pp. 281—324.
given by Heath in his Memoir ‘‘On the Dynamics of a
56 Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
33. But to determine the coordinates (A, B, C, F, G, H) of the 1 line AC or BD,
and the coordinates of the points A and C or B and D of the points in which it meets
the lines XY and Y respectively, I employ a different method.
We consider the lines (a, b, c, fi 9, h), (a, b A Gy hy):
and their reciprocals (f.' 9, h, @ B50). Gia Gis hs. Gs Os Gs
A line (4, B, C, F, G, H) meeting each of these four lines is said to be a per-
pendicular. We have (A, B, C, FG EE (G; "0, (CG, 7a ge 1) 0!
: (fg, h, a, b,c) =0,
ia (6,005) Gye: Fan Gs) =O;
23 (f» Iv h,, a, b,, ¢,) = 9,
equations which determine say A, B, C, F in terms of G, H, and then substituting in
AF+BG+CH=0 we have two values of G: H; i.e. there are two systems of values
(A, B, C, F, G, H), that is two perpendiculars.
The equations may be written
(A+F)(a +f)+(B+@ (6 +9)4+(C+ A) (c +h) =0,
(A+F) (a,+f,)+(B+@ (b,4+9,)+(64 HZ) (ce, +h,) =9,
(A=F)(a —f)+(B-G)(@ -g)+(C—H) (¢ —h)=0,
(A —F)(a,-—f,) + (B-@ (b,-g,) + (C— 4) (e, —h,) = 9.
Hence we have
A+F = B+G : C+H,=
(b+g)(c, +h,)—(b, +9,)(C+h) : (CHh)(a, +f,)—(@t fe, +4) : (@+F)(b, +9)-(G +f,)(b+9),=
Ata : B+, : C+;
A-—F : B-G : C-H,=
(b—g)(c, —h,)—(0, -—9,)(e—h) : C—h) (a, -—f) — (af) (,—y) + (@-f) (6, — 9) -(, -F) O- 9), =
Aa : wB—pB : C-y;
equations which may be written A+ 7, B+G, C+H=2r (At+a, 3+, €+y),
A-F, B-G, C-H=2%(A-a, B-B, O-y),
where A=be,—be+gh,—gh, %=bh,—bh—(cg,—¢,9),
B=ca,—cathf,—-hf, B=cf,-¢,f—(ah,—ah),
C =ab,-ab+fo,-f.9 y=49,-—%9-(bf, — 5, Ff).
34. We have
(A+a)*+ (33+ P¥ + (C+y)*= (at f+ b+gt+(ct+hy} (a, +f) +, 4H) + (+h)
—{(at+f) (a, tf) +(b+9) b, +9) +(C+h) (+h),
(A —a)*+ (3B —f)*+ (C—y)?={(a-f)? + (b—g)' + (c—h)} (a, -A +O, -— 9, +, — 2)
—{(a—f)(a,-f,) + 6-9) (b,-9,) +(e -f) (, —h,))*5
’
Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY. 57
or puttin P= 4h 4+ 4+f%+g +2,
P g Pp us
p; =a, +b? +0, +fPtgy +h’
o,=aa,+bb,+cc,+ ff,+ 99, thh,,
o=af,+bg,+ch,+a,f+b,g+e,h,
the foregoing values are =pp,—(c+0,), pp, —(c—a,)’.
Hence
A? + B+ 0? + F'+G@ 4H? = 40 (p22 — (o +.0,)} = 4p (o'0? — (o — 0);
or we may write 2° = p’p,?—(¢ —a,)’, or say N= J/p*p—(a—c,)’,
w= pp, —(o+0,), #=—Jp'p, —(¢ +0,"
Making a slight change of notation, if we put
uw ait on tht+af+bg+oh _ o
’
Ja? + &e. Ja,? + Ke. PP,
uy ae t 8, + 06, +f, +99, + hh, _ o,
ie Sey os 2 iitpp
Ja? + &e. of, a, + &e. PP1
then the values are
X=rr,V1—(M— Iy, w= —rr,V1+(M + My.
And, this being so, the two systems of values of A, B, C, F, G, H, are
A(A+a)+u(A-a), | A(@+a)-yw(A—-a),
(43+ 8) +4 (—P), | (88+) —p(As— 8),
A(EC+y)+u(C—y), | ~a(C+y7)-u(€C—y),
AAt+a)—w(A-a), | AA+a)+w(A—a),
A (83 +8) — (83-8), | (B+ 8) +4 (38-8),
A(C+y)-w(C—-y), A(C+y7)4+n(C-y);
viz. the two perpendiculars are reciprocals each of the other.
35. Before going further I notice that if
a,t+f, +49, eth, ir Os sa Bll AU HNC
epg on, © ar Cee. ean
then the four equations for (A, B, C, F, G, H) reduce themselves to three equations only :
and thus instead of two perpendiculars we have a singly infinite series of perpendiculars,
(see ante 15).
To explain the meaning of the equations, I observe that a line (a, b, ¢, f, g, h) will
be a generating line of the one kind or say a “generatrix” of the Absolute if
a+f=0, b+g=0, c+h=0:
Wor XV. Parr 1 8
58 Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
and it will be a generating line of the other kind or say a “directrix” of the Absolute
if a—f=0, b—g=0, e—h=0. Or what is the same thing, we have
(a, b, ce, —a, —b, —e) where a’+b*+c’=0 for a generatrix,
and (a, b, c, a, b,c) where a’+b’+c*=0 for a directrix of the Absolute.
Consider now two directrices (a, b, c, a, b, c) and (a, b,, ¢, a, b,, ¢,): if a line
(a, b, c, f, g, h) meets each of these, then
(a+f)a+(b+g)b +(c+h)e =0,
(a+f)a,+(b+ 9) b,+(c +h) ¢,=0,
and consequently
at+f:b+g:c+th=be,—b ec : ca,—ea : ab,—ab,
and similarly if (a,, 4,, ¢,, f,, 9,» 2,) meets each of the two directrices then
a,+f, : b+ 9, : ¢, +h, = be,—b,c : ca,—c,a : ab, —a,b,
that is if the lines each of them meet the same two directrices of the Absolute, then
ath, 4+% _ th,
at+f b+g cth’
and conversely if these relations are satisfied then the lines each of them meet two
directrices of the Absolute.
In like manner if the lines each meet two generatrices of the Absolute, then
ie = 91 mee as h,
a-f b-g ch’
and conversely if these relations are satisfied then the lines each of them meet the’ same
two generatrices of the Absolute. In the former case the lines are said to be “right
parallels” and in the latter case “left parallels.”
A line (a, b, ¢, f, g, h) meets the Absolute in two points, and through each of these
we have a directrix and a generatrix: that is, the line meets two directrices and two
generatrices.
Through a given point we may draw, meeting the two directrices, or meeting the
two generatrices, a line: that is, through a given point we may draw a line
(4,; b,, C,, te Iv h,)
which is a right parallel, and a line which is a left parallel to a given line. That is
regarding as given the first line, and also a point of the second line, there are two
positions of the second line such that for each of them, the 1’s of the pair of lines,
instead of being two determinate lines, are a singly infinite series of lines.
36. Reverting to the general case we have found (A, B, C, PF, G, H) the coordinates
of either of the lines 1 to the given lines (a, b, c, f, g, h) and (a,, b, o,f, 9h):
supposing that the + intersects the first of these lines in the point the coordinates of
which are (#, y, z, w) and the second in the point the coordinates of which are
(@,, Ys 2%, W,)
Pror. CAYLEY, ON NON-EUCLIDIAN GEOMETRY. 59
then we have for each set of coordinates a fourfold expression; the choice of the form
is indifferent, and I write
e:y:2:w=cB-bC:aC-cA:bA-aB: fAt+gBr+hC,
2 :Y,:%:w=¢eB—-bC:aC—cA :bA-aB: fA+g,B+he,
and we have then for the distance of these two points,
ead iat? ck
So = Jz, — y2) + &.
Ja y+ etu Jett yi +27 +4;
J@ty tatu Jat yit+ei+u,
where #=5 or 8, according to the sign of the radical X : « contained in the expressions
for A, B,C, F. G; Hi.
cos f = » sing=
I have not succeeded in obtaining in this manner the final formule for the deter-
mination of the distances: these in fact are, by what precedes, given by the equations
sin sin = M, cos 6 cos @= M,.
For then, writing ¢ to denote either of the distances 6, 0, at pleasure, we have
uk s ia
sin®d cos*h
that is cos! @ + cos’ d (M,? — M*+1)+ M?*=0,
or cos’ $ = 3{(M?— 1? +14 /M'+ M*— 2M 7M? — 2M? — 2M" + 1},
which is the expression for the cosine of the distance.
In the case where the two lines intersect M=0, and if 6 be the 1 distance which
vanishes, then 6=0, and consequently cos@=M,: the last-mentioned formula, putting
therein M=O and taking the radical to be = M,?—1, gives cos*$=WM,’, that is $=0,
and cos*@= MM; as it should be.
37. I verify as follows, in the case in question of two intersecting lines,
(af, +b9,+ch,+a,f+b,9 +¢,h=0),
LL, + YY, + 22, + WW,
the formula. | cos 6 =—— ‘
J@+yt2+u? Joe +y2 427+;
We have here
A=Q= be, —b,c + gh, —9,h,
B=B=ca,—cathf, —hf,
C =C=ab,-ab+fo,-/g,
F=a =bh,—bh—cg, +c,
G=6 =cf,—¢f—ah, +ah,
H=y =ag,—4,9- bf, +b,h.
I stop to notice that these formula may be obtained in a different and somewhat
more simple manner: the two lines (a, b, ¢, f, g, h) and (a,, b,, ¢,, A, J 4) mtersect ;
hence their reciprocals also intersect: the equations of the plane through the two lines
and that of the plane through the two reciprocal lines are respectively
(gh, — g,h) w+ (hf, — hf) y+ 19: — Of) 2 + (af, +89, + ch,) w = 0,
(bc, — b,c) w+ (ca, — ¢,a) y + (ab, — a,b) z + (fa,+ gb, + he,) w= 0,
8—2
60 Pror, CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
the line (A, B, C, F, G, H) is thus the line of intersection of these two planes, and it
is thence easy to obtain the foregoing values.
From the values of A, B, C, F, G, H we have to find a, y, 2, w and 4, y,, 2%; W
by the formule given above. We have
x=cB-b0= ca,—cce,a+chf,—ch, f
— abb, + a,b* — bfg, + baf,
= (bg +ch)f+a,(b? +c’) —b,ab —c,ac — bf, — fh,
=—f(af,+bg,+ch,) + a, (0? + c’) —b,ab—c,ac;
1
or writing here a,f+b,g+c,h in place of —(af,+bg,+ch,) this is a linear function
of a,, ,, ¢,, and similarly finding the values of y, z, w we have
x=a,(b+c?+f")+b, (fg—ab) +.¢,(hf—ca),
y=a,(fg — ab) +b, (ce? +a?+g")+ ¢, (gh — be),
z=a,(hf —ca) +b, (gh— be) +¢,(a°+0'+h’),
w=a,(bh —cg) +b, (ef — ah) +¢,(ag— Df).
And in like manner (I introduce for convenience the sign —, as is allowable)
—«,=a(b? +07 +f") +b(fg,—a,),) +¢(h,f, — ¢,4,),
—y,=a(f,9, -— 4,b,) +b(¢2+ a+ 9,7) +0 (g,h, — b,¢,),
—2z,=a(hf, — ¢a,) +b(g,h, — bc,) +¢(a?+b7+h,%,
—w,=a(bh, — ¢,9,) +b(¢,f, —a,h,) +¢(a,g,— 6,f,).
38. Write for shortness
p=e+hic’, p=fr+get+h’, —w=a,ft+bg+c,, and therefore
q=aa,+bb,+cec,, QM=ff,t+99,t+hh,, -o=af,+bg,+ch,.
r=a,+be+e7, r=f7+92+h,,
We have
L=a,p—aq+fo, £,=—ar+aqt+fo,
y=bp—bg+go, y,=—br+bq+ 9,0,
z=¢c,p—cqtho, 4,=-er+ceqtho,
\ aes Agh Wr LARC h, )
a, b, ¢ ay (oh, ©
a,,b,,¢, a,, b,, ¢, |
from which we easily obtain
a+ y' + 2* = p(pr — q’) + (p, + 2p) ?,
and by expressing w* in the form of a determinant
w =p, (pr —q’) — po’,
Pror, CAYLEY, ON NON-EUCLIDIAN GEOMETRY.
we obtain
a+yt+2+w=(p+p,)(pr—g to’),
and in like manner
oo +yl +2) +? =(rt+r7,) (pr-¢ +o’).
And again
we, + YY, + 22, =4 (pr—q’) + (g, + 2q) @%,
and by expressing ww, in the form of a determinant
ww, = 9, (pr — g°) — qu’,
we find
we, + yy, + 22, + ww, =(¢ + 4,) (pr—-F +o’).
Hence substituting in
LL, + YY, + 22, + ww,
J@+y+2+u Jar tyi+2 + we
cos 0 =
the factor pr—q’+q° disappears, and we have
q+
cos § = 4 4___- y ,
Jp Dele Ae
the required result.
61
III. On the full system of concomitants of three ternary quadrics. By
H. F. Baxer, B.A., Fellow of S. John’s College, Cambridge’.
§ I. Summary.
Tuts Essay was undertaken to find the concomitants of three ternary quadrics. As
such the net result is given in § III. For completeness I have given also a consecutive
account of the present theory, § VII. It is possible that some of the concomitants
given are themselves reducible, for some reductions effected have not been arrived at
at all easily. With a view to rendering the process of obtaining them readable, I have
studied extreme brevity—and it would seem quite practicable to apply the same abbreviated
method to four conics. § IL is an explanation of the method; § IV. its application ;
§ V. investigates a quasi-reducibility of 18 types of forms, reducibility on multiplication by
uz; § VI. gives some necessary identities; § VII. contains a connected account of the
theory as given by Gundelfinger, Rosanes, and in Clebsch’s lectures; and finally § VIII.
gives some notes on the geometry of the forms—though apparently any competent expression
thereof requires the establishment of new geometrical ideas. § IX. gives a list of memoirs
on three conics.
§ II. Explanation of the method.
The method here followed for obtaining the system of concomitants of a system of
ternary forms in terms of which all others can be expressed as rational integral algebraic
functions is based on the remark, due to Gordan or Clebsch (Ueber ternire Formen
dritten Grades, Math. Annal. 1. 90; Ueber biternire Formen mit contragredienten Vari-
abeln, Math. Annal. 1. 359), that if, in the symbolic expression of any concomitant, con-
taining one point variable # and one line variable u, wherein any letter a (symbol of a
form az”) which occurs, can occur only in the combinations a,z, (bea), (baw), we omit the
power of a, (which occurs, say, f times), change (baw), (caw)... into bz, cz... (say g such)
and (bea), (b’c’a)... into (bew), (b'c'w)... (say h such), we thereby both eliminate the letter a
and also obtain a new invariantive combination; namely, we deduce a concomitant of one
degree less than the original (and which in fact has its order lessened by f—g and
its class by g —h).
As then every concomitant of any degree r can be thus treated, it follows conversely
that if we take every possible concomitant of the (7— 1) degree, then in any one such
choose among the components of the type b,, cz... (any one of which may be repeated)
a certain number g(} 7) and replace them by (bau), (caw)... respectively, a being a symbol
* I am indebted to the great kindness of Professor Cayley for several suggestions tending to help the
intelligibility of this essay.
CONCOMITANTS OF THREE TERNARY QUADRICS. 63
(of the form a,”") which does not occur in the concomitant of the (7 —1) degree, and at
the same time replace some (say h, g+h}$n) of the type (bew) by (bea), and then
multiply the result by a,"%—"=a,/ (thus obtaining a concomitant of the 7 degree) and
do this for every value of f, g and h, and for every possible selection of the components
acted on, and for the symbols a of every form of which the obtained combination is to
be a concomitant, we shall obtain finally every possible concomitant of the r® degree.
And under the title of “every possible concomitant of the (r—1) degree,” must of
course be included all forms capable of arising by the process of the first paragraph from
forms of the rt degree, and therefore all products of the (r—1)™ degree obtained by
multiplying forms of lower degree. If however a form K of the (r—1)™ degree can be
written as the sum of products and powers of forms of lower degree, and of products of
forms of the (7-1) degree with powers of the identical covariant u, (namely is, as we
say, a rational integral algebraic function of other forms), then, as each constituent of the
sum must necessarily be also of the (7 —1) degree, the process applied can only result in
giving, from K, forms which are themselves sums of other forms of the rt" degree (some of
these being, possibly, products). Thus, if in our enumeration of forms of the (r—1)" degree,
we include simple products, we can exclude forms which are rational integral algebraic
functions of other included forms, and we shall obtain a series of forms of the r* degree,
in terms of sums of multiples (by numbers or powers of u;) of which, all forms of the
wth degree are expressible and which are therefore by the same reasoning competent to
give the similar system of the (7 +1) degree. It is this sufficient system for the algebraic
rational integral expression of all other concomitants which it is our aim to obtain for
every degree.
Thus far with the general theory. For the case of three ternary quadrics, a,*, b,7, ¢,’,
the method is considerably simplified. Here the derivatives from any concomitant of the
(r —1) degree are obtained by only five distinct operations. (1) (The # operation.) Leaving
uw untouched and replacing one z by the point (vaw)a,=0 or (vbu)bz,=0 or (vew)c,=O0 [Le.
replacing «; by (au); az = (aju;, — ayuj) dz, ete.]. (2) (The w operation.) Leaving « untouched
and replacing one w by the line ayaz=0 or byb,=0 or cyc,=0 [ie. putting for u;, aa, or
bibz OF CiCz]. (3) (The wa operation.) Leaving w untouched and replacing two a’s, that
is, writing for mgnz, (mau) (naw) or (mbu) (nbu) or (mew) (neu). (4) (The we operation.)
Leaving x untouched and replacing two w’s, that is, putting for wytg, dpdg or dybg or CyCy.
(5) (The wu operation.) Replacing one # and one wu, that is, writing for mrp, (maw) ap
or (mbu)b, or (mcu)c,; and upon any form each of these five operations, in their three-
fold method, must be applied in all possible ways. And i is not necessary to consider
products of the (r—1) degree in order to obtain all the requisite forms of the rt” degree.
For first to clear the ideas it may be remarked that, since the number of places in which
a letter a can be introduced, by changing either u, into a, or mz into (mau), cannot
be greater than two (for the second degrees of @ are real coefficients), there is no utility
in considering a product of more than two factors, for one, at least, of these factors will
remain unchauged and be a factor in the result. Further there is no utility in either
of the two first of the ‘five distinct operations, as applied to products, for either of these
will only modify one of the factors of the product and not really bind the two together.
64 Mr H. F. BAKER, ON THE FULL SYSTEM OF
And finally any form obtainable from the product by any of the three remaining operations
of the five can and will arise among the derivatives of each of the factors alone. This
is best explained by example—the root of the matter lies in the remark that a simple
invariantive product (of symbolical factors) involving a quantity 7 once, can be obtained
by continued application of the two processes of changing u; into a; and a; into (au);,
(each time multiplying by a, if requisite), from the single term r,. So that the application
of any one of the ‘three distinct operations’ spoken of to a product of two forms, which
must, to bind them together, introduce a single letter, say a, into each, gives a result
obtainable by taking one of them, introducing one @ and multiplying by a, and then
operating continuously on this az, until the part of the result due to the other form is
obtained. For example take the product a,*.b,? giving rise to a,b,(acu) (bew) and note that
we can proceed thus: a,°, a, (acu) cz, az (acu) (ebu) bz; or take (abc) dzbzcz . (b'e'u) (c'a’u) (abu),
giving rise to (abc) (adw) b,c, (b'c'd) (c’a'u) (a’b'u), (where d=a or b or c), and we can pro-
ceed thus: (abc) dzbzcz, (abe) (adu) beezdz, (abe) (adu) bye, (db’u) by’, (abc) (ad) byez (db’C’) Cx'be’,
(abe) (ad) be, (db‘c’) (cau) (b’a’u), making the form arise from (abc) dzbgc,: and it also
arises from (b’c’w) (c'a’u) (av’b'u). And this reasoning remains valid in case particular com-
binations of the letters are abbreviated by the use of other letters. To see this we may
suppose the original letters explicitly reintroduced, in which case the form will generally
be replaced by a sum of forms and a product of two forms replaced by a sum of
products. But, for example, (A+B+C)(D+#+F) gives for its derivative the sum of
the derivatives of (A+B+0C)D, (A+B+C) £, (A+B+C) F, which latter derivatives are
proved to be also derivatives of (A +B+(), as is also, therefore, effectively, the derivative
of (A+B4+C)(D+#+F).
Passing now to the mode adopted of conducting the method thus justified—the three
conics are written aZ=a,2=a,72=..., b2=b,2=by"=..., Ce =.... and their ‘clusters’ of
tangents, namely (aa’w)?, (bb'u)*, (cc'u)? are written Ug? = Ug? = Ua"... Up? =...) and Uy? =... 5
or say, we write (aa’);= aja, —a,a/ =4;, etc. Then it is to be noticed that the factor a,
in a form involves always the real factor a@,2—for daze = }da.Uz; also a factor (aa'u)
[unless the form contain also (aa’a’) in which case it would be written immediately
Mu,a,” and not need the reductions in question] involves always the (real) second degrees
of a, a, a;. For (adu) f (a) =— (aa'u) f(a’) =43 (acu) {f(a —f(@)}, and, in f(a)—f(@),
a, a only occur in the combinations (aa’); and a, a’ in the whole expression can be
replaced by 4, 4, %, occurring to the second degree. So a factor (aa’x) in an expression
(wherein (aa’a”) is, possibly, not another factor) shews that the expression is reducible to
a form containing a, a only in the second-degree—combinations of the three (aa’),, (aa’).,
(a2’),. And these are reducible, for (aa’x)?=4a,?.a,° and therefore (aa’x) (aa’y) = $da° . dzMy-
In fact u,? =(aa'u)?, whence (aaa)? = (aa’. va’)? [where, as always, (ab.«y) is used for
a a a, be Ly @y |=Agby—Aybe=| G, a. 3 |=| (ab), (ab), (ad), |],
| b, b, b, A) Yo Ys b, b. by Dy Ly an
(wy) (xy)o (xy)s n Ys Ys
or (aa)? = (gly — Ag z)* = 2g? . Ag? — 2Az da ZA a = 2AzPUa* — Zz? Aa? = $Me". An's
CONCOMITANTS OF THREE TERNARY QUADRICS. . 65
uw
(and in case the expression does contain (22’2) (aa’a”), this is 4d. AzMa"). So that in our
investigations where we are seeking to retain only terms which do not contain real factors
of lower degree, we can always omit terms containing a factor (aa’”), since this involves
the real factor a,°. Such terms in (aa’z) are often, here, shortly written aa’; and in fact
in any expression containing a and a we may interchange any a with any a’, the result
being only the neglect of reducible terms. For Mu,vy = M [uav.+ (uv. aa’)] = Mug, + 20’,
which is generally written here Mu.v. = Muav., the sign = meaning, generally, “equal to
except for terms containing real factors of lower degree,” and always “may, in our tables,
be replaced by.” A particular case is when a form reduces entirely to products of
forms of lower degree: this I write =0. A further aid to brevity consists in only writing
down, when there are several forms similarly arising from the different conics, only a re-
presentative one, for example w,b.bz is used to represent the six forms wababz, UsCaCzs UpCgCz,
UplgAz, Uyly1z, Uybybz. The various forms of a fundamental identity, used, are
(abc) dz = (bed) a; + (cad) bz + (abd) cz,
(abe) (def) = (bed) (aef’) + (cad) (bef) + (abd) (cef),
(abu) (bev) — (abv) (bew) = (abc) (bur),
(uab) (ucd) = (uac) (ubd) — (uad) (ube),
(abu) (cdv) — (abv) (cdu) = (bur) (acd) + (wav) (bed),
{(cad) bz + (abd) cz}? = {(abe) d, — (bed) a,z}?;
which, since the squared terms, on expansion, are immediately interpretable as real terms
(for the case when a,’... are conics) gives
(cad) (abd) b,c = — (abc) (bed) azd;.
Further bzb,’ = bybz' + (bb’. xy),
Drbybz'by! = 4 (bz. by’? + by? . bz?) — (bb’. wy)’.
And as typical, the following examples may be given,
1. (aBy) bycpba’bz' btn = (28) batpba'by bate + (aBry) (bb'. yx) cpba'babe
=b,?. (aBry) Caba’by’cz + (ary) (bb' . yx) Cada’ bla = (aPry) (bb' . yx) Caba'brez
= 4 (aBry) (bb’. yx) (beba’ — babz’) Caer =} (aBy) (S’yx)(B’xa) eger
= 4 {(By2z) ca + (yar) cg + (48x) c,} (B’yx) (B’xa) cp
2 (Byx) (Biya) (B'xa) cata + 2 ep° . (yar) (B'yx)(B'aa) = 3 (Byx)(B’y2) (8'xa) coc
+$0,*. (8x) (B’Bx) (B’aa)
4 (Byx) (B’aa) ca {(Byx) ep’ + (B’Bx) cy + (B’yB) cx}
4 (Byx)*. (aP’x) caca’ + BB’ =4(Byz)?.(aB'2) cace’
=0;
and the second column will be, in the work, omitted. The thin lines ————, underneath,
indicate the associations of the parts.
Vou. XV. Part I. 9
66 Mr H. F. BAKER, ON THE FULL SYSTEM OF
2. (ubc) (ub’c’) (abc’) (ab’c)
= (beu) (be'a) (ub’c’) (ab'c)
= |(bec’) (bua) — (bea) (buc’)} (ub’c’) (ab‘c)
= (bec’) (bua) (ub‘c’) (abc) + (abc) (ub’c’) (c’ub) (b’ca)
= 4b,b, (aub’) (abu) + (abc) (ub’c’) {(ubb’) (c’ca) + (be’b’) (wea) + + (c'ub’) (bea)}
= (abo)?. (ub‘c’)? — } (abu) by’ (ab’u) by — § {(woa) op + (ce'a) up} {(cab) (c’ub’) — (cab’) (c'ub)}
= (abe). (ub’c’) —4 (abu) by’. |(b’ub) a, +(uab)b,’ +(ab’b) uy} —4 { (wea) cs’ + (cca) up} {ag (co’ uw) — cp (ac’w)}
= ete.
= (abo)? (ub'c')? — 4 (abu)*. by? = (cau)? o9”* — Fagayuptty + } (pay — rap)"
= — AgayUgt,.
3. (bew) (abe’) (ab’c) (b’a'u) (c'a'u)
= {(ab’c) (c'a’u)} {(abc’) (b’a’u)} (bew)
= |(ab’c’) (ca’u) + (b’ce’) (aa’u) + (cac’) (b'a’u)} {(ab’c’) (ba’u) + (be'b’) (aa’u) + (abb’) (c'a’u)} (bow)
= ete.
=(ab'c’) . (ca'u) (ba'u) (bow) — § (wab) a,b, . (wa’b’)? — § (uc’a’) cg'ag’ . (uca)? + F (aBy) Uatiptty
=1}(aBy) uatary.
4. (abe) agbyuguycr = |(ubc) a, + (auc) by + (abu) cy} agbyupce
= (ubc) ayagbyupcz + b,?. (auc) apuipcr + dey. (abu) agbzug = (ube) aga,byuper
= {(uba) eg + (wac) bg + (abc) ug} aybupex
= (uba) ayby « Cgcxup + 4 bp? . (wac) ayuyCz + Ug? . (abc) a,byCz = (uba) a,b, . CatieCe 3
i.e. (abc) agbyuguyc, = 0,
and the second column would be omitted in the work.
Note. In verification of the theory given, it is worthy of remark that though Gordan
(Math. Annal. 1. 90, ‘Ueber ternire Formen dritten Grades’) does not apparently recognise
that it is not necessary to consider the derivatives of products of forms, yet this is really
not so—the arrangement only is different. As a fact all his 34 concomitants do occur,
independently of the products, in Tables 1—xxIx. (pp. 103—106), except the last one
ufufuy’ (spt) (page 102, 12% Ordn.), which occurs on page 128 as equivalent to 7 of page 127,
namely w,?u’c,d,(cdu) (bew) (abu)? (adu). This last form would however, in accordance with
our theory, arise also, independently of products, from w’. For putting
uv, = (a’b'c’) (b'c'v) (c'a'u) (abu),
CONCOMITANTS OF THREE TERNARY QUADRICS. 67
and then v=c, the form in question is
upd, (edu) (a’b’c’) (b’c'c) (c'a’u) (a’b’u) (beu) (abu)? (adu),
and we should have the following series of derivatives:
of degree 6. u/d,d,’,
» » 7. UPdd, dau) a’,
a 8. u2didz (daw) (abu)? . bz,
5, 3 9. wd, (deu) (daw) (abu)? (bew) cz,
“ » 10. ud, (deu) (dau) (abu)? (bcw) (cb’u) bz”,
e » Ll. ud, (dew) (dau) (abu)? (bex) (cb’c’) (b’c'u) bz'cz’,
fr » 12. ufd; (deu) (dau) (abu)? (beu) (cb’c’) (B'c'a’) (b’a’u) (c'a’u).
which is the form in question.
The form here of seventh degree u/d,d, (daw) a, does occur in Gordan’s work as the
heading of Table xvil., page 111, under the form u,a,a, (abu) b,?: and in our arrangement
there should occur under 3 of that table the form u,2a,a, (abu) (bew)? cz, which is the same
as the form above of 8th degree. But this form it is unnecessary for Gordan to write
down since it arises from the product [wa,a,7.¢,? = u;2apAz (AxCz”) Cz] of two forms included
in the table, § 4, page 101 (viz. under 1 Ordn. and 6 Ordn.), namely by changing 2; into
(bu); and getting u,’ayaz (abu) (chu)? c;. Our arrangement, if longer, possesses the advantage
that all possibilities are exhausted in the course of the work—at any stage it is exhaustive
so far as it has gone—while Gordan’s arrangement is not trustworthy until the examination
is completely finished.
§ III. Statement of the system obtained.
zero degree. uz =(011) (1 form)
degree 1. a,? = (102) (3 forms)
2 © (22) =(Gew)ibzez U_* = (220), (bcu)? = (220), (9 forms)
degree 3. (300), =a,° (8) degree 4. (410) = (bew) bac. (=4Fuca,?) (3)
(300), = b.? (6) (402), = bacabzex (3)
(300), = (abc)? (1) (402), = (ya (3)
(311), = uababz (6) (421), = (bow) bacxtla (6)
(311), = (abc) (beu) az (3) (421). = (bew) byczuy (6)
+(303) = (abc) azbzez (1) +(421), = (a’bc) (uea) (uab) az’ (3)
(330) = (bew) (caw) (abu) (1) (421), = (Byx) ugu, (3)
68
Mr H. F. BAKER, ON THE FULL SYSTEM OF
degree 5. +(501), = (abc) azbaca (= Fy xp) (3) degree 6. (600) = (aBy)?
(501). = (By) aa, (=beb,= Sexe) (3) (611), = (ay) (By2) we
(520) = uguyaga,y (3) +(611). = aga,b,bxtip
(512), = (yx) agazty (6) +(630), = (aBry) warlprty
+(512). = (abc) aguighzcz (6) +(630), = (abu) agbyuprty
(512), = (Byx) crCprty (6) +(630), = (bew) wgu,bycp
7. $(710); = (@By) apaytta (3) (603), = (Bry) (yaa) (482)
+(710), = (bow) aga,bycp (3) +(603), = (Bry) azbsapby
+(721) = (aBy) babstepily (6) +(603), = (Bry) brerbyCp
8. +(801), = (By2) bycpbaCa (3) » 9. +(911) =aga,bbacaCztle
+(801), = (a’be) agayb,caaz’ (3) » 10. +(1010) = (a’Bry) bycpbaCatta’
$(812) = (@By) (vax) (@Bx) ve (3)
its own reciprocal or its reciprocal appears in the table.
The 18 forms marked + are reducible when multiplied by w,. Each form is either
sponding to any type is given by the number in brackets which follows.
§ IV. Establishment of the system. First and second degrees.
The first degree forms from which we start are a,’, b,”, cz’.
From a,’ = (102), Az (au) az = 0,
az (abu) bz,
(aa’u)? = u,2,
(abu)*.
Thus the second degree forms are typified by
(212) = (beu) bez,
(220), = u,2,
(220). = (bcew)?.
Third degree.
From (212) = (bew) b,c, we proceed to shew that we get (300), =a,’,
(220), = w.2 (300). = 6.2,
(220), = (bew)? (300), = (abe)?,
(311), = uababz,
(311), = (abe) (beu) az,
(303) = (abc) azbzez,
(330) = (bew) (caw) (abu).
The number of forms corre-
CONCOMITANTS OF THREE TERNARY QUADRICS. 69
Derivatives from
(212) = (bew) bz cz. From u, = (220). (220). = (bew)?.
1. (bew) by (cau) dy = (311)s. 11. uabab; = (311),. 13. (beu) (bea) a, = (311)s.
2. (beu) bz (cb’u) bz = (311), and (800),. 12. b,? =(300).. 14. (bew) (bec’) cz’ = (311), and
3. (beu) bz (cc’u) cz) = (311). 17. a2 =(800). (300).
4. (bea) axbzCx = (303). 15. (bea)? = (300);.
5. (bec’) Cx bzCx —()) 16. (bcc’)? = (300),.
6. (bceu) (baw) (caw) = (830).
7. (bcw) (bb’u) (cb’u) =0.
8. (bea) b,, (caw) = (311),.
9. (bcb’) b, (cb'x) = (311), and (300),
10. (bec’) by (co’u) =(311).
Of these 1. =— (abc) (abu) cz . Uz =(311)>.
2, =—(beu)? . by? + (bow) b’z . {(bb'n) cx + (cb’b) ue}
= d Cytlg (Uple — UrCp) — 4 UxCp (Wale — UxCp)
=4 fe? . Ug? — Quy CyCpig + Ux? . Cp?| =(311), and (300),.
3. = u,b, (bux — byu,) = (311).
7. =4(bbu) { | =;
9. =—4 Cg (gery — UxCg) = (311), and (300),.
14. =D, — b,u,) = (311), and (300),.
Fourth degree.
We proceed to shew that from
(311), = Wababz we obtain (410) = (bew) boca,
(311), = (abe) (bew) dz (402), = bacabxCe
(803) = (abe) asbace (402), = (Bya),
(330) = (bew) (caw) (abu) (421), = (bew) baexttas
(421), = (bew) byCxtty,
(421), = (abe) (uca) (wab) ay’,
(421), = (Byx) ugtty.
70 Mr H. F. BAKER, ON THE FULL SYSTEM OF
From (311), = wababz. From (311), = (abc) (bew) ay.
1. taba (bau) az = (421)s. 8. (abc) (bow) (aa’w) ay = (421),.
2. aba (bb'w) b’, = (421). (410)
3. Uaba (be) cz = (421),. 9. (abe) (bex) (abu) b’, = {ian
4. ’.b'zbabz = (402). (421);.
5. CaCzdabz = (402),. 10. (abc) (bea’) a’,az = (402),.
6. b’aba (bb'w) = 0. 11. (abc) (beb’) b’,az = (402),.
7. Cabs (bew) = (410). 12. (abc) (bea’) (aa’u) = 0.
13. (abc) (beb’) (ab’u) = (410).
Beir (abe) agBees From (Gan (eeayenu)
1A. (abe) (alu) bee = 0: 18.,. (Gea!) (cau) (abu) a’, = (421),.
Tee (che) (aban er 0a 19. bec’) (car) (abu) vam teat
1G, (abo) a. (hat) (ent G21), 20. (beu) (caa’) (aba’) = (410).
17, (Ghe) a, (leu) (edu) = 420 1. * (beu) (cae!) (abe’) = (410).
Of these
2=4 ug (B22) u. =(421),.
4 = 0,2. b,2— (bb. 2a)? = (402),
8 =}(bew) Ua (Cabs — Cxba).
9 = (abc) (b’cu) (ab'u) bz + (abe) (ab'u) {(bb’u) cz — (cbb’) uz}
= (abc) (b’cw) (ab’u) b, + 4 ugez (acu) ag — 4 cguz (acu) ag
= (410), (421), and (421),.
10 = (bea')? . a2 + (bea’) az . \(aa'c) bz — (baa’) cz} = 4 Cabs (bal -- Vala) — § Dalz (Dax — bzCa)
= DeC20 707.
11 = —4 ga, (ager — azCp).
13 = + 4p (cau) ag.
14 = $ uabzCr (Cabr — Cxba).
15 = (abc) czb’, \(bb'u) az — (abb’) uz} = kupertic( Apex — AxCa) — 4 Urliper (ApCx — Aga) = + duly . UpCpdxCr-
17 =4 u,a, (aw) b,.
19 = 4), (abu) (uaz — Uz).
21 = $4, (aub) by.
CONCOMITANTS OF THREE TERNARY QUADRICS.
Fifth degree.
We proceed now to shew that from
(410) = (bew) baca, we obtain
(402), = bacabsers
(402), = (Byx)’,
(421), = (bew) bacztla,
(421), = (bew) byczu,,
(421), = (abe) (uca) (uab) u's,
(421), = (Bye) wary,
From (410) = (bev) baca.
1. (bea) dybaCa = (501). 3. baCa (baw) axl, = (512),
2. (bec’) ¢'xbaCa = (501),. 4, daca (bb'w) b/c, = (512),.
B bios Geu)ie'scs = ee
6. bala (bau) (caw) = 0.
7. dala (be’u)(cc'w) = (520).
From (421), = (bew) dacxtlas From
12. (bow) betta (caw) a, =O. 27.
13. (bew) bata (chu) b', = (520). 28.
14. (bew) bata (cc'u) cy = (520). 29.
15. (bew) baCrb'ab’x =(5i12)) 30.
16. (bew) dacyc'aC'x =(501), and (512), 31.
17. (bea) agbaCxtla = (512) 32.
18. (beb’) B'xba Cra = (512), 33.
19. (bcc’) cba Crt =(); 34.
20. (bew) ba (cb’w) bn = = (520). 35)
21. (dew) b, (ecu) c’, = (520). 36.
22. (bea) bart, (caw) = (0) Sie
23. (beb') batta (cb'w) = (520). 38.
24. (bec’) batta (ec) = (520) 39.
25. (beb') bace ba - 40.
26. (bcc’) baCxC'a = (501) 41,
From (402), = bacabzCx.
501), = (abe) azbaCa.
01), = (By) pay.
= Ugly pty.
12), = (By) agar.
17) = (abe) AgupbxCe.
12), = (Bryx) CxCptty.
From (Byz)? = (402)s.
8. (Byx) (By .
9. (Byx) (By .
10.
LU (Styacw): =0.
(421), = (bow) byex2ty.
(bow) byt, (car) a, =0.
(bew) byuy (cb'u) b’, = 0.
(bow) byw, (cen) ce’, = 0.
(bew) byay CxQe =(512),.
(bow) bybyCxb'x = (512),.
(bea) byity CxQx =((5112),.
(beb’) bytly Cab’ =(512),.
(bec) byuty Cx" a
(bow) bya, (cau) =O.
(bew) b,b', (chu) =0.
(bea) byw, (caw) =0.
(bcb’) byw, (cb'u) == 0.
(bec’) byu, (ecu) == 0.
(bea) byayex =(501),.
(bcb’) b,b’ex = 0.
71
au) dz = (512),.
CU) Cy = (512);.
(By . aw)? = (520).
ro
‘=
From (421), = (Sy) upty.
Mr H. F. BAKER, ON THE FULL SYSTEM OF
From (421), = (abe) (uca) (uab) a’.
2. (By . au) Ugttyaz = 0. 49. (abe) (wea) (uab) (a'a"'w) a”, = 0.
43. (By . bu) uguybe = 0. 50. (a'be) (wea) (wab) (a’b'u) b’, = (520).
44. (Byx)aptydz =(512),. 51. (abe) (uca) (aab) az", = (501), and (512)...
45. (By. au)aguy = (520). 52. (abe) (uca) (b'ab) a’ xb’ =(512),.
46. (Bry. cu) cauycr = 0. 53. (a’be) (uca) (c'ab) a'xc'x =(501), and (512),.
47. (Bryx) apy =(501),. 54. (a’be) (uca) (a"ab) (wa'u) = =0.
48. (Bryx) carvtycx = (512),. 55. (abe) (uca) (b’'ab) (abu) = = (520).
56. (abe) (wea) (cab) (a’e'u) = = (520).
57. (abc) (a"ca) (a ab) a’, =(501),.
58. (abe) (b’ca) (b'ab) a’ = (501),.
Of these
= 4 (yar) byba.
3 = (bac) Uabatzlr or say (abc) upgdpb,cz.
4=1 (Bar) upcacz or say =4 (Byx) upayaz.
5 = bacat's {(cc’u) bz — (ce'b) Uz} = $ Uydabs (yar) — $ Ugdyba (yan).
6 = (abc) (bew) data = } (bow)? . a”.
7 =4 uyba (Uyda — Uady).
8 = (Byx) agazu, — (Byx) ayazu.
9 = (By2) CaCxtly.
10 = — 2agayuguy.
11=0.
12 = (cau) data (bea) Uz + (cua) bz + (uba) Cz} = Uz (Ca) ba (abc) Ua + Crtla (uba) (caw) ba
=— nu, (beu) (abu) dala + Crtta (uba) (bar) C2, = 0.
13 [= — bbatta - (b'cu)? — ]
= (bew) uab'z (bb'U) Ca = $ Catlgla (pC — UxCp).
14= hu, (byuz— bay) bata
15 = (bew) (bb’ . ax) c,b’. = 4 (Bax) (ugla — Ualp) Cr.
16 = (beu) (cc’ . wa) bac’, = § (yaa) (byuz — bztty) ba.
18 =—4 cg (Par) Crtta.
20 = (beu) b’, {(bb'u) Ca — (cbb’) Wa} = 4 UpCa (UpCa — Ualp) — 4 Calla (UpCa — Walp) = — WallaCaCp-
21 = b uyba (bya — Datty).
22 =— (abu) (beu) ca, = 9.
23 = — 4 Celta (Upla — Walp)»
CONCOMITANTS OF THREE TERNARY QUADRICS.
24 = uyb bata.
26 = 4b ba (yaa).
27 = (bew) ayu, (chu) az = — (beu)? . ayazy = 0.
28 = (bew) b' yu, (chu) b', = — (beu)? . b'yu,b’, = 0.
29 = (bew) cyu, (be'u) ce’, + (bew) c'yuy (cbu) ce’; + (bew) uy? (cc’b) c', = 0.
30 = (bea) byuyeraz-
31 = (bew) (bb' . yx) byez =4 (Byx) (upey — Uy Cp) Cx.
33 = — dcp (By2) u,Cz.
35 = — (abe) (abu) u,c, = 0.
36 = (bew) u,b’, (cb’b) = — 4 cpu, (ugcy — U,Cg).
37 = — (abu) (bew) cya, = 0.
38 = — dcprly (Upcy — u4Cg) = 0.
39 = u,byuy by = — (ub. yy = — 407. (ube).
AD = 0 (= ug? ... —U,?...).
43 aye - Dy,b;,.
45 = — Ayllptgtly.
46 = (catty — CyUlg) CpulyCx = 0.
49 = 4, (uca) (uab) (Cabs — Crda) = 4 Ua (uc) (Wac) baby — 4 Ua (ba) (wab) CxC, = 0.
50 = (uca) (a’b'u) b’, (bua) (bea’) = (uea) (a’b'u) b’, {(buc) (baa’) — (bua’) (bac)}
= 4b, (buc) b', (ub'c) va — (uca) (bac) (a’b'w) {(b'ua’) bz + (bb'a’) uz + (bub’) a'z}
73
= dua. (ub'c) {(bb’c) uz + (bub’) cz} — 4 (a’bb’) (uca) uz {(acb) (ua’b’) — (acb’) (ua’b)}
+4 (ubb’) (uca) a’, {(acb) (ua’b’) — (acb’) (ua’b)}
= 4 uaa (Cala — Calg) — $ UplaCx (Cpa — Calla) — + a'piz (uca) {eg (aua’) — ag (cua’)}
+ f upd’, (ued) {cp (aua’) — ag (cua’)}
= — Hux . CaCalalip + $ Uzalp (Calla — Cala) + + U20’g (uca) {(caa’) ug — (waa') cp}
— 4} uaripcp (Caller — Crtla) — + up’, (uca) . {(caa’) ug — (waa’) cp}
Ill
— uz . CaCpatlg + $ UxCaCplallp + 4 UxCallp (Calg — Cala) — $ UzCpla (Callp — Calla) — $ Ux » CaCpUlallp
+ 4 UatipCp (Calls — Cela)
— 2Uz . CaCplallg.
51 =4), (whe) a’z . (Cals — Cxtla) = 4 (abc) Azbala » Uz — 4 (abc) CzA ada.
52 = 4 ag (uca) a’, (apc, — W'xCp) = 4 (aa’ . Bx) (uca) a gly = 4 (482) Cy (Calg — Cpa).
Vou. XV. Parr I.
— 4 Uz . CoCplalp + $ Ux - CaCplallg — Ux » CaCpllalla — $ Uz . CaCalallg — $ Uz » CaCplallp
+ § Uz « CaCplatlg
10
74 Mr H. F. BAKER, ON THE FULL SYSTEM OF
53 = (a’be) (uca) c'z « {(c'a’b) az + (c'aa’) bz} = (be'a’) (wea) az « {(be'a’) cz + (bec’) a’, — (a’ce’) bz}
+4 cabre'x (ube) Ca
= hb,a,a',{(auc)(a'be’)—(auc')(a’be)} — § a yaxbe {(auc) (wbc')—(auc’) (wbe)} + c'abx (ube) (ce! . ax)
= hbyaza’z {uy (aa’b) — a, (ua’b)} — 4a’ yazbs {uy (aa’b) — ay (wa’b)} + F (yar) by (dytta — datty)
=—Lb,a,(aa’ . yx)(ua’b) — $ dabdsty (ary) + 4 (aa! . wy) bra, (wa’d) + $ (yar) bybztta — F (yaar) Dadztly
=—1), (aye) (dattz — brtta) — F (yar) dadztly + F (yar) de (Darly — dy ta)
+4 (yan) bybatta — £ (yar) Dabzity
54 = —4 (uca) va (abc) ba = $ (bew) (abu) dala = — $ Ga . (Dew)? °
55 =4(uca) ag (cua') a’g = $ (uca) a's {(caa’) up — (waa’) cg} = + cattg (Catlg — Cptla)
—F Wace (Calg — Cptta) = — 4 CaCplarlp.
56. Consider it under the form (uwbc)(ub’c’) (abc’) (ab’c). This is given as example 2 of
§ IL, where its value is written down. It is = wguyagay.
57 = (a’be) We (Mp
58 = } ag (aa'c) cpa’, = 4 Ca (Ba) Cp.
This completes the establishment of the fifth degree. We have arrived at all the
forms written down on page 71 and no others.
Sixth degree.
We proceed now to shew that from we obtain
(501), = (abc) dabaCa, (600) = (aBy)?.
(501), = (By) aga, (611), = (@By) (Bre) te.
(520) = ugu,dpay, (611), = agaybybziig.
(512), = (Byx) apaztly, (630), = (aBry) Watlatty-
(512), = (abc) agugbrer, (630), = (abw) agby uprty.
(512), = (Byx) CxCptty, (630), = (bew) upuyb,ce.
(603), = (By) (yar) (afr).
(603), = (Byx) azbyapby.
(603), = (Byx) brexbycp-
From (501), = (abe) azbaea. From (501), = (Byx) agay. From (520) = wpuyapdy.
1. (abc) (aa'u) bacaa’, = 0. 3. (By . au) aga,a', = (611),. 5. Wg xy pdy = (611).
2. (abc) (ab'uv) bacab', =(611),. 4. (By. bw) agayb, =(611),. 6. C'gC'gttyapdy = (611),.
7. a'pa'agdy = (600).
CONCOMITANTS OF THREE TERNARY QUADRICS.
From
(512), = (Bry) apt.
(By . wu) dpaza’zu, ==0.
(By . bu) agazb zu, =0.
10. (By . cw) agazeztly =0.
11. (Arya) ag (aa’'u) au, =(611),.
12. (Byx)ag (abu) bu, =0.
13. (Bryx) ag (acu) cxty = (611),.
14. (Byx) agaza’',a'z = (603),.
15. (Bryx) agazb,b, = (603),.
16. (By. wu) ag (aa’u) u, = (630).
17. (Bry. bu) ag (abu) u, = (630),.
18. (By. cu)ag(acu)u, =0.
19. (By. au) agaza’y = (611).
20. (By . bw) agazb, =0.
21. (Byx)ag(ad'u)a’, =(611),.
22. (Byx) ag (abu) by = (611).
From (512); = (By) cxCpuly.
38. (By . au) cycguya’, = 0.
39. (Bry. bu) excpu,b, =0.
40. (By. cu) cxcauye’, =0.
41. (Bryx) (caw) cguya, = 0.
42. (Bryx) (chu) cgu,b, =0.
43. (Bra) (cc'u) cgu,c’, = 0.
44, (Bryx) CxCpd,Az = (605),.
45. (Bryx) cxcabybz = (603),.
From
23.
24,
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
(512), = (abc) agugbyce.
(abc) agug (abu) cra’, = 0.
(abc) agi (bb'u) czb', = 0.
(abc) agus (be'w) exc’, =0.
(abc) aguigb, (cau) a’, =0.
(abe) agugb; (cb'u) b', =0.
(abc) apuigb, (cc'u) ce’, =0.
(abc) aga’ ga 2bxCx
(abc) age’ gbrCx
(abc) aga’g (ba’u)cz = =0.
(abc) age’s (bev) cz += == (611).
(abc) aga’gbz (cau) =(611),.
(abc) age’gbz (cc’w) =0.
(abc) agug (baw) (ca'w) = (6380),.
(abc) agug (bb'u) (cb’u) = 0.
(abc) agug (be’w) (ce’u) = (630),.
(By . au) Crtpty =(611),.
(By . bu) cxeaby =0.
(Byx) (caw) cpa, =(611),.
(Bryx) (cbw) cab, =0.
(By . au) (caw) cgu, = (630),.
(By . bu) (chu) cguy = (630),.
(By . cu) (ecu) cpu, = 0.
L= tue (Cabs —Crba’) DaCa ANA Ua'Ca'DadzCa = Ua'DxCa (be . aa’)=4b, (cw . aa’) (be . aa’),
2 = (abc) cab’, {(wab) b’, + (ab’b) ua} = (abe) b'xb’a {(abc) Ua + (uac) ba} —4 apCatta (ApCx — Axle)
= (uac) (abc) b’. (bb . ax) + 4 CaCpdpdztla = 4 (Bax) (wac) (pCa — Mala) + 4 CaCadpxla
=h\uUp Ua Uz | ApCat 4 CaCpdgdza = $ WalpdxtpCa + CaCaMpMxla = CaCpUpdzWa = (611),.
Ap A, Az |
Cp Ca Cz
3 =(a'gu, — a yug) agaya’, and a’gu,aga,a’, = uya_d’, (aa’ . y8B) = — § (aBy) (aBe) uy.
4 = byugaga,b,.
10—2
=(603)..
= (603).
75
76 Mr H. F, BAKER, ON THE FULL SYSTEM OF
5 =a’ ua, (aa’ . y8)=— 4(aBy) (a8) w.
7=(a8y). For =4(a,?. a+ a/a's’)? — 4 (aa. By).
8 =(a'guy — ays) Apr ry = — AzMglip . Wy’ zlly.
9 = dyugtyagarby = apazig . bybrity.
10 = cgdgcza, . Uy.
11 = 4 (Bya) ue (@Bx) uy = — 4 Uz (4B) (722) Up.
12=|as3 ay dz| dgbrtly = Up (Aybz — Ady) Apbzty = bz? « Uptlypty — AzAprlp . bybzty.
lbp b, b
Us Uy Uz
13=|dg Gy Gz | UpCzlly = — Cg (Aylz — AzUy) ApCzlly = — Uz » CpApCzMyy + Uy? . UrCxMpCp-
PE es
|p Uy Ue |
14 = (Byx) aga’, (aa’ . wy) = ¢ (Byx) (yar) (a8z).
15 = (603)..
16 =(a’guy — a’ jug) (aa’u) aglty = — 4 Uallpily (487).
17 = (abu) byuigagity.
18 = cad, (acu) . up.
19 = agryatga_a’, = a's24yQz (aa! . By) = (@By) (xB) 4,
20 = bj? . ugagar.
21 = 4 (Bye) (ay) ua.
22= dg a, az| agb, = uga,b,agb,.
‘iy tne
| tg Wy Uz
23 = (abc) a’ gug (baw) cza’, = (abc) (bau) cz . a’guga’z.
24 = (abc) cb’ zug {(b'ua) bg + (uba) b's + (bb’a) ug}.
25 = (abc) (bau) c'gugczc’, = (abc) (bau) cz . c'puipc’z.
26 = (abc) uga’, {(a'ua) cg + (uca) a's} = 4 watgbzCp (Cabz — Cxba) = — 4 Dabrila « Clap.
27 = (abu) agegb, (cb'u) b’, = {(ab’u)'b, + (abb’) uz + (b’bu) az} apcgbz (chu) = (b'be)bz . (cb'u) . azapug = 0.
28 = 4 u,agiigh, (bya, — bray) = 4 uybyb, . Updpaz.
29 = (abc) a'gb,c, (aa’ . Br) =4(aBer) (cabg — caba) bxer = — 4 (482) cababzer.
30 = (abc) age’ bz (cc’ . 8) = 4 (Byx) agBz (baz — bay) = 4 (Bye) agbyazb,.
31 = (abc) uga’'s (ba’a) Ca = — $ batgla (Cadg — Caba) = 0.
32 = (abc) (abe’) cc’ pup = (c'be) (abc’) cyagiig + (ab’c)? . cxlaitp = } byagug (yb, — azby) = 4 agayugb,be.
33 = (aba’) agegb, (ca’w) = — 4 bacabs (Wala — UpCa) = § CaCaudrba.
34 = tb,u, (bag — bgay) ag = 0.
CONCOMITANTS OF THREE TERNARY QUADRICS.
35 = (abu) apes (ba’u) (cau) = (abu) (cau) cp {(bau) a'g + (ba'a) up} =—4 Cprmpba (bow) Ua
36 = 4 updgita ca (uca) = $ (au . BB’) upce (uca) = 0.
37 = $ u,aguab, (aud).
38 = (a'pity — a’ yp) CxC ply = Uy? « ApdzCzCp — CxCplg » UyAzlly.
39 = babyy . CaCzip.
40 = uy? . C'pc'xCxCp-
Al = Cp Cy Cy | CptlyAg = Cy (Apdly — Ayilp) CptlyAn = — UpCpCe » UyLyMy «
dp ty Az
Up Uy Uz
A2=|cg Cy Cz | Cptlybr = Up (Cybz — Cxby) Cpttydz = — CrCpiip . Dzbytly.
bp b, bz
[Up Uy Wy
s3= 03 CG | Cpe gy = Cz (C'plly — Cyt) C'xtly = 0.
én Gy Oe |
Up Uy Uz
46 = (Aptty — AyUg) CxCpdy = ApdyUyCxCe-
AS = byuipcrCab, = 0.
A8=\Cg Cy Cz | Cady = Cz (Aptly — Aytlg) Cady = Ap CaCzlly.
| Gi, hy Whe
Up Uy Uz
AD=|Cp Cy Cx | Cady = Up (Cybz — Cxby) Cab, = 0,
Bapeloe (0;
Up Uy Uz
50 = (agity — dyig) (caw) Cgly = — (Cart) UguydyCp.
51 = byugcput, (chu).
52=c'pcp . u,?(cc’'u) =0.
Thus justifying the system of the sixth degree.
Seventh degree.
We proceed now to shew that from we obtain
(611), =(a4By) (By) Wa, (710), = (4By) apgayita.
(611), = aga,bybzug, (710), = (bew) bga,b, ce.
(630), =(4By) Uatipity, (721) =(aBry) badrrtgrty.
(630), = (abu) agbrgity,
(630), = (bew) wpu,bycp ,
(603), =(Byx) (yar) (a8),
(603), = (Byx) arbragby,
(603), = (Byz) breabyep,
Th
Mr H. F. BAKER, ON THE FULL SYSTEM OF
From (611), =(@8y) (By2) Ua.
1. (aBy) (By . at) Uae = (721).
2. (aBy) (By . bu) uabz = (721).
3. (ay) (By) babz =0.
4, (aBy)(Sy.bu)b, = (710).
From (630), =(@@y) Uatiptly.
12.
13.
(aBry) babruiptty = (721).
(a8y) vatgty =(710),.
From (abw) agbyuguy = (630)..
14.
15.
16.
ilv@
18.
nD:
(aba’) agbyugtytz = (721).
(abb’) agbyuguyb’z = 0.
(abc) dgbyuguyer = 0.
(abu) agbya’guya’, = (721).
(abu) agbycavycr = 0.
(abu) agb,uga’a’, =(710), and (721).
(abu) agbyugb',b’, = 0.
(aba’) agbya’gu, vanishes.
(abc) agbycaty =(710),.
(aba’) agbyuaa’y =(710),.
(abb’) agbu,gb’, vanishes.
(abu) agbya’ga’y =(710),.
From (603), = (Byx) azbzagby.
37.
38.
39.
40.
41.
42.
(By . au) az dzbzagb, =0.
(By . b’'u) b'azb,agb, =0.
(Bry . Cu) CrAzbzapb, =0.
(Byz) (aa'u) b,a'apb, =0.
(Bryz) (ab'u) bb’ ,agby = 0.
(Byx) (acu) bzc,apb, =.
(By2) a, (ba'w) a’ gb, = 0.
(Byx) az (bb'u) b’,agb, = 0.
(Byx) az (bew) cagb, =.
46.
47.
48.
49.
50.
51.
52.
53.
54,
From agaybybztig = (611)..
5. apayb, (baw) ag'ug = 0.
6. dga,b, (bb'u) b’zuig = 0.
7. apayby (dew) cyttg =0.
8. agaybyb,a'pa’, =0.
. ApdybybaCpex = 0.
10. aga,b, (bau) ws + =(710),.
11. agayb, (bew)cg =(710)..
From (630), = (bew) upuybycp.
26. (bea) aztigtybycg = (710),.
27. (beb’) b’uguybycp = 0.
28. (bew) aguyazb,cg = (710).
29. (bec’) c'pttyc’zbycp = 0.
30. (bca) aguybycs =(710)s.
31. (bcc’) c’pu,bycp =0.
32. (bew)agaybycs =(710)..
From (Sy2) (yax) (48x) = (603),.
33. (By . aw) (yan) (48a) az = 0.
34. (By . bu) (yax) (@Bx) bz =0.
35. (Byx) (ya. au) (a8. aw) =0.
36. (Byx) (ya . bu) (a8 . bu) =(721).
(By . au) (aa'u) byagby =(710), and (721).
(By . b’u) (abu) bzagh, = 0.
(By . cu) (acu) byagby =.
(By . au) az (baw) agby = (721).
(By . b'w) az, (bb) agb, = 0.
(By . cu) az (bow) agby = (710)..
(Byx) (aa'u) (ba'u) agby = (721).
(Bra) (ab’u) (bb'u) dpb, = 9.
(Byx) (acu) (bow) agb, =9.
CONCOMITANTS OF THREE TERNARY QUADRICS. 79
From (By2) bzerbycp = (603);.
55. (By. au) AzbrCxbycg = 0. 60. (By. au) (baw) cxbycg = 0.
56. (By . bw) U'xbrCxbcp = 0. 61. (By. bu) (bb) cxbycg = 0.
57. (Byx) (bau) azcxbdyca = 0. 62. (By. cu) (be'w) Crbycg = 0.
58. (Bryx) (bb'u) Cxb'xbycp = 0. 63. (By«) (baw) (caw) bycg = 0.
59. (Bry) (be’u) CxC'xdyCa = 0. 64, (Brya) (bb'w) (cb'u) bycg = 0.
Of these
1 = (aBy) (ag — ayitg) Wall
2 = (aBy) watlgbyby.
=— (yar) (aBx) bab, = 0.
4= (aBry) upbyba.
5 = Up Az Ug . ayby (bar).
6 = ug? . aybybz' (bb’a).
7 = cadyb, (baw) cxtig = ayby (baw) . CxCartp.
8 = ag (aa’. yB) bybraz’ = — $ (aBy) (aBx) bybz = } (Byx) (yar) babg = 0.
9. For this consider (abc). (Aya)? = (agbyex — UpbxCy + Ayb2Cp — Aybper + ArdpCy — ArbyCay?
= (agbyCx + dybzCg — dzbyCp) = 2agayb,cpbrCe-
10 = wgayb, (ba’a) ag’ = — dugh,ba (ayf).
14 =— } (aBx) b,bauguy = — 4 (aBy) drbarigrty.
15 = (ubb’) ipdyCtpttydy, =0.
16 = (ubc) apbyupayc, = (uba) cpbyuigdyCx = (wba) bydy « CarttpCe.
17 = (aba’) aghyugu,ac’ = — § (aR) bybatipuy = — 4 (aBy) babztiptly.
18 = (abe) agb,ugu,c, = 16 = 0.
19 = (abu) a,b, . wadg'ae’ + 4 (aBy) byip (ade — Uada).
20 = (abw) agupb, (bb' . yx) = } (aptly — yup’) (B’yx) pig = (Byx) uguy . dp? + BB’
=
— (Byx) aga, . Up? + BB’ = 9.
22 = (bew) aga,bycg.
23 = — 4 (ary) b,barig.
25 = (aba’) agb,uga,’ or 23.
26 = (abc) ugiiizbycp = (ube) agiiytxbyCp = Uz (abc) UpttybyCg + (uca) cpg . Drbyity + (uba) aguyexb cp
———— | eo ————————
= Uz. (whe) agdybycp + (cha) upityaperbdy = Uz. (ubc) agdybycg + (cbw) upayaperby
—— —
= Ug. (ube) apdybycp + (abu) ayby. Caigcr = Uz . (whe) apayb,ce,
a reduction not at all obvious,
80 Mr H. F. BAKER, ON THE FULL SYSTEM OF
27 = (bub’) bzcguybyce = 0.
28 = 26.
29 = (ucc’) cp'bycx by Cp.
31 = bY. (ucc’) cpce’.
33 = (aguy — ayup) (yar) (a8x) az = (Bax) (aBx) ayuyd, — (yan) (aryx) Apigdx = 0.
34 = (yar) (a8) bybyup = (yar) (yBx) brbatig = (yBux) babz (Bax) Uy + (ya) Ux}
= —(aBy) (Bye) babs» Ux + (48x) Uydz (Bye) by = (yaw) (aBir) bgby . Ux-
3d = Ua". (Byx) Upay.
36 = (Byx) (bytta — batty) batig = (By) bybattattg = (Bye) bybatlartp.
BT = (Mtg tly — Ay’ Uig) Azz byAgby = Uybybz « Up'Uz Updiz — Uppy « dy Ae byby.
38 = by upbz'azbzagby or brbyby‘bx' . updpaz.
39 = CptlyCatxzbzagby OY CgCzMpiz . Uydybz.
40 = } (Byx) (a8x) Uabrb, = 4 (By) (yBx). Uababrs
41 =) ag ay Gz | debs agby = Up (Aybx’ — Azby’) brbz/agby = — azdpuig . byby/dzbz’.
| bp’ by b, |
| Up Uy Uz
42 =| ag dy Gz | byCxpby = — Cp (Aylz — Uzly) brCxtaby = — apdybycpbsCx + UpCptzCz » Uybybz = 0
Cp Cy iGz | (see 9).
| tg Uy Wz |
43 = | bp by Dx | Axtte dgby = by (ap'Uy — Ay'Uig) Azz Apby = dzbyity « Up Ag Aap — Uptighz . dy Wy bydy.
| Te SS (oles |
| Up Uy Ux
44 = ug (bybz’ — by/bz) Arby! apby = — Uptiptiz . (dybz’ — by bz) bybr’.
45 = dg by, dz | dzCottpb, = bz (Cptty — Cyulp) AgCatgby = Dybytly « CadpCrMz.
Cp Cy Cx
| Up Uy Ux |
46 = (cg'uy — Oy Up) (aa'u) bragb, = — § (aBy) Uatigbrd, = — 4 (aPBry) Uatigbaby
=—4 {(aBy) upbyba . Uc + (w@Ba) Upgttybaby}
=—4(aBy) Upbyba. Uz +4 (aBry) uptybab, = (710), and (721).
47 = bug (abu) bagh, = (bb'. wy) (ab'u) byagup = 4 (Bary) (updy — apily) apiig.
48 = cpu, (acu) byagb, or (acu) pcp . Uybybz.
49 = (cg! Uy — Oyig) (ba) aagb, = (ba’a) ugdpu,azb, — Upp, . dy'b, (ba'w)
=— 4), (axB) upu,by = — $ (ay) babstipily-
50 = b,/ugaz (bb'u) aghy = by upd, (bb'a) upb,.
51 =cgu, (bew) azagh, = cpu, (bea) azupb, or 26.
52 =4u, (Byr) (Uabp — Upba) by = — } (By) Ualgbyba = — 4 (aBy) waripbybr.
53= tug (By2) (Upy — Uyjtp) Ope
CONCOMITANTS OF THREE TERNARY QUADRICS.
54 =| bg db, bz | (acu) agh, = bz (cay — Cyup) (acw) agb, = (acu) cpg . brbyuy — (acw) agcyugbrby.
| Cp Cy Cx
| Up Uy Ue |
55D = (dgily — Ayulg) AzbzCrbyCg = Cad pCrAx . brbyuy — AydyArbx . UperCea-
56 = b,/ugbzbzcrbycp = Cxcpuip . byb,/bxbx’ .
5T=| bg by bz | azexbycp = bz (aptly — Ayg) AxCrbdyCp = AxtgCaCz . bebyuy — dybyazbz . UpCaCe-
Gg Wy Az
| Up Uy Uz
58 = dup (Byx) (B’yx) exp = 0.
————
59 =| be by bz | CxCr'byca = be (cp'Uy — Cy Up) CxCx byCp = bxdyily » Ca’ Cx CaCe-
Cp’ Cy Cx
Up Uy Uz
60 = (agu, — ayug) (baw) crbycg = (baw) crbycgaguy — Upegez . (baw) byay = (bac) ezbyugagu,
81
= (buc) Crbyupagdy = (bua) czbyugcgdy = (bua) bya, . Crrpcea-
61 = by/up (bbw) Cxbycg = b,/ug (bb'c) Cxbyug, Which vanishes.
62 = cp'u, (be'w) crbycp = Cp'Uy (be'c) czbyug = by? . (uc'c) cp'crdyutp.
63=| bg by bz | (caw) bycg = by (aguy — aug) (ca) bycg = bybzuy (cau) cag — (car) cabyayugbz
dg Ay Gz = — (abc) ug dybreg = — (cub) aguyaybzce = — (cua) aguybybrce
Up Uy Uz = Uybybz . Cadp (cau) = 0.
64 = (Byz) (bb’c) (chu) byug = Sep (Byx) (ugcy — Catty) Up = 0.
The seventh degree is therefore established.
Enghth, ninth and tenth degree. End of the system.
We proceed now to shew that from we obtain
(710), = (aBy) apayuta, (801), = (By2) bycpbaCas
(710), = (bew) agaybycp, (801), = (a’be) aga,bycpax ,
(721) = (aBy) babsupuy, (812) = (a’By) (vax) (aBz) Ue
and thence and thence
(911) = agaybybaCaCxrl,, (10.1.0) = dycgbacatta (By),
and that this is the end of the system.
From (710), = (aPy) agayita.
1. (aBy) aga,b.b, = (801),.
From (710), = (bew) aga,b,ce.
2. (a’be) azaga,b,cg = (801),. 3. (beb’) bz'aga,bycg = 0.
Ven, XV. Parr 1 il
82 Mr H. F. BAKER, ON THE FULL SYSTEM OF
From (721) = (a@Sy) babsuiptly.
4. (aBy) ba (daw) aztgtly = 0. 10. (aBy) babztigb,’bz’ = (812).
5. (aBy) ba (bb'u) bz'ugy = 0. ll. (aBy) ba (baw) aguy = 0.
6. (aBy) ba (bow) czipuy = 0. 12. (aBy) ba (bew) cpuy = 0.
7. (aBry) babsttatytz = (801);. 13. (aBy) ba (baw) uga, = 0.
8. (aBy) babseptlyCx = 0. 14. (aSy) ba (bb'w) ugby = 0.
9. (aBry) babdstiptyte = 0. 15. (aSy) babsigay = (801).
From (801), = (Byz) b,cpbaCa-
16. (Sy. au) azb,cpbaca = (911). 17. (By. bw) bz'bycpbaCa = 9.
From (801), = (abe) agayb,cpax’.
18. (a’be) agayb,cg (a’a’"w) az” = (911). 19. (a’be) agayb,cp (a’b'u) bz’ = 0.
From (812) = (a’By) (yx) (a8) ua.
20. (a'By) (yar) (a8 . aw) Azle = 0. 24. (a’Bry) (yar) (a8 . bw) ba = 0.
21, (a’By) (yar) (a8 . bu) bata = 0. 25. (a’By) (yar) (aB . cw) ca = 0.
22. (a’By) (yar) (a8 . cw) Crtta = 0. 26. (a’By) (ya. av) (a8. av) us = 0.
23. (a’By) (yar) (a8e) bab, = 0. 27. (a’By) (ya. bu) (a8. bu) ua = 9.
From (911) = agaybybaCalxtip:
28, agdybybaCaCeip Az = 0. 32. agdybybaCa (c0'U) Cz'Utp = 0.
29. agdybybaCaCzxep Cr = 0. 33. agdybybaCa (cau) ap’ = 0.
30. dptybybaCa (Ca) Az Up = 0. 34. dpdybybaCa (cc) cs = (10.1.0).
31. agdybybaCa (cb'u) brug = 9.
From (10.1.0) = (a’By) b,¢abacatla’.
35. (a'By) bycabaCaba’ br’.
Of these
1 = (ax) agaybab, or say (Byx) baCabyce.
3 = (bab’) b/a,b, . cg’.
4=|b, be by,
Ue Op Ay
balhzllipily = — dytaigbaliztlplly OL — Uplgtlr . Uybybatla:
| Uq_ Up Uy
5 =} (aBy) (Sar) upuptly = dup". (aBy) (Rar) uy, + BR’.
CONCOMITANTS OF THREE TERNARY QUADRICS, 83
6 =| ba bp dy | dacztiptty = by (Catlg — Cala) baCatlptly = — Calpe « Vadadytly.
Ca Cp Cy
Ua Up Uy
7. Making one cyclical change forward this becomes
(4181) byeatabate = (Bry) byoptabaty = te (84) Dytgbate + (ay) Baby . peace + (0/Ba) byoprtybat
= Uz . (ZB) byCpdaCa + (BA) bregtiybaCe = Ux » (wBry) byCpbaCa + (yBx) becpybaca
= Uz. (wBry) bycpbaCa + (4B) CaCp . bzdy tly = Uz (Pry) byCpbaca (cf. 26, p. 54).
8 = (aBy) babaCptlyCa = (ZRH) CpCa . Dyybz.
9 = (a8) babyugdydx = (ayx) baby. UsUpla.
10 = (ay) (bb. a) byuigdy = 4 (aBry) (B’axr) (B’ny) up.
11=|b. bg dy | dattgrty = Ua (day — byag) batiptty = dabatigtlylatty = 0.
Ga Ap ty
Ua Up Uy
12=|b, bp by | bacptty = dy (Calg — Cpa) balay = UpUydybaCaca = $ (aBy)®. (ube).
Gruicn, Gy For (ube)? . (aBy)? = (Uabpcy — UabyCa + UpbyCa — Upbaly + UyDale — UybpCa)?
Ua Up Uy = (— UabyCg + Upb,Ca + UydaCp)” = 2uguybybaCaCe-
13 =| bd. ba by | daripay = by (daig — Apia) batipdy = — Uallgigdybyba = — 4 (uab)?. (ary).
Ma, Ug Wy For (wab)? (aby)? = (Watipby — Uattybg + Updyba — Upttaby + Uytabg — Uytpba)®
Ua Up Uy _ = (Ualtpby + Updyba — Uyiigda)® = WatlpagdyDyba .
14 = § (a8) up (B’ay) up = 4p. (ay) (Bay) + BB.
15 = (aBz) b.byaga, = (801),.
16 = (aguy — ayuig) AcbyCpbdala = AxtplpCadabyy — AzlybybaCaCaua ;
both represented by c,Cababytyagu%e or (911).
(Bary) (B’yx) Upcaca = (B’ary) (ay) Upcacer
= (Bary) (ary) up . cp" + BB
Ill
17 = by/uighz'bycabaCa = by UpCpbala (bb' . yx)
18 = $ugtiptybyCp (Cab — Cxda) = 4 bxdyAypCpCala — $CxCpgdybydaUla.
19 = {(b'bc) ag’ + (a’b'c) by + (a’bb’) cx} apa,bycg (a’b'u) = {(a’b'c) by + (a’bb’) Cx} apaybyce (a’b'u)
= (w’bb’) cxaptb,cp (a’b'u) + (a’b’c) bzagayb, {(cb'u) ag’ + (a’b'c) ug}
“71 - _ S
= hag Cr pdyCp (Updy — Uydg’) + (a’b'c) (cb’w) agby (aa’ . yB) bx
=4 (aa. yf’) aga, Cregg + 4 (ay) (ba Cp — bg’ca) (cb) bybx
=} (ary8’) (a8) cxcptig: — § (4/8) ba'cab, (cb'u) be
=H (ayP) (aBy) . CxCeup + BB’ —$/ca cp Cy | bby’ cabs = $b,’ (Cattg — Carta) byCaba’ dy
ba’ bp’ by
Ua Up Uy
= bby ba CaCpuipbybe = (bb. wy) ba'Cacpigby = 4 (B’axy) (B’ya) CaCarlp
—
=} (yaw) (B’ya) cacatts = BB’ = 0.
11—2
84 Mr H. F. BAKER, ON THE FULL SYSTEM OF
20 = (a’By) (yaw) dgtladztla’ = Ua. (aBy) (yar) aps + aa’.
21 = (a’'By) (yar) babrtigtta: = 4 (aBy) (yar) up . badrttar + 4 (a’Ba) (yaw) b,byuiptia.
22 = (a’By) (yaw) (Catlg — Cala) Cxtlar = (aBry) (yaw) Up . Ca'Cxtla + aa — (aBry) (yor) Cpcr . Ua — aa’,
23 = (a’Bx) (yar) (a8) barb, = aa’ + (a/Be) (your) (af Bx) Baby.
24 = (a By) (yax) batigba = (aBry) (yar) up . ba? + aa
25 = (a’By) (yar) (Catlg — Cptla) Ca’ = aa’ + (aBry) (yar) Up. Ca? — aa’ — (a’ Bry) (ya'x) Catala
= = (By) (yolB). cattata = (By) (YB2) Catlaca = Oa = 0.
26 = (a’Bry) ayaliptatla’s
ed
- , , Sey
27 = (a’Bry) (bya — batty) dartptla = (a Bry) dybatlartpitar = ac’ = 0.
SSS")
28 = aybybaCalxtp’ (aa’. Bx) =4 (a’Bx) (a’'yB) bybacacr = 4 (a Bry) (a'y8) . drbaCalx
+ 4 (yBe) (a'y8) babatate = bu* - (YB) (ary) Cate + a
29 = agdybybaCaCz (cc’ . 78) = $ (Bry‘x) agaybyba (yaw) = heyy’ +4 (Byx) apa, . byba (yan).
30 = cgaybybaCa (aa) Az'Up + dyDybala (CHU) . Ap'Ay ip + AybyDala (CHM) Az’ . Up?
= buy (aya) Cpbybacatig = 4 (ayx) byba . CaCa’iplla’ + aa’.
31 = agayby bala (cb'v) byUg = Updylyiip « DaCabz’ (cb'b) + agayby baCabx'uig (chw) = agdyby byt . baCa (cbw).
32 = duy (y'ax) apa,bybatig = vy +4 (yar) byba . dypiiptly.
cae,
33 = agdybybaCa (cau) Ug’ = Updyip dy’ « baCa (Cb) + AgayybaCa (ca'b) ap’ = (aa’ . yB) apiybaca (ca’b)
———— ———————————
= 5 (ay) update (Dep — bata) =} (ay) UybaCpCada = b)a* . (ary2) CaCptly + aa.
34 = hu, (xa8) a,b agba.
35 = (aBy) bycabaCaba’ bx’ = (aPBry) byCpda'Carda” + (bb'. aa) (aBry) byDa''CpCa
= 4 (Bax) (B’ya’) (aby) Caca = — 3 (B'ax) Cpca . (aBry).
This completes the system.
§ V. Forms reducible on multiplication by Uz are
(303) for u,. (abc) azbzc, = (bow) bry « Ag? + (CAN) Coy « Dy? + (ADU) Azdy . C22.
(421), for u,. (abe) (uca) (uab) ay = a2 . (bew) (carn) (abu) + (can)? . (abu) azbx
+ (abu)? . (caw) Cry + Ua? . (bow) bree — $Uz {(bew) battaCe + (dew) Cattabz}.
(501), for uz. (abe) azbaca = (abc) arb taCa + (abC) AxxUaDa
— {2a_2. (bow) bree + ba? . (CAN) Cxtz + Ca? . (ADU) Ube — Ay? . (bow) baCa}.
(512), for uz. (abc) agugbxer = artigitg . (dew) dyer + ba? . (CA) Aptiper + Cx? . (abu) apupbr.
(611), for wz. aga,b,big = 4 (abu). (Byx), save for products.
(630), for uz. (aPry) Uatlpty = Ua? (yx) Uptly + Up? . (YL) Uytla + Ua? . (48H) Uap.
CONCOMITANTS OF THREE TERNARY QUADRICS. 85
(630), for wz. (abw) agbyugity = — (abu). (Byx) up,
+ up? . {(abw) uyaybz — (abu) aybyax} — up . (abu) agugbz.
(630), for wz. (bow) wgityb,cg = — (bow)? . (By) upity + (bew) bycxtty « Up? + (bow) cabxttg . U4’.
(603), for wz. (Byx) dzbzapby = — (Byx)*. (abw) dzbe — a2 . (Byx) brb up
+ bi? {(Byx) Axtyp — (By£) Axlpity}.
(603), for wz. (By) brexbyca = — (Byx)?. (bew) bee + (BYZ) CaCxity « bi? + (Byx) bybatip . Cx’.
(710), for wz. (ary) apdyta = (AB) Uallpdatly + (AB) Uylatedp — (ABx) Uatlp . Ay?
— (yar) Uya . Up? — 207. (Byx) pty.
(710), for wz. (bew) aga,bycg = — (bew)?. (Byx) apdy + ug? . (abe) a,bycx
+ u,?. (abc) Cpagby — (caw) Catia . bytyb, — (abw) ayby . Cpiigex-
(721) for wz. (aBy) babxttpity = (Byx) Upily . Uadabs + Up? . (yeu) Dadatly + Uy? . (Bx) babzrt_.
(801), for _ (Bryx) b,CabaCa = — (Bryx)?. (bew) bata + 0,2. (aBry) Calptly
+ ¢:7. (aBy) babyuip — (yar) byba . CaCxtig — (48x) CaCp . bybztty.
(801), for wz. (a/be) agaybycadx’ = Az" . (bow) agaybycg + (abu) a,b, . CxtxCpdg + (Car) Cpdg . AzbzAyb,
—4 {(yas) byba . UpCaC: + (aBx) CaCg . Uydybz}.
(812) for w,.(a’Bry) (yar) (aBx) Ue = Ua? . (Byx) (yax) (aBx) + $a," . dz? . (Bye) Uprly
+ (aBx)?. (yar) Uyita + (yan). (aB8x) Walp — 2007. Ux {(Byx) Apdztly + (Byx) a,axu_}.
(911) for wz. agdybybaCaCaa = — (COW) ApCxttg . (your) Dyba + yt? . da. bp? . CY — ey? . Azdpiip . Uababx
+ 72. Uattgdybatlay — AzMglig . bydattyCaCz — 4 Ua? . bp”. Uy? - Cx’
(10.1.0) for wz. (a By) bcabaCatla’ = Ua? . (By) byCpbaCa + (yar) byDa » CaCpliallp
+ (aBx) Cacp - Dybdatlytla — 2a? {(abw) ayby . Cpiper + (caw) Catt . bytybz}.
Thus all but (421),, (501), and (710), are expressible by products of terms of lower
degree, and these are expressible by forms otherwise occurring in the list of forms.
In regard to the previous table we may remark that, multiplying still further by
Uz, we have
Ug? . (501),
ux? . (710),
ug . (801), + still further reducible, say are “doubly-quasi-reducible,”
tig? (10.1.0) |
U_ . (911) J
and there are, of the 18 forms just given, 13 which are only “singly-quasi-reducible,”
the reduced forms being expressible by the following 13 “whole” types of forms
(abe, b2, a2, O22, (bow) beer, (beu)*, Uababe, (abe) (beu) az, (bew) (caw) (abu),
(aBy), Ua’, (BYx) Upity, (Byx)’, (aBry) (Byx) Ua, (Bryx) (yan) (a82),
(bow) bata, OaCadaCe, (beu) daCxtla, (bew) bycrty,
(Byx) Opty, Uplyipdy, (Brya) Apdzly, (Bryx) CrCprp.
86 Mr H. F. BAKER, ON THE FULL SYSTEM OF
Further, of concomitants of two conics, there is one which is reducible multiplied by
uz, namely (630); =(bew) wguybycs, and its reciprocal (603); = (Byx) bycxb, cp.
(Geometrically these represent angular points and sides of self-polar triangle of the two
conics.)
Proof of the reductions by multiplication by Uz.
(303) is obvious.
(421), (a’be) (wea) (wad) Mz = dy? . (bew) (caw) (abu) + (ca’u) (cau) (abu) bya’ + (abu) (abu) (cart) cya’
where (ca’w) (cau) (abu) bya’ = (cau)? « (abu) azbx + (ca’u) (abu) by {(caa’) uz — (uaa') cx}
=(ca'u)?? . (abw) azby + 4 (duc) Uabz {Cate — Cxita}
and (a’bu) (abu) (car) Cxtz’ = (abu)? . (caw) Cre + (a’bw) (car) Cy {(waa’) be — (baa’) uz}
= (abu) . (caw) Cray + 4 (dew) Uae {Uadz — Uzda}.
(501);
(abc) azbytala = Va (ubc) dzbxCa + (abu) Azdz . Ca? + (AUC) AzbxdaCa
=ha,2. (ubc) byer + (abu) dzbz « Ca? + Uz « (ALC) AzdaCa + (AUD) AxCabaCa + (bUC) bala + Ma?
=ha2. (bew) byez + Ca? . (ADU) Azdz + Uz (ADC) AzdaCa — (ALC) AgCxdatla + (CHU) Cote » Da”
+ (bow) AzCxbada — (be) bala » Ae"
or Uz. (abc) dgdaCa = (abc) Azbzlala + (abc) AgCrdatla + Ax? « (bow) bala — $a" . (bow) dare — (CHU) Cre » Da”
— (abu) Arby « Ca?
(512), is obvious.
(611), (abu)? (Byx)? = {agbyuiz — apdzity + aybyttg — Ayptlz + Agdpily — Axdyup}?
= {apbyuz — agbztly + Aybztp — Azbyuia}?
= Quy, . Agdyb bug + 2apagtlp . bzbyuty = Wz « Agdybybzr.
(630), is obvious.
(630),. Consider (abu)* . (Py) uptly
= (abu) ugty| dz by Ux | = ug? . (abu) wy (bety — bydz) + Uy? . (abu) Up (Azbg — bap)
dp be Up | + Ug (abU) Uptly (agby — aybg)
ly Dy Uy
= Uz . (abu) agbyuguy + ug? . {(abu) Uydybz, — (abu) Uybyaz} — Uy? » (abu) apugbe.
(630),
(beu)? . (Byx) Upity = (beu) uptty| bg by be |= bp (dou) uptry (Cyl, — Cxtly) + Cp (Dztly — Dyuig) (bow) uptly
Cpl Cy\ Cz + Up? . (byez — bycy) (bow) Uy
Ug Uy Uz
=p’. (bew) byCy + Uy . (bow) eabzttg — Uz . (bow) upuyb,cp.
oo
~I
CONCOMITANTS OF THREE TERNARY QUADRICS.
(603),
(Byx) . (abu) drbe = (Bye) Azbz| Ap dy dz) = Uz (Bye) apbytizdy — Uz (Bye) dzbxdybp
Dp by be| + az. (Brya) be (bptty — by) + bx? . (Byx) dz (ayitp — gly)
Up Uy Ux
= Uz « (Ryx) AgbyAzdy — Az? . (Bya) brbyuig + by? « (By) datyup — (Byer) AllpUly} «
(603),
(Byx)?. (bow) bree =| bp by dx | (Bye) bre, = bp (Bya) bata (Cyl — Cally) + (Dy tise — bycptx) (Byx) bate
GainGy Cz + by? . (Byx) Cx (Cply — Cyttp)
Up Uy Ux
= (Brya:) bybrtip « Cx? + (BY#) CpCatty « Ox? — (Byx) beCadyCp « Une
(710), We - (AB) Apdyia = (By) Apdy . Wa? + (yHar) Uplatpy + (482) Uatyllply
and (yx) Upglallpty = (4Br) Ayala + (Bax) Ugtta . dy? + (yBa) UpWalaly
= (APY) UatlpAaly — (48x) Uatlp. Ay? — ka? . (Bye) Upity
(482) Uglyply = (4B) Uylalndp — (Ha) Uy, . Up? — bag? . (Bryx) Ugly.
(710),
(Byx) apdy . (bow) = (bow) apay be by by |= bp? . (Cytle — Cxtly) (ACW) dy + Cp (Dytly — byt) (ber) aay
py Cy. Cn, | + Up (byCx — byCy) (bow) agay
Up Uy Ux
= $)p. (caw) Cytyty + $e? . (abu) byagiig — Uz . (bow) agayb,cg + (bew) agayb,ttycg + (bow) Ag Cxdyil_,
while (dew) agityaabaca = Uy? . (bea) caab, + (baw) cyuyagbrce + (acw) byu,dpbace
= uy? . (abc) Cgdgb, — 4 (abu) agugb, . cy? — (car) cpag . byuydz
and (bow) agugayCxby = ug? . (abc) byte, — (can) ayttyCz . bg? — (abu) bya, . Carper.
(721) is obvious.
(801),
(bow) bata . (Brya)? = (Bry) Baca |p by bx | = dp (Cytte — Cxtty) (Bry) daca + Cp (Datly — byttz) (By) bala
G3 Oy Ge + up (byCx — byCy) (Byx) bala
Up Uy Ux
=$ dp? . (Cytiz — Cally) (Ayr) Ca— (yx) DaCadyCp » Ue—ACy? « Uipbz(Baa) da + (Byx)drDalaCp y+ (Byx) baCaCxDyip ,
of which 1b,” . cyt; (ayx) Ca = }bg? . Cy? . Uz (aan) = 0;
and (Bry x) DrbaCaCptty = b;? . (Byx) CaCplty + (ayx) brbpCatptty + (Bax) bed, CaCp ty
= b,2 . (aPry) Cabpuly — Lg? . (yaar) CxCatly — (482) CaCp . Dydztty
(By) CxCaDabdyiip = Cx? « (By) dabyiig + (Bax) CrCydabyug + (4x) CoCpbadyitp
=¢,° . (4B) babyug — Fe? . (482) babstig — (yar) byba . Caerup.
88 Mr H. F. BAKER, ON THE FULL SYSTEM OF
(SO1), uz . (a’be) azagaybycg = (ubc) agaybyca . dz’? + (abu) Cuz! dyapbycg + (auc) bzAty’Apcadyb,
and (a’bu) ext dyagbycp = (abu) ay’by . CxdzCatg + (aa’ . yx) (a'bu) czapbycp
= (abu) ay by « AzCxdpCa — 4 (you) (datig — bptla) Cxbyep
= (a’bu) ay'by . AzCrpca — 4 (yaa) byba . UpCpCz + | dp . (yar) UaCxCy
= (abu) ay'by « CxtzCaag — 4 (yar) byba . UpCpCe ;
also (a’uc) bytz'agcadyby = (auc) ap 'cp . Azbxdyby + (aa’ . Bx) (auc) aybzcaby
= (a’uc) p'cp . Azbxtyby + 4 (4Bx) (Waly — UyCa) bzCpb,
= (a'uc) dg'cp . Axbzayb, — 4 (Bx) Cacg . Uydabdy + be7 . (48x) Uabrbg
= (auc) ag'Cg . Axbzdyby — 4 (aBX) Cacp . Uybzby.
(S12)
(2/By) (yar) (48x) Uae « Ue = (Bryn) (yt) (aB.x) « a? + (/Ber) (ya) (4.x) tatty + (ary) (ya) (482) Uap
and (a'By) (yax) (@Bx) Way = (4 Ba)? . (yax) Uyla + (@'Bx) (yaw) (Aa'x) Uptly + (yar) (a’Bx) (4Ba') Ugly
=(a' Bx)? . (yar) Uyla + $(aa'x)ugu,(Byx)(w20')—4(Baa’)(Byx)(waa’)uztly
= (a' Bry)? . (vax) Uya +2 Ae? . Az? . (Byx) Ugly — 2 a? . (Byx) Apigtly . Ux,
while (any) (yar)(@Bx) Uap = (yXx)? . (aBa) Uattp + (ya'x) (yaa’) (aBx) Uxp + (yar) (a'ax) (28x) Ugly
=(ya'x)?. (48x) waipt 4 (yaa’)(Bay)(xaa’) uxug + 4(alax) (Bay) (vaa’)upity
= (ya'x)?. (a8) Uatlg + 3a? . Az? . (By@) Uptly — $a? . (Byx) Azdylg . Uz.
(911)
(cau) apex, . (yar) byba = bybadperilp|Cy Ca Cx |= fCy. dabzptlp (aUz— Aza) —Ay (Call — Cxla) dybatpCatlp
Ay Ae Wy + Uy (Catz — Cra) dybatipextia
by, the, Ue
= gytg? . da. de®. cP — hey. ArAgilp . Uadabe + Cx” » Uatiptgtybyb, — aragitg . bybaCaCrtty — $a”. bp®. wy? . Ca?
— dpttybybaCaCxlp » Wx
proving the theorem.
While further for Uallgpa,byb, square (uab) (aBy).
(10.1.0) (a’Bry) bycabaCatla’ - Uz = (HBy) bycgbaCa . Ua? + (aay) bycpbaCatla’lg + (a'Bxr) bycpbaCatla’lly,
and (avy) byCabsCatlaia = (4x7) Dba . CplipitaCa’ + (a ary) byCpdaCxtta'ig + (awa) bycpbaCyUalp
= (yar) dba. Cala ipa —4 (aay) (bu.ac’) by . Cpigert+i(aa'x) babatlaip.Cy?
=(yax) byba . CpCaUpa’ — 30" . (abu) ayby . Cartpcz,
and (a Bx) bycpbaCatla ly = (a Bx) bycpbatty {Cala + (cu . ac’)}
=(a'Bx) Carp . bybatlytta + $ (cu. act’) {(Bac’) by — (aac’) bg} bycguy
=(a'Bx) Cale . bybattytla + % Ua" . (CUM) Mcp . barby,
omitting —} (cu. aa’) (waa’) cyuy . by? = 0,
completing the reduction of the 18 forms on page 54.
CONCOMITANTS OF THREE TERNARY QUADRICS. 89
§ VI. Identities and examples.
The following are given, some because used, others because noteworthy.
1. The invariant t= 7.° of Gundelfinger.
To establish the identities — 1 7,° =[(abc)*} + 3 (ay)? — (ap? . a2 + bY . ba? + Ca” « Cp”)
(a’b’c’) (a’bc) (b'ca) (c'ab) = [(abe)?]? + $ (aBy)? — § (ap? . ay? + 0? . ba? + ca" « Cp”)
where we put nx = 6 (abc) azbzCr,
Ug? = — 6 (bew) (cau) (abu),
and these are the definitions of the symbols y,° and u,°. These give
—UgUoWe = (bew) (cav) (abw) + (bew) (caw) (abv) + (caw) (abv) (bew) + (caw) (abw) (bev)
+ (abu) (bev) (caw) + (abu) (bew) (car),
*, —49.°=— (abc) acbatg = (a'b'c’) (a’be) (b'ca) (c'ab) + (a'b'c’) (abe) (c'ca) (bab)
+ (caa’) (abb’) (bec’) (a’b’c’) + (caa’) (abe’) (beb’) (a’b’c’) + (a’B'c') (aba’) (beb’) (cac’)
+ (a'b’c’) (aba’) (bcc’) (cab’),
and
(a’b’c’)(bea’)(cac’)(abb’) = 4 ag(ace’) {(ca'b)(a’c'b’) — (ca’b’)(a’c'b)} = — Lagag'ayay = }(aB8y) — $a". ay
(a’b’c')(caa’)(bec’) (abb’) = 4 ag (caa’) {(cc’b) (c'a’b’) — (ce’b’) (c'a’b)} = 4 age’ (caa’) (cc’a’)
= 4 cp'Ca (Cala — Cala’) = £6p°Ca® + § (ay)? — F Ca°Cp® = 5 (AB)
(a'b’c’)(caa’)(abe’)(beb’) = 4c4(beb’) {(be’a)(b’c'a’) —(be'a’)(b'c'a)} =4h- cuca’ (beb’) (bb'c’) = (ay)? —4 ea? - Ca
(a'b’c’)(aba’)(beb’)(cac’) = 4b, (cbb’) {(U'c'a’) (c’ea) — (b'c'a) (c’ca’)} = 4 daca’ (cbb’) (b'cc’)
= } baby’ (by ba — baby) = 3 batby? + 4 (ay)? — 4 bbe? = § (ay)?
(a’b'c’)(aba’)(bec’)(cab’) = $b, (aba’) {(ab‘c) (a’b’c’) — (ab’c’)(a'b’c)} = — $b, by baba’ = F (aBy)? —4 bY - ba’,
from which the result above given immediately follows.
Further
(a'b'c’) az'by'cz — (a'b'c’) az'b,/ey + (a'b'c’) bz'eyaz — (a'b'c’) by’ez'dy/ ae (a’b'c’) cz'ay/bz’ — (a’b'c’) cx'a2/b,/
=(a'b’c' . (xyz).
Put herein 2;, y;, 2;=(bc);, (ca);, (ab);.
Then (a’d’c’)? . (abc)? = (a’'b’c’) (a’be) (bea) (c'ab) — (a’b’c’) (a’bc) (b’ab) (cca)
+(a'b’c’)(b’bc)(c'ca)(a’ab) —(a'b'c’) (bbc) (c'ab) (aca) +(a'b'c’) (c’be) (a’ea) (b’ab) —(a'b'c’)(c’be)(a'ab) (bea)
from which, by the results given, the above formula follows.
2. To find the value of ude where, as in 1, u,* =— 6 (bew) (cau) (abu).
We have — 4 w,v,? = (bew) (cav) (abv) + (caw) (abv) (bev) + (abu) (bev) (car),
whence —}1,0,7=(bcw) (caa’) (aba’) + (caw) (aba’) (bea’) + (abu) (bea’) (caa’)
=— (bew) bata — $ ba {(wea) (bea’) — (uca’) (bea)} + 4 cq {(bua) (bea’) — (bua’) (bea)!
= — (bew) daca — 4 daa (ube) + $ Cada (buc)
or Ugg? = 4 (bow) bala,
Won XV. Panr I. 12
90 Mr H. F. BAKER, ON THE FULL SYSTEM OF
namely, with Gundelfinger, Up = BUgHy? = 12 (dew) baca
(where Gundelfinger uses u. for uy).
So Ug = 3Ugb,? = 12 (cau) cgag,
Uy = BUgCo? = 12 (abu) a,b,,
and these are the definitions of the points p, q, Tr.
»
3. To sind the value of (qrx) in terms of our concomitants.
ch (rz) =|(ca’), (ca’), (ca’)s | cpag’ayby’ =| (cab’) (a’ab’)
| Cpdp ayb,’ = (ab'c) az'caag ayy
(ab’), (ab’), (ab’)s| YS 8a
+ (aa’b')cxcaits'dyby,
2D, ae Zs
= (abe) azcgaga,y'b, + $ ba’ (ayB) cxcaby’ = (abe) az'agaybycg + (aa’ . ary) (a’be) cgagh, — 4 (aBy) bybacxep
= (a'be) az'agaybycg + 4 (yan) (bata — Daca) byes — $ {(w@By) baCabycp + (yaw) byba . Ca? + (ax) bybacyce}
= (abe) az'agaybycg — § (yar) CaCy . bg? — 4 (48x) babg . cy? — $ (Byx) bacabyep
= (a’be) a,/aga,bycg — 4 (By2) bacab,ce.
It is then expressed by the two straight lines (801), and (801),.
4. To shew that the invariant (pgr)=s is expressible by our concomitants.
It is afterwards shewn otherwise, after Gundelfinger, that it is =8— 12S (m3),
S(m) meaning the quarticinvariant of the ternary cubic 7,%. But by definition
= (be), (be), (be’)s | baca'Cgag'ayb,’ = | (bea’), (bab’) | baca'cadg a,b, = (a’be) (ab’c’) baca Cap yb,
(ca’), (ca’)o (ca’)s \(c'ca’), (c’ab’) — (abb') (a’cc’) baCa Cad dyby
(ab’), (ab’), (ab’)s|
and (abb’) (acc’) baCa'Cgdp dyby = + ag (B’ay) ay (y’Ba) ayag’ = $ (aB'y) (aBy’) apayayag’
= t {(aB'y’) ay + (ayy) ap: + (7'B'Y) aa} {(@B'7') ap + (@BB’) ay + (B’By’) a} aycte’
which is reducible ;
also (a’be) (ab’c’) baca'Cpag'ayb,’ = {(a’b’c} b, + (b’be) ay’} (ab’c’) baca' cpap ay
= \(a'b’'c’) c, + (a’c'c) ba’} (ab'c’) bybacade’ ay — & Cprty’Ca’ Cap’ Ay (C'p'Va — Ca Mp’)
=(aV'c’)\(a'b'c')ag+(ab'a’) cp’} CabybaCpdy—4 0 yba'bybattp' dy (dybp’ —agb'y)—4 cay Ca’ Cpttp ay (C' pa — Cad’)
= — $B a (Wacy — by c'v) Cp Cabybaca — $ a’ yba bybadg'ay (aybg’ — apb’y) — 4 cpray'Ca Cap dy (Cpa — Ca Ap’)
= $V b/c ap Cabpata + 4 a’ yda' bybatg ayapb'y
= 4 (bv. ya’) baby Cacpe'acp’ + 4 (bb. ary’) ba’ byagayag’a’' y
=} (B’ya’) (Bary) catpe'aes’ + ¢ (B'ary’) (B’ya) apa,ag'a’y
=} (aya’) (B'ary) carcac’ aca’ + 4 (yay’) (B’yx) agaga' ga’ y
= U.
This indicates how its value and thence that of S(m,*) can be actually found in
terms of the 11 fundamental invariants.
CONCOMITANTS OF THREE TERNARY QUADRICS. 91
5. Putting (par) fe = (qra) a2 + (rpz) b,? + (pqe) ez’,
(this is the cubic of which the conics are first polars, as will be proved)
and Up = 12 (bew) baca ete.
(Byx)? =X,’ ete.
bxCxDle = Tx ete.
(bew) agayb,cg = u, ete.
(aBry) Uap, = u, ete.
Then it may be shewn that
sh (pqr) f2 = de G2 (4 Up - A? + Ug. te? + Uy; . 827) + two similar terms
+ (a7)? . up + (627)? . wu, + (C27)? U, — De7Cx? . U, — Cz? . Az?Um — Az? . bz? . Un,
which expresses the cubic in our forms.
6. To find u,*a.dz
— 4,70, = (bev) (cau) (abu) + (cav) (abu) (bow) + (abv) (bew) (car),
*, —4$.U,°gMy = (be) (caw) (abu) az’ + (caa’) (abu) (bew) az’ + (aba’) (bow) (caw) ay’
= (abc) (caw) (abu) az’ + 4 (bew) baeztta + $ Cabztia (bow) — Ux . (bow) baCa-
7. (a’be) (ca) (c'ab) az’b,’ ¢ = (abe)? . (abc) agbycz — $ ap? . (yY¥Z) Ardy + $d? . (yxz) byby
—4tc.2 . (Bary) catz + $ (Byx) apay . dzdy — $ (yaz) byba . brby +4 (Bay) cata . Cal:
+ $ (By) (Bay) (ya2) — § (28x) (Bry) (y22)
+ § (aBy) (2x) (@By) — § (yax) (aB8y) (By2)
+ (48) (yy2) (482).
8. Thus (a’bc) (b’ca) (c'ab) azbz'cx’ = — £ (Byx) (yax) (&Bx) + (abc) . (abc) drbaCy :
+4 (Bryx) aga, . az? — 4 (yan) byba . bz? — $ (aBx) cacgcz’,
or say (Byx) (yax) (48x) = — 4 (a’bc) (b’ca) (c'ab) az bz'¢z’ = AeboCodxbxCe.
9. Miscellaneous.
(beu) (car) battadz = — (ca)? . babstta + (abU)? . Caza — Ua? . (abc) (abu) cy
—4tuz{ta,?. (bow)? + (abu). c.2—(cau). b.2 — (bea)? . ua},
(bew) (bea) (b’c'a) bz’cx’ = (bea)? . (bow) brer + 4 (uab) agby . by — } (uab) a,by . b2
+4 (wea) Cry . Ce? —4 (uca) Cadp . Cx? +4 Uz . (Bya) Apay — F (Byx) ardpty — t (Byx) a2Ayup,
(abc) agbyCrlgty = (ubc) agbyupd,cr = (uba) cabyuiptycz or (uba) a,b, . caCrup = 0,
(abc) uguyazbyce = 0,
(abe’) (ab’c) (ubc) (a’b'u) az’cz’ reduces to the forms (a@By)(Pyx) ua, save as to products of forms,
Ug*AgbxCx (abc) = (caw) (abu) (a’be) (a'b’c’) bez’ = 0,
(dex) Ugdabg = WaCaCpU pz;
(abe') (ab’c) (ubc) (b'a'u) (c'a'u) = (abc) (bew) agua’.
12—2
92 Mr H. F. BAKER, ON THE FULL SYSTEM OF
§$ VII. An account of the theory of three conics as given by Gundelfinger, Rosanes, and in
Clebsch’s lectures.
§ 1. Establishment of the cubic of which the conics are first polars.
For a ternary cubic f=f=g,=hS=... I write the Hessian, after Clebsch,
H =(foh) frgziz = H = ete., the quarticinvariant S=—(fgh) (ght) (fi) (fyi), the sexticin-
variant 7 =—(f’g/h’)?(fgh) (Fgh) (g/hf) (Xfg) and the Cayleyan
us = — (Fgh) (ghu) (hfu) Fgu),
then we have the known equations
fife. = tu, fe=S8, HZ=T.
And since a system of three conics is determined by 35 =a fifteenfold arbitrariness,
while a system consisting of a ternary cubic and three points is given by 9+3:2=also
a fifteenfold arbitrariness, it is to be expected that from a system of one kind we can
uniquely determine a system of the other: in particular, in order that three conics
a,2=a,2=..., 62, cz should be the polar conics of a ternary cubic f,° in regard to
three points p, q, 7, it is sufficient that
ag=fefrp, b& =f2fy Ce =Sfifrs
leading to (par) fe’ fe = (qr&) aa* + (pgéy ba? + (pg) cx’,
which gives u,°=— 6 (bew) (cau) (abu) = — 6 (ghu) (hfu) ( fox) frggir = 6 (ghu) (hfu) (Sau) fogrhg
= — 3 (ght) (fu) ( fgr) fo (Galen — gle)
= —3 (hfu) (fgu) (ght) gp (hg fr — hr fa)
=—3 (fav) (ghu) (hfu) hy (So9r —Fr9)
=, by addition, (pqr) u;',
or, a> = (pqr) Us’,
and therefore Vg" Ug = (pgr) VsUs,
or in particular Ag Ug = (par) agus =(pgr) fe fils =} (pqr) 8. w,
and similarly bette =} (par) S . Ug, Coo = § (pgr) Su,,
so that the points p, qg, r must in fact be the points a,*u,=0, bou>e =0, Cots = 0,
and we may take the arbitraries so that up=3a.°Ue, Ug =Sde'Me, Up = BCo*Ue, (pgr)S=1;
while conversely if (par) f2 = (qr) Gf + (rpx) 2 + (pga) Og? ..2...escrecceesen ase (i),
then 3 (par) fe fo = (grp) a2? + 2 (rpax) baby + 2 (pg) Cxlp + 2 (Grex) AzMy,
and (as already shewn) u,=12(bcw) bac, so that byb, = 12 (beb’) bz/baCa = 6 (4B) Caly = AzMy,
and CaCp = 6 (yar) byba = axAr,
therefore 3 (pqr) f2 fp =(pqr) a2? + 2az {(qrx) dp + (7px) Aq + (pqe) ay} = 3 (pgr). az’,
or WEE: = a.
Whence Fi fa=b2, fi fr=ca, (par) fife = (qré) ad + (pf) ba? + (p98) cx’
So that equation (i) properly determines the cubic in question.
CONCOMITANTS OF THREE TERNARY QUADRICS. 93
$ 2. Expression of the cubic.
The cubic (i) may be expressed by our concomitants, for we have shewn
T4a (qr) = (abe) azagaybyop — $ (Bry) bacabycp ;
. aha (gr) f= az? « {(wbe) ay! agaybycp — 4 (Bye) dacabya} ++ eeceeeceeeee (ii).
Or again it may be expressed, after Gundelfinger, in terms of the discriminant, in
regard to a, of wat pb2+ p07, =d2 say, the discriminant being defined as
(dd’d”)? = 6 dy dy» dys
ds, doy dog
dz dz d. 33
For putting Ei= Pit t+ Gita + Tits, (t= 1, 2, 8),
so that in fact p,, 2, mw, are the co-ordinates of the point & in regard to the triangle
P, % 7, we have, solving for p,, M2, mw; in terms of € and substituting in the definition
equation of d,2,
Y 1 . 4 " ate
da = Copy (gre) aa? + (mp) be? + (pg) ca} = fz fr
and thence the discriminant (dd'd”)? is equal to H, namely to the Hessian of f°,
while (dd’d”) is in fact the cubic
Oy? = a? « pa’ + Dp? . a? + Cy? « fs? + BBs? papa + 3a? thr’bly + Bea ps py + Barp%u,2u, + Ba,2u eu,
+ 3D," ps"o + 6 (abc)? paploptes,
and therefore, remembering that the Hessian of the Hessian of a ternary cubic is
ay S°*f-1TH,
we see that the Hessian of (dd’d”)? in regard to p, namely H(d,°) is equal to
(par). ts SHf — 3 THs}:
namely [as (pqr)S=1
and x = 6 (abc) azb,cz = 6 (Fgh) f2Gaha frGqhr = (pgr) H',
whence bao = (par) ne = (per). Ae =(pgryD) voc vcnsvccnsesscocsenevecos (B),]
BesPaye H (d,)= ay fe — dtd,
that is yee Aas ads at, e...28 (iii),
which gives the value of /,° (referred to p, q, 7 as triangle of co-ordinates and) expressed
in terms of the discriminant in regard to w of p02 + pobz? + pez’.
And the 10 invariants a,?,... b,2,..., (abe) are expressible by the cubic,
for dJ=Hf =(H,. + H,. w+H,. p;)',
so that a.°= H,... b= H,°H,... (abc) = H,HH,,
with which compare a,?=/f;' ...a2=fyfip ... dq, = brbp = CyCg = fp fa Fr
Further the conic a2 being in fact (apd +a do +G,Az)? (where MAA, are the current co-or-
dinates) when referred to the p, g, r triangle, it is seen that the 18 coefficients of the three
conies are in fact only 10, corresponding to some extent to the simplification when two
conics are referred to their common self-polar triangle.
94 Mr H. F. BAKER, ON THE FULL SYSTEM OF
While also, remembering that the quartic and sextic invariants of the Hessian of a
ternary cubic are in fact IP-AS, ZST-3T,
it follows from df=
that S (d,2) = (par)! {8 T?— fy 8} = 30 — ah (par),
where S, 7 are, as previously, invariants of 7%,
and dB) = nee = Lt (pqr)-28;
and therefore (pqr) = 88-128 ge
and 2¢ — 9tS (d,8) — 18T (d,8) =0
So that any invariant is a rational function of the ten a,’,...b,%,... (abe) and of t.
The previous mode of expression is Gundelfinger’s. Otherwise we may say
ng =(pqr) do = (pqr) {aa? « pad + «+. + Bde" pipe + -.. + 6 (abc)? taplops},
giving the equation of »;° referred to Gundelfinger’s triangle,
and H (ne) = (pqr) H (dv), S(n®)=Sa,!), Tae) = Td?)
(pgr) = 86 — 12S (8), 26° — 9t8 (ne) — 187 (n°) = 0,
(par) fe = (par) {wiG2 + pad,2 + os6,2} = 12H (ne) + 4tnP..... cee (v),
giving the expression of f° in terms of n¢.
And we may see the exact significance of the cubic satisfied by ¢, by putting
S (ne) a 392,
T (n#) > 3 9s,
and 4u3 — gs — gg=4(u— &) (u — e) (U— @).
Then the cubic solves and we obtain t=— 3e; and therefore from (v)
(pqr) fe = 12 {H (ne) —e « ne},
namely by a known theory f# is one of the three cubics of which 7; is the
Hessian, which is right; or, say, f is a sub-Hessian of 72°.
And (pgr) = 6 {12e? — g.} = 24 (e; — e) (e; — ex) = 12p"a;,
where pu is Weierstrass’ elliptic function, with g., gs as invariants, and w; a semi-period:
and the interpretation of (pgr)=0, t=0 can be deduced.
Note too, the resultant of the three conics, vanishing with the discriminant of the
cubic, or S*—67?, vanishes with igh the Feaik
(pqry (pry?
namely with (pqr) — 68,
which is therefore the resultant of the three conics.
CONCOMITANTS OF THREE TERNARY QUADRICS. 95
There is a third way in which we may express the equation of f,°.
For (par) fz’ fe = (qr) ag? + (mp) bs? + (pg) cz, where uy = 8a,%ug
Oe tO Gs Pl |e deh) OE where (wv); = &;
eGR Ta FE Ao, ba, coo, 4 (uw),
Ware Gil te E, Ago, bo, Co’o, 4 (WwW),
Ps 9s Ts & Geto (eet, (eter 3 (wv),
=9 |? a’o, Ago, Ag2o3||0 wy uw U3/=9 PENS lijslie, GEUiA|lecanpenee (vi),
b2 b,*c, ba, b?o,| |0 VU, Uz, U3 Caen (Ou Ue Cells
Ge (Co407 wGa7Gg) Cs*0,|||1 0! 0 0 Gs Ws of
a form which will be afterwards obtained geometrically. But now using the equation, of
which the proof may be momentarily deferred :—
Ga? Dy? Cx? | = 3 (xyz) nenynz — 4 (y20)(zaa)(xyo),
dy? by? cy?
la2 62 c2|
where nz = 6 (abc) azbzce,
Us = — 6 (bew) (car) (abu),
we have 4 (par) fe fe = 3 (£00) nrloVeNene + 4 (aa'a”) (cox) (0x) Ugg’
= }(#a0’) (Eoo') nenene +4 (a0'0”) (0'0"E) (0's) (a2)
for &;= (uw),
and thence 4 (pqr) fb =4 (aoo'! nnene —$(ae'a”) (c'a"x) (aon) (oo'xr) ......... (vii).*
[We may prove the value of | a,” b,* c,°| quoted, as follows (after Rosanes, Math. Ann.
| OF lhe OF vi. 279)
OR lie CF
from 7,3 = 6 (abc) azbzCz 4 n2'ny = (abc) {dybzez + byezdz + Cytzbz}
Ug’ = — 6 (beu) (cau) (abu)) — 4.0,7u, = (be) (cav) (abv) + (cau) (abv) (bev) + (abu) (bev) (cav).
Therefore 4 {uy . 72°71, +(ryo)? Us} consists of terms like
(bow) {dy7bxC2 + AybyAxCz + AyCyAzbz + (ab . xy) (ac . xy)},
or (bew) {a,7bz6z + dybyAzlz + AyCyAzbz + Uz? . dyCy = AyCy . Azdz — Ayby » AxCr + wy? « DrCr},
or (beu) {2a,7bzcz + az7byCy}.
That is
& [Uynz?ny + (zy)? Us} = (bew)(2ay*bxCz + Az*dyCy) + (CU) (2by7Czdz + b2Cydy) + (abu)(2e,7azbz + Ct yby).
* [From which it follows [since (pqr) S=1, us*=(pqr)u,3, | and thence (ss'x)? H,H,Hy =} S*f-47H,
n,=(pqr) H,3], that it must be possible to express aaa (ss's") (s's""z) (s"'sz) (ss’z) = —4S°f- 9TH,
ternary cubic in terms of its Hessian, Cayleyan and quartic- Giese usta 4095 15
invariant, in the form §
4 S°7,3=4 (ass’)? HH,H, — i (ss‘s") (s's’x) (s'’sx) (ss’x), | sothat 3S?f,S=4S*f-4TH-(-4S°f-3TH),
and indeed (zss’)? wu, =(HH’u)? HH,’ +48 . (fou) f925 which is right.]
96 Mr H. F. BAKER, ON THE FULL SYSTEM OF
So
4 {urmynet (eyo)? Us} = (ber) (a,7brez + 2az*bycy) + (cau) (d,2crdz + 2b,*Cyay) + (abu) (Cy2axbz + 2Cz2dyby),
whence } {uz . 9,272 — $ Uy « M2 My + $ (zyo) Ue} = az? . (bow) bycy +b, . (cau) Cydy +c? . (abu) dyby,
whence § {2uznynmz—4 Uy» NaN: — $ Uz - Nx ny + (eye) (wze) Uo}
=a,’ . (beu) (bye: + bey) + b2 . (cau) (Cyaz + C2dy) + cz? . (abu) (ayb, + azb,),
or putting u;= (yz),
4 {2 (xyz) nenynz — (yzo) (zac) (wyo)} =|a2 b2 ¢,3).]
Gi lie Oy
Cano aco
Theory of conjugate systems.
There is also a theory founded on a relation of a locus of points of the second
order (say, shortly, a conic) to a cluster of rays of the second class (say, here, a cluster)
[which is an extension of the relation of a conic to two points conjugate thereto or of
two lines to a cluster in regard to which they are conjugate], under which relation
{either curve may be said to be conjugate to the other or better] the locus may be
said to be circumscribed to the cluster and the latter inscribed to the former. It is
that poristic relation under which a single infinity of sets of three of the rays of the
cluster form a trilateral self-polar in regard to the locus (so that the cluster-conic is in
fact inscribed, viz. in a trilateral), and a single infinity of sets of three of the points
of the locus form a triangle self-polar in regard to the cluster (so that the locus is in
fact circumscribed, viz. to a triangle).
If a2, bZ be two conics, the cluster of tangents of the latter bemg uw?=w?=...=0,
to the former w,?=w.*=...=0, then the tangents to b,? from the point v,.u.=0, which
is the pole of the line v in regard to a,’, are (Awa) (Xwa') v.04 =0, which are conjugate in
regard to a,?=0 if (Aan) (Aa’a”) Vava = 0.
But 0 = (Aaa) (Aa’a”) vada = $ Va (Naa) {[(Aaa’”) Ve — (a’aa’”) va}
= 3 a,? {Vo°A,? — AaVaAVa} = ¥ Aa? {Va°An? — $ GaVy?}
gives in general the cluster v,°a,?—1a,°7,°=0, of which the common tangents of a and
b, form part, which cluster coincides with that of the tangents of b,2=0 provided
a,7=0, and then we have b, inscribed in a single infinity of self-polar triangles of a’,
and also, as may be similarly shewn, a,’ circumscribed to a single infinity of self-polar
triangles of b,.
Or a,2=0 is the condition that a, be circumscribed to wa’.
And it is useful to bear in mind that
1. A conic is circumscribed to a two point cluster provided the points be con-
jugate in regard thereto—a,* is circumscribed to u,zu,=0 provided a,a,=0, which is the
condition for conjugate points.
2. In particular a conic is circumscribed to a point cluster repeated, when the
point is on the conic.
CONCOMITANTS OF THREE TERNARY QUADRICS. 97
3. A cluster is inscribed to a two-line locus provided the lines be conjugate
in regard to the cluster—p,qz is circumscribed to u,? provided pag. = 0.
4. In particular is inscribed to a line-locus repeated, provided the line be a
ray of the cluster.
And, as an example, the equation f?f,u,=4Su,, quoted (p. 92), shews that the
polar conic of any point in regard to a ternary cubic is circumscribed to the polar
cluster in regard to the Cayleyan of any line through the point.
From which we derive the interpretation of equation (vi) of page 95—for if
MG? + Agdz? + Asx?
be the polar conic of a point & where two lines u,=0, v,=0 intersect, in regard to a
cubic, it must be circumscribed to the polar clusters of u,=0, v,=0, in regard to the
Cayleyan u,’= 0.
Therefore Apes + Ade a + Asle"Ue = 0, Ae" + Ande2Va + AslCo2Ve = O,
from which the equation follows.
Now to be given that a cluster is inscribed to a conic is equivalent to a single
linear relation among the six coefficients in its equation, so that a cluster is determined
by five circumscribing conics (in particular by five tangents). A ‘swarm’ (schaar) of
clusters (the single infinity fu,?+gu,), similarly, by four circumscribing conics, and finally
a ‘web’ of clusters (the double infinity g,w,?+ g.u,2+ 93u,2) by three circumscribing conics,
or, say, by a circumscribing ‘net’ of conics fia,?+ frb2+ free (since a,?=0, b?=0, c,°=0,
require also g,4,?+ gob,? + 9s¢,7=0), and every cluster of the web is circumscribed to
every conic of the net.
The equation of the cluster of this web which is also inscribed in the two
arbitrary conics v,°, w, (which we may take to be repeated elements of the cluster, viz.
vz is a straight line as also w,), is got from
a? = 0, b,?=0, cv’ =0, ur? =0, v7 = 0, Wy — 0;
and is therefore O=|a,? a? a? Ass A3% GM, | or say, (abcvwu) = 0,
b? bE bj bby dsb, did, |
CC,” Cy" C3" CxCg C30, CC
Ui Va Us? VsVs | Vey Vive |
= |
We We We WW; WW, WW,
Un? Us Us? Uglls Ugly UUs |
where a,2... are the coefficients of the first conic.
But then from a,?=0, b,?=0, c,7=0, u*?=0 alone, we see that we must have
Grr? + Got? + gy,” =(abcvwu),
Vou. XV. Parr I. 13
98 Mr H. F. BAKER, ON THE FULL SYSTEM OF
where u,*, u,2, u,? are determinate, and g,, 92, g; unknown, with also
MUN + Gade + Gv? = 0,
wr’ + Ge +P gw = 0,
and therefore after determining a numerical factor
(abevwu) =|\u? w2 u,? |.
lo? 02 a!
we wy w,?
Just so we shall find
»
— 8 (Apvayz) =a? b2 ¢,*|,
dy? dy? ey?
ae EY OF
and in general the relations between the net and web are mutual.
And we notice another method of writing the equation of the web. The polars
of 2, Ardy, bzby, CxCy are concurrent if (abc)azb,c,=0 and then in (be);byc,. This point
is then conjugate to # in regard to all the conics of the net—namely, one of the
inscribed web is the two-point cluster w, (bew) byez.
So we may therefore write the inscribed web, y, z, ¢ being three arbitrary points on
Nx? = 6 (abc) azbzC, = 0,
th . Uy (bow) byCy + Jo . Uz (beu) bc, + gy . Ur (bow) bre, = O.
The Jacobian and Cayleyan of three conics.
‘We proceed to consider some relations between two derived curves of the net and
those of the web.
Defining the Jacobian of the net as the locus of the point « whose polars in
regard to three and therefore all the conics are concurrent, we obtain as its equation
n2 =6 (abe) azb,c,=0, the polars of # meeting in (bew) b,c, =0 or (car) czaz = 0 or (abu) ab, = 0.
But also there is a single definite conic of the net which consists of two straight
lines meeting in 2. For fiaZz+fib2+f,e2=0 satisfies the condition, provided simul-
taneously f,a,a; + f.b,b; + fiexe; = 0—giving the same locus for «—while also
Sit fot fa=(de)ibete + (00) iCaAe > (ab)iaxdx,
and the line pair intersecting in @ is
(be) bier . 42 + (Ca)iCrte . bi? + (ab)idedr . c° = 0,
(t being the variable)
and therefore making 7=1, 2, 3 this line pair is equally
(be . qr) byez . a? + (ca. gr) Cx, . b2 + (ab. Gr) dzbz . ¢ = 0,
CONCOMITANTS OF THREE TERNARY QUADRICS. 99
where u, = 3ugt,* = 12(bew) bce as formerly, and 6,b, = CxCy =6 (Byx) aga, and therefore
(ca . qr) Ceaz = (be . rp) bylz, (ab. gr) azbz = (be . pq) brez, 80 that the line pair is
(Gry) a2 (Gp) (gu iGs—Oleen senees sete. esseceec ee: (a),
where y is the conjugate point of a, namely (bcu)b,c,=0, as of course is obvious
from our previous determination of the cubic of which the conics are first polars. But
the theory of the Jacobian should be independent of the theory of this cubic.
So the line pair intersecting in y is
(CPE) 0 Cie ar (GOD) 6 REA (7a0 29) ¢ GPO coseacoccnon loonoosnopoosur (8).
And if & be any point on the join of «, y the conic
(qrk) a? + (rpk) b? + (pgk) c2 =0.......... Scere Siar ere vee ae (y),
—since k is a linear function of « and y—is a linear function of the other two, (a)
and (8), and therefore passes through the intersections of these. In particular when /
is on the Jacobian (namely is third point of intersection of wy therewith), this conic
becomes the line pair through its conjugate point and can therefore only be the diagonals
of the quadrilateral formed by other two line pairs, and z their point of intersection
must be the conjugate of this third or ‘complementary’ point of #y on the Jacobian.
Also in general the polar line of & in regard to (qr&) a+ (rp&) bf + (pq&) ¢? =0 is
(qr€) agce + (rp) bybr + (pg) Err = 0,
namely as (qr&) asa, = (qrt) ag 4- (7 Et) aeag + (Eqt) EP Ree.
and (rt) deg = (r&t) bed, = — (Ert) bby
This polar is simply
(qrt) ag + (rpt) b2 + (pat) eZ = 0.
Thus from harmonic properties of the quadrilateral, the equations of the lines yz, xz are
(grt) az? + (rpt) bz + (pgt) ex? = 0
(grt) ay? + (Tpt) by? + (pqt) cy? = 0
Therefore the line pair through the ‘complementary’ point /, the conjugate of z, which is
(qrz) a2 + (rpz) b? + (pqz) ce? = 0,
must pass through z and y and thus contains xy as one part.
which intersect in 2.
13—2
100 Mr H. F. BAKER, ON THE FULL SYSTEM OF
Thus, purely from the theory of conics, we arrive at the third property of the
Jacobian, that the join of every pair of conjugate points thereon is itself part of one
of the line pairs contained in the net. And through every point on the Jacobian
there pass two line pairs of the system, one having its central point there—but in
general through any point of the plane there pass three line pairs, as may be easily seen.
Consider now the Cayleyan—it is the envelope of the joins of conjugate points on
the Jacobian, say the envelope of a line cutting the conics in involution, and therefore,
from the theory of binary quadratics, its equation is
(beu) (caw) (abu) = 0.
But it is, by the theory just given, also the envelope of the lines, or say better,
the cluster of lines, into which the polar conics of the system break up. As_ such
however its most natural form of equation is given by a determinant of six rows and
columns. Namely we eliminate from equations of the form
Siig + Fabig + frcag = Uadj + YV;,
the quantities Py dba dim Oia hy Oh
which determinant is however given from the previous definition by noticing that the
conjugate points, considered as a two-point cluster, are inscribed in the conics
Chey Wy (ay ayy Cia, Oy.
We have in fact the following noteworthy identity, after determining a numerical
factor :—
Ug =—6 (beu) (cau) (abu) =
BG eke GAY ZUR OLR Ps || | GE BR aie LNs LyL, 12,
b,? b.? b? 2b,b, 2b,b, 2b,b, 2a WHY. ZLsYs LYzyt VsYo LsYit MYs MYo + Voy
C7 Cs? Cs Qos 2exc, 2re,|| 2 Yr Ys" YoY Yt Ye
9
=—3)a,? a’ as? 2a, 2a;a, 2a,a.| (where (xy); = u;).
[D2 2 bj? QWyby Lyd, Ayby |
CeCe 1 1Cr2) 2CaC5 0) SCs BAC iCa |
i On OO Us Uy
WO OR Ge Th 6 ih 0 |
So for the Jacobian, if 2;=(wv);, 2 and its conjugate are not only a two-point
cluster described in a,?, 6,2, cz’, but also in uz, ugdz, V2’,
and therefore
na = 6 (abc) able =6| a, ay? a3? O20, O3%, O,0,||2, 0 0 0 a w|
b? dj? bz Byby bid, did, ° a 0 m 0 a|
|G? Ce? Cs Oils G:C; Cre |)0 0 ay 2, a, 0
CONCOMITANTS OF THREE TERNARY QUADRICS. 101
=6| a? a as AAs 30, a,
i RIDE bib, bb, b,b,
c,° C.? C3" C2C3 C3C; C, C2
Ue ER eS UUs Ugly WUy
ZY, WetyVy 2VUgz Ugg + Us, Ug, + UyVy WV. + UV,
0,7 Oe ae UqUs Us; VV»
Let now u,', %,* represent the Jacobian and Cayleyan of the web. The former
will be the envelope of the joins of two-point clusters of the web:—two conjugate
points on 7,° are however such. Thus w,? will be the same cluster as U,*, and similarly
= as n°. And, in fact, if in
8 (abewww) =8 uy? w,2 Uy (page 98),
lon? Og ON
WwW, WwW, w,?|
we put Ur=%m%, UZ=0, uZ=0, 2uu,=0, 2u uw, = 0, Quy = 0,
WO Gat, w=, 2,0; = Us, 2uv,=0, 2v,U, = th,
w?=0, we=0, we=Us, 2Zww,=Uu, 2ww,=%, Qww,=0,
and use the identity of the previous page, we obtain
— 4g? = | APH WA Ag + . eee | = Suyuptly | MAA, | = Suauyre, (Aur),
namely Ug? = — 4u,.
So Ne = 2,3
Resultant of three conics (see also page 94).
If the three conics a,*, 6,%, c,?7 meet in y, the point cluster repeated w,2=0, is
inscribed in all the conics of the net, namely is one of the point pairs occurring in the
inscribed web, with however the speciality that the points coincide. Thus the Jacobian
(of the net) must have a double point at y, and therefore all its first polars will pass
through this point or be circumscribed to it. Namely each of the conics 7,2,, NxN2, NxNs
will be circumscribed to (u,?= ) gy? + gou,2 + gsu,2 = 0,
so that GM + Gonem + 9sy7n, = 0,
JM. + JM N2 + Ys. = 0,
NNN Ns + JonNs + GN": = O,
are consistent. These give
m° | or say $(97'n")|\mx*° 2 7° |= 0.
A a We 1?
M9 9 9)
Bx Bg By |
2
row 2
M273 |x De
/ 2 o. 2 ,
Dx Dp Hy
iA 2 ur 2 ” 2
| n pa n bh’ n y
102 Mr H. F. BAKER, ON THE FULL SYSTEM OF
But from the identity proved at the bottom of page 95, this is
2 (ULW) Ugly Wy — § (VWE) (Wut) (wvE) = 0,
where u,v%, w=, 7, 7,
and u,3=—}u,° is the Jacobian of the web and =,2=47,° is the Cayleyan of the web,
namely, is (UW) UgVoWe + (VW) (Wu) (uvy) = 0.
So that the resultant of the three conics may be written
CL! 2 NT ETT
(nnn)? No oN + (nn'n”) (n/n) (nm) (gyn) = 0.
And as verification, since w.°=(pqr)u;’, 72°=(pgr) Hz* (as proved), it should be possible
to write the equation and discriminant of a ternary cubic
CH) zy =9,(Ft),
while as H(H)=),S:f-1T7H, 8(M)=3T?- 4S, frws =S, Hao = T,
this becomes } S*— 7, which is right.
And, as for the net, so for the inscribed web, we can write down a class cubic
whereof the first polars coincide with the web.
§ VIII. Notes on some of the concomitants.
1. We can find a class cubic of which the clusters w,’, ug’, u,? are the first polars
in regard to three straight lines. For the polars of q and r in regard to f, and f,
respectively are the same straight line, namely,
bxb, = CxCq = 6 (Bry) apay.
Put then bb, = Clq = 6 (Byx) aga, = 5a
Cxlp = Aztt, = 6 (yan) byba = 3 Bg? .
. ln
zg = bzby = 6 (ABZ) cep == 5,
3 ¢,?
so that Mylly = BDz? . UyCyCp = Dg? . Cy? . Up = Nu,
and take (Imn) u;’ =(mnw) ug? + (nlu) ug? + (Imu) uw. Then as previously w;*l, = u,%, ete.,
and really (mnu) = 9bg'c, (aa’u) aga’, = Dbgcy?ua (gra) = Ibp*c,? (bow) bye,
= —} 9bp'c,? {(a’By) UabaCabyCp — % da? . (Ube) baCabyce}.
Thus the cubic can be expressed by our concomitants, or in terms of (Byx) (yam) (aBx)
and (aBy) va%aly as before.
One form of its equation is
Uae Up? U,?
Pay (qr) Ua + ig . (rpB) up + re - (pay) Uy = 9.
CONCOMITANTS OF THREE TERNARY QUADRICS. 103
2. The conic bdzcrbaca=0 {or (402),=0} is the locus of a point whose polars in
regard to b,? and c,? are conjugate in regard to a,’.
These polars meet in the point (bcu)b,cz=0, or say ug=0, and we have through
this point three lines conjugate in pairs in regard to the three conics. Namely, y being
the variable, the pairs are
(z€y) =0 and b,b, = 0 conjugate in regard to b,?, or harmonic in regard to tangents from € to b,’,
(w&y) = 0 and CxCy = 0 ” ” ” Cy ” ” ” ” » ”? ” Cx",
Drb, 7a 0 and CxCy = 0 ” ” ” Ay? ” ” ” ” ” ” ” Ay.
3. In general the condition that the conjugates through &, (S£y) ug=0, (yEy) u, = 0,
of a line w, in regard to 6,? and c,? should be conjugate in regard to a, is
(aE) (aE) uatly = 0,
namely, « touches a conic and there are two such lines w through &.
Putting herein, to connect with (2), & the conjugate of « or v4; =(bev)b,cr, we obtain
(a8 . be) (ary . 0'c’) brerbz'CzUptty = — DaCabrCz {bybatly . CpaCztig + 4 uz? . de® . 0,7}
+ 4uz [c/ {b.? . b2 —4 (@Ba)*} . cacztia + bp? . {c2ex? — $ (yax)} . b,bru,],
and if # be on b,¢abzcz=0 the cluster is two pointed, one point being, as predicted,
x and the other on the join of the points b,b,u,=0, cxcgu3=0 (whereof the former is
the pole in regard to c,’ of the polar of « in regard to b,’).
And as # moves on b,¢,b,c, =0 its conjugate (bcw) b,c, =0 moves on
0 = (aBx) (aya) bycpbrez = — (Bryn) . baCabaee + } (bz? « Cy? (a8)? + C2 . Dg? (yaw)?}
+} (Byx)? . (b2 . Ca? + Cz? . ba?) — 4b? . Cp? . (yar)? — 4,2 . b2 . (aBe).
4, Further in regard to the cluster (aB&)(ay&) ugu,=0 [which reduces to the con-
comitant (611),], the polars of € in regard to 6,* and c,? are among its rays and for
the conjugate through & (in regard to b,*) of bybs=0 we must take the join of & to the
point (ay&) b,bv,z=O0—which point is the pole of the join of & to bjbw,=0 in regard
to a2—(it is the concomitant (512),).
For consider the locus of the poles in regard to b,* of the rays of the cluster
(aB&)(ayE) ugu,. Its equation in y is 0=(aBE) (ayE) dg’b,'b,b,, or say (ayE) (ayé) b,b, = 0,
which certainly passes through &, and putting py;=£&;+.«z; and then (z&);=4u;, we obtain
(ay&) uabybr =0 (for x =0).
5. Consider further the conjugates through & in regard to b,? of the rays of the
cluster (aB&)(ay&) ugu, through & They are the joins of & to the two points given by
b:b, = 0)
(ay&) (ay&) bby = 0
104 Mr H. F. BAKER, ON THE FULL SYSTEM OF
Putting py;= &+ «2; in both, we obtain
be + «bib, = :
’
and (az&) (ay&) byby + « (a2€) (aE) bb. = 0
wherefrom (az) (ayé) bybe + bb, — (az&) (ay&) bb, . by = 0,
and therefore (az&) (ary&) (bb’ . Ez) bby’ = 0,
or finally (a2&) (BzE) (aE) (Byé) = 0,
so that the conjugates sought are rays of the cluster (ay&) (Ay) vaus=0, which is of the
same form in regard to c,” as the original in regard to a,’.
We have then through &€ six lines OP, OP’, 0Q, OQ’, OR, OR,
such that OQ, OR are conjugate in regard to a,?, as are OQ’, OR’,
OR, OP 5 + x 5 Oa 5s, OR OR.
OP, OQ ” ” » ”» Cx", ” OU, OY.
6. Cacrua=0 is the pole in regard to b,* of the polar of « in regard to ¢,,
b,b,u,=90 is the pole in regard to c,? of the polar of x in regard to b,’.
The join of these points is (Byy) b,¢pbxcz = 0.
Conversely if this join passes through a fixed point y the point « lies on a conic;
which conic is harmonically circumscribed to a,* (or u.*=0) provided y lie on the line
(Bry) bycabaca = 0, [This is the concomitant (801),],
and then the points # form a single infinity of sets of three, each forming a triangle
self-conjugate in regard to a,’.
CONCOMITANTS OF THREE TERNARY QUADRICS. 105
From the equation of this join we derive the equation of the self-polar trilateral
of the two conics: namely 2 must lie on this join. giving the equation
0 = (By) bycgbrex for (603), = 0].
Further, the pole of this join in regard to a,’ is the point
(aBy) ab CpbxCx = 0,
and if this .pole lie on a fixed straight line, # describes a conic, which 1s harmonically
circumscribed to a,’ (or uw’) provided this line pass through the point
(a’By) byCabaCala’ = 0 [which is (10.1. 0)=0].
7. The point (bew)b,c,;=0 is conjugate to « in regard to both 6,’ and ¢;,’, say is
the ‘conjugate’ of «. Conversely the locus of the conjugates of collinear points is a
conic, the conversion being allowable because the conjugate of the conjugate of a point « is
(bow) (bbc) (cbc) 20x02 = — 4 Uz . (Byx) bycpbrex,
namely is « itself—the factor (Syx) b,cgb:c, representing the common self-polar trilateral
of the two conics.
The locus of a point « whose joins to its conjugate always pass through a fixed
point y is a cubic curve
(be . xy) bate = bz" . Coly — Cx - bzb, = 0,
which passes through the intersection of the conics, through y itself, through the con-
jugate of y (this being in fact the “tangential” of y on the cubic), through the points
of contact of the tangents from y, and in general may be generated as the locus of
the points of contact of tangents from y to the bundle b? +2Ac;7=0. And thus, in
fact, from a known property, any cubic curve can be thus generated; as also follows
from the fact that three collinear points «, y, 2 on a cubic f,°, whereof H,* is the
Hessian, satisfy
Srfyfz=9, H,H,H,=0
(as follows from Salmon’s identity (/Hw)* = 0).
Vou. XV. Part I. 14
106 Mr H. F. BAKER, ON THE FULL SYSTEM OF CONCOMITANTS, &c.
8. The conic (bcu) bc, =0 is also the locus of the poles of the line w in regard
to the conics of the bundle b,?+Ac,"=0: and, in fact, the pole of (q7z)=0 in regard
to the general conic (qr&) a,” + (rp&) bx? + (pgé) cz’ =0 is the point (bew) bec: =0, the
point conjugate to & Thus if & move on a line v, its three conjugates (bcw) bycz = ,
(cau) ceag=0, (abu) asb;=0 move on three conics (bev) byc,=0,... and these three conics
correspond also to (qrz)=0, (rpv)=0, (pqv)=0 respectively, im regard to the general
conic (gr&) a,” + (rp&) b,” + (pqé) ex =0, which is now described about a fixed quadrilateral.
9. Lastly the conic (bcw) b.cp=0 for the line (qrz) =0 is
(be . gr") baCr = babs . CrCz — (CeCz)’,
namely touches bbz, ¢,cz the polars of g, r in regard to b,’ and c¢,” respectively, on
the line cc, = 0.
10. The conic ay’. (bcw) dyer + by? . (caw) cra + Cy? . (abw) azbz = 0,
y being the variable, is the conic of the net for which a is the pole of wu.
If the line w be (grx)=0, then since
(ca . gr) Cx@z = (be . rp) beer, (ab . qr) azby = (be . pq) brez,
the conic is ay (qr&) + by? (rp&) + ey" (pgé) = 9,
where & is the conjugate of 2 in regard to b,” and c,’.
In general the conic passes through « provided
Ux « (abc) Abe, = 0.
Take u,=0. :
Then the conic touches the line w at the point « It is a line-pair provided
is tangent to a class cubie (for the discriminant of a cubic is of the third degree in
its coefficients). Thus through any point w there pass three line-pairs of the net, which,
touching the tangents to a certain class cubic at this point, must either have their
double points at 2 (which is excluded) or have the three tangents to the cubic as part
of themselves. Namely the class cubic is the Cayleyan.
§ IX. The following list of memoirs may be added:
1. Gundelfinger. Crelle, LXxx. 1875, 73.
2. Sylvester. Camb. and Dub. Math. Journ, t. vil. p. 256. (1853.)
3. Cayley. Crelle, hvu. 139. (1860.)
4. Hermite. Crelle, Lvu. 371. (1853.)
5. Darboux. Bulletin des Sciences Math, t. 1. p. 348.
6. Rosanes. Math. Annal. vi. S. 264.
7. Schriter. Math. Annal. v. 8. 50.
8. Smith. Proceedings Lond. Math. Society, 11. (1868.)
9. Lecons sur la Géométrie. Alfred Clebsch. Lindemann. Traduites p. Adolphe
Benoist, Paris, 1880. Vol. 11. 248.
January, 1889.
IV. On Sir William Thomson's estimate of the Rigidity of the Earth. By
A. E. H. Love, M.A., St John’s College.
[Read April 28, 1890.]
THE question really propounded in the articles of Thomson and Tait’s Natural
Philosophy devoted to the discussion of the Earth’s rigidity is this:—Swpposing that for
purposes of discussion the Earth is replaced by a homogeneous elastic solid sphere of the
same mass and diameter, what degree of rigidity must be attributed to such a solid in
order that ocean-tides on the sphere may be of the same height as the actual ocean-tides
on the Earth? This rigidity is called the “tidal effective rigidity.” As is well known
the tides to be considered are the fortnightly tides, as being of sufficiently long period
to be capable of adequate discussion on the “equilibrium theory,” and at the same time
free from certain difficulties which beset the observation and discussion of annual and
The actual amount of the fortnightly tide on the Earth appears to
be still to some extent matter of dispute.
semi-annual tides.
For the purpose in hand the estimate of
it employed is one made by Professor G. H. Darwin founded on a series of observations
chiefly made in the Indian Ocean.
fortnightly tide is little less than 3 and certainly much greater than 2 of the true
Now, in the articles of the Natural Philosophy referred to, it was
shown that if the Earth were replaced by a homogeneous incompressible elastic solid
sphere of the same mass and diameter, and of rigidity equal to that of steel, the height
of the ocean-tide would be reduced by the elastic yielding to about 2 of the equilibrium
height, while the reduction would be to about 2 of that height if the rigidity were
It was concluded that the tidal effective rigidity of the Earth
is nearly that of steel, and the conclusion was held to disprove the Geological hypothesis
of internal fluidity.
The present paper is not occupied with any attempt to review the evidence used
by Professor Darwin as to the amount of the observable fortnightly tide, or to criticise
the conclusion of Sir William Thomson from the great tidal effective rigidity of the
Earth to the improbability of the hypothesis of internal fluidity*. Its purpose is merely
to discover what difference would be made in the tidal effective rigidity if the elastic
According to this estimate, the amount of the
equilibrium height.
equal to that of glass.
* [Note added Sept. 1890. It is proper to mention that
Professor G. H. Darwin has in a recent paper, Proc. Roy.
Soe. Lond. Noy.1886, expressed an opinion that it is probably
impossible to obtain a correct estimate of the Earth’s tidal
effective rigidity. In all previous calculations it had been
supposed that the fortnightly tide obeys with sufficient
accuracy the equilibrium law, but it is there pointed out
that oceanic tidal friction is probably too great to allow of
the application of the equilibrium theory to the fortnightly
tide. Sir W. Thomson’s estimate of the Earth’s tidal
effective rigidity is based on such an application.]
142
108 Mr LOVE, ON SIR WILLIAM THOMSON’S
solid replacing the Earth were not assumed to be incompressible, but to have its modulus
of compression and its rigidity in the same ratio as most hard solids have. It may be
premised at once that the difference is very slight. We find ourselves confronted with
a particular case of the following problem—A gravitating solid elastic sphere of any finite
rigidity and compressibility is subject to the action of bodily forces derivable from a
potential expressible in spherical harmonic series, it is required to determine the resulting
displacements. Certain problems of the same kind, but less general than this, are solved
by Thomson and Tait. These authors consider the case where the elastic solid has any
tinite compressibility and rigidity but is free from its own gravitation, and the case where
the solid is incompressible and gravitating and of any finite rigidity. The solution of
the general problem is here obtained, and it is noteworthy that it cannot be derived
from these solutions by any method of linear synthesis.
Let W, the disturbing potential, be expanded in a series of spherical solid harmonics
in the form W= Wy, where 7 is an integer, and suppose the equation of the deformed
free surface expressed in the form r=a+e¢Qii., where e; is a small quantity and Qj,
is a spherical solid harmonic of degree (+1), then among the bodily forces acting at
any point are included the attractions of the inequalities. These are derivable from a
potential of the form =V;,,, where V;,, is in like manner a spherical solid harmonie.
The other forces to be taken account of are the attraction of the nucleus and the forces
whose potential is W. It is easy to obtain, by using Thomson and Tait’s solutions, a
general solution of the equations of equilibrium under these sets of forees in a form
adapted to satisfy boundary conditions at the deformed surface. The conditions to be
fulfilled are those which express that this surface is free from stress. Such solutions contain
complementary functions, and particular integrals depending on the bodily forces, and, inasmuch
as the harmonic inequalities contain terms depending on the complementary functions, the
bodily forces, some of which arise from the attractions of these inequalities, contain
similar terms, and thus the particular integrals contain unknown harmonics which occur
in the complementary functions. This is one important difference between the present
problem and those considered by Thomson and Tait. A second consists in the fact that,
the attraction of the nucleus being very great compared with the other forces concerned,
it is not sufficient to estimate the surface-tractions to which it gives rise at the surface
of the mean sphere, but they must be estimated at the surface of the harmonic in-
equality. This is done by a method I have employed in a previous paper (Proc. Lond.
Math. Soc. x1x.). When the complete expressions for the surface-tractions at the deformed
surface arising from the complementary functions and particular integrals have been
obtained, it is easy by equating them to zero to deduce the expression of all the unknown
functions that occur, and thus to express the displacements at any point in terms of
the disturbing potential. One result is that the harmonic inequality arising from any
spherical harmonic term in the disturbing potential is proportional to that term and
contains no other harmonic.
The application to the tidal problem is made by supposing the disturbing potential
to consist of a single term which is a spherical solid harmonic of the second order, say
W., and thus by taking 7=1. We have also to take p the density of the solid equal
ESTIMATE OF THE RIGIDITY OF THE EARTH. 109
to the Earth’s mean density. The elasticity of the material composing the sphere will
be defined by two constants m and n such that m—4n is the resistance to compression,
and n the resistance to distortion. By supposing m to become infinite, and n to remain
finite and comparable with gpa, where a is the radius of the sphere (taken equal to
the Earth’s mean radius), and g is the value of gravity at its surface, we fall again on
the case of incompressible material treated by Thomson and Tait, and obtain the same
results. This serves as a partial verification of the analysis. If however we suppose m
and n both finite and comparable with gpa, and connected by the relation m=2n which
holds nearly enough for most hard solids that have been submitted to experiment, we
get a different case. Now it is shown in this paper that in both cases the harmonic
inequality is expressible in the form eW./g where ¢ is a number, and that ¢ is a
rational function of a second number S=4gpa/n. This number % is such that (3S)?
is the ratio of the velocity of waves of distortion in the material to that due to falling
through half the radius of the sphere under gravity kept constant and equal to that at its
surface.
When n/m=0, as in the first case, the numerator and denominator are linear in S.
When n/m =4, as in the second case, the numerator and denominator are cubics, neither of
which has a positive root. It appears on calculating the values of the two functions for
positive values of $ that the values of ¢ in the two cases are always very nearly equal
for the same value of 3. When the rigidity is not less than that of glass S is $ 5
and it appears that for all such values of S the value of e given by the second sup-
position is slightly greater than that given by the first, fur some value of % greater
than 5 they become equal, and subsequently the value of ¢ given by the first is slightly
greater than that given by the second. The differences are always very minute. Thus
for the purpose of estimating the tidal effective rigidity of the Earth, Sir William Thomson’s
method is sufficiently exact. For this purpose we must consider a third case of the
problem, viz. we must find the tidal distortion in a sphere of homogeneous liquid of the
same mass and diameter as the Earth. his is also expressible in the form eW,/g and
e is the fraction §. If then the values of ¢ found by either of the previous calculations
be multiplied by 2 we shall have the ratio of the elastic solid yielding to the fluid
yielding. The fraction obtained by subtracting this ratio from unity is the ratio of the
height of the ocean-tides on the yielding nucleus to the true equilibrium height. As
mentioned before, this fraction is about 2 for a tidal effective rigidity equal to the rigidity
of steel, and about ? for a tidal effective rigidity equal to that of glass,
1. Let W be the potential of the external disturbing bodies, and suppose that for
space within the sphere W is expanded in a convergent series of spherical solid harmonics
in the form
Suppose that by the action of the external forces the sphere originally of radius a’ is
strained so that the equation to its surface becomes
ao
r=a+ €:Qix, eisai eteletailen clas tice ttece eect cas cea (2),
110 Mr LOVE, ON SIR WILLIAM THOMSON’S
where ¢; is a small quantity and Qj,, a spherical solid harmonic of degree 7+1. Then
the harmonic inequalities ¢;Q;;, will exert an attraction on the mass whose potential we
may denote by V, and this potential will, like W, be capable of expansion in a convergent
series of spherical solid harmonics in the form
If p be the density of the solid and y the constant of gravitation the bodily forces will
be derivable from a potential
ARBRE TUN ct Wn snp veaiant suaene shen ee ane (4),
which we shall denote by Y, and the general equations of equilibrium will be three of
the form
08 OY _ a
ma +nV?a+p = =) 5 act Sastacaveinaeerre eet ice cee Rae (5),
where a, 8, y are the displacements in the direction of the axes of 2, y, z, 6 is the cubical
dilatation 02/dx + 08/dy +0y/dz, and m and n are two elastic constants.
2. The solution of the system of equations (5) consists of particular integrals and of
complementary functions which satisfy a system identical with (5) when Y is left out. The
latter are given in Thomson and Tait, Art. 736 (e), in a form adapted to satisfy conditions
at the surface of a sphere r=a and this form is equivalent to
7 nt OW; Ovi
ay SE 2. sey} pgeey te
a==, [45 +07 Mii. an eM; in | a SSoisies shaw aedssacEee ee (6),
where we have picked out the terms of order 7 in a, y, 2 § and y are to be derived by
cyclical interchanges of the letters (A, B, C), (, y, 2), Ai, Bi, C; are spherical surface
harmonics, and at the surface
CLAS BIB isa, Cotas cowtarenseieutercccecss detec tee (7),
M; is the constant Geir CTS» ae ambi agne NS (8),
0 PVG PN O fn 0
and Yia=e (4; ) 7 (2, a) +5, (0 =) ue. See (9),
which is a spherical solid harmonic of degree i—1.
3. For the expression of the surface-tractions at the surface of the mean sphere r= a
we have to introduce a new function $_j;-, defined by the equation
ra) qth ra) qt fa] aH
waeraee aon) + = (Bi cra) + a (Ce Gers) vseeseeceeessneee (10),
then $-;. is a spherical solid harmonic of degree —i—2 and differs from Thomson and
Tait’s ;,, only in being divided through by 7***/a‘t. The surface tractions parallel to
zx, y, z at any point of the mean sphere are calculated in Thomson and Tait, Art. 737,
and are equivalent to F, G, H, where
z a
Fran [G-DAn— 357 i
i ] QO pptits pats 9 qzits
, (ee $+) = Bits waits ap (Yen saa) ae lenuy (11),
=
ESTIMATE OF THE RIGIDITY OF THE EARTH. 111
and we have picked out the terms containing surface harmonics of order 7. G and H are
to be derived by cyclical interchanges of the letters (A, B, C), (a, y, z), and E;,, is the
constant
1 m(i+4)—n(2i+8)
21+ 5 m (i+ 1) +n (2% +3)
4. We have now to consider the particular integrals of (5). We shall treat first the
term of order zero —3aypr*. The purely radial force —4mypr hence arising produces a
purely radial displacement U whose amount can easily be shown to be
Ui Ar Pr 4% ssncsaaeee eae eens ee (13),
where A is an arbitrary constant and
— 14
Bs eT a alt ie (14).
The six strains e, f, g, a, b, ¢ referred to the axes of «, y, z depending on (13) are
given by such formule as
G=Jél Ges ip) tera ney oar) =U ELIA ceoy'o00 sdoonodeanaoneenseneon or (16))),
as shown in my previous paper (Proc. Lond. Math. Soc. xix. p. 185), and the surface
tractions at the surface r=a+eQi,, are of the form AP+pU+7,...,..., where (A, w, v)
are the direction cosines of the outward-drawn normal to the surface and
P=(m—n) 6+ 2ne,...S=na,...
are the six stresses as calculated from the formule (15).
Now neglecting ¢,7, X is given by the formula
(a 2! z Q OQi+1)
Lr
A=Z+ Sef ee iy || oo anaes ecaco car (16),
and for » and v we have similar expressions, and we find without difficulty for the part
contributed to F.7r, neglecting ¢;*, the form
2 { Ha? (im +n) + A (3m —n)} (1 +> a) + 2HareQ:..(5m+ n) x
—ade; Ea (5m — 3n) we + 4nH (t+ 1) aQisn + (8m —7n) A al Racism lit):
We shall shew hereafter that the term «[Ha*(5m+n)+ A (3m—n)] is the only one not
containing a spherical solid harmonic with a small multiplier, like ¢,Q;.,, and thus this
term will have to vanish, and we find
5m+n
SAS eer ae Oe SLO COO ODOC OH ODOOHOOCONNOADOOOOGMOCEOF (18).
This with (13) and (14) gives the mean radial displacement, a matter which need not
detain us here.
Using now (18) to simplify (17) we obtain for the typical term contributed to F’.;
2 {5m —(21+1)n} HaeseQess + 4nHare, 2 Scie tates MEANS as (19),
112 Mr LOVE, ON SIR WILLIAM THOMSON’S
or at the surface of the mean sphere r=a, we find by using the identity
— = 1 2 OQis1 2t+5 0 ge) 9
2Q34. = G43 ( an —7 am (S8 Bee eA tices woe eee ee (20),
that the typical term contributed to F.r may be written
2 Hare, met t+ 5) OQist gg WM — AFD M gixs 0 (22) eS ea
a 2+3 Oa ao 2+3 Shona A248) 12 is ee
as in my previous paper, p. 187, equation (44), with a like verification to that on p. 188
of the same paper.
5. Take next the term of order 7+1 in (4) and write
Ve — i+1 + Wis eee cece eee ene e ee eee sees st sesssseares (22, .
The particular integral will be found as in Thomson and Tait, Art. 834, by taking
6 2g _b | _0b »_o, 5
3 Serr oe © = Vig eee hence ces ccdeaeseenene (23).
This reduces equations (5) to the form
i. FN) VAG pV Fa 0 oa Sock Sock ctee toes testes ate (24),
and a solution is
week? ES ee ee... 25
p= ae) ey cepeeorpagonaoagedao: (25),
since Y;,, is a solid harmonic of order i+1.
Hence the particular integral for @ is of the form
Oa 2) SP Pe oh (26),
m+n 2 (245) ox
or by using the identity (20) with Y in place of Q we find for the typical term of the
particular integral for @
Li ah iat i, Pg! (a2) -4; a al (27)
hs | GEES) (G5) as eRe oe | 27),
and those for 8 and y are to be found by cyclical interchanges of the letters (a, y, 2),
and the complete value of a is to be found by adding the expressions in (27) and (6).
This practically agrees with Thomson and Tait’s Art, 834, equation (1). The surface-tractions
that are contributed by the solutions such as (27) are calculated also in Thomson and
Tait’s article and the typical term contributed to F’.r can be written in the form
m+n(i+1) OV in, (2i+5)m—n
pee) /faiein
x == a) | cwesene 28).
f (m+n) (21+ 3) z de? (m +n) (21+ 3)(21+5) ta ioe (8) ee
6. We have now to find V. This is the potential within a sphere of radius a of
a distribution of density on its surface equal to the product of the volume-density p and
the radial displacement (22+ 8y+-+yz)/r calculated for the surface r=a. The part con-
tributed to the surface-value of ax+Py+yz by the complementary functions (6) contains
ESTIMATE OF THE RIGIDITY OF THE EARTH. 113
a typical term which is seen to be
1 1
Cy ae A = F 2
a E 5 Wis H41 p.| PTS aaa koa od nooeeoere (29),
where we have picked out the terms containing surface harmonics of order (i+ 1).
This is obtained by using (7) and observing that in virtue of an identity similar
to (20)
pi +1
7 Oe Cz ~ = eee ple = S| | SORDOOU OOOO OSD OOO ODOOOr CE 5 .
F(A, r+ By +Cz)= == Shae aH $.| (30)
The part contributed to ax+f8y+yz by the particular integrals (27) has a typical term
whose surface-value is
th (ae aaa
m+n 2(21+5)
1p ERO 0 ok OL oe (31).
Hence the surface-density of which V;,, is the internal potential is
1 1 p Gare =
py a en NI A 32).
Pa ioe 5 Yin al WS m+n 2(2i+ 5) } in) ; 2)
We may easily deduce an equation for V;,, in the form
_ 4arypa? {1 Ty ys p t+2
if ae eer a Ele 33
y in = 243 3 Ee 5 Vian a ae 1 q7ts —i-2 m+n 2 (2% +5 5y Mi i+1 a Wis) (33).
Hence
4arypa*
214+ 3 ie gee i ee
ee — ————— < er ss, —— ee
View 2ryp*a* a4+2 Ee +5 Vin 21 fat eae me +n 2(2i+5) Wins. (34),
m+n (21+3)(27+5)
an equation which may be written
es — a; Wiss + babies =f Cron Reta Teatercicteinia cteYoie vs stersiaicictelaisteiste earete (35),
and then Vern CURE Gy) W ics ir OMe at FG Pit soe cccsoccc canes stwsmodet (36).
Thus the potential of the bodily forces contains terms hoe on the complementary
solutions of the equations (5).
7. The unknown harmonics 4;, Wi, P-i-: are to be determined by adding together
the terms contributed to the surface tractions and expressed in (11), (17) and (28) and
equating the result to zero. Observing that in (11) and (28) all the terms contain
surface harmonics multiplied by small quantities of the order of the amplitude of the
harmonic inequality, we see that (18) holds and (17) may be replaced by (21). Also
by (29) and (31) we have
6Q:4 = (ax + By + 2)/r
= ul pete
ssid lace vem eae > m+n 2 (2145) Yin),
= 1 Peni MRE pe 1 p AES |
or eQin=a Fes Vin (1 m+n 2 bi ' 7 iP plgss az 7a 1) ats a5 mon (ie 5) ¢ if
sa ea a ;
aK +n 2 (20 + 5) qa at a;) Was | sence eeee (37).
Vout. XV. Parr I. -
114 Mr LOVE, ON SIR WILLIAM THOMSON’S
We substitute this in (21), add together the terms of (11), (21) thus modified, and (28)
modified by using (36), and equate the result to zero, and find a surface-condition which
may be written
. C = i+1 oi+5 iH 1 Orbis 2i+5 Vin
s|nG- 47 +P; + ing, 2 (Ben) «py Oe + r2t5Q’ a (ee)
" C) ot gi+5 0pi- = Qe
P; ae Q BTS hee) + Q; ? - nee | al rancinneeionshiae scene (38),
when r=a. The values of the coefficients P;, Q;, Pi’, Qi, P:”, Q:” are given by the equations
i m+n(i+1) , m+ (A+5)n,. 5
Se mite) | 22+ 3 we (25+ 8) (i+ 5) "+ )
“m+n (224+ 3)(Qi+5) +” (+3) +5)
ae Bea) | (21+ 5)m—n Hat? 5m — (21 + 1) Lote. 2)| |
pr. ap 2, eee Ha 2m Cer a)n (642) 4 Has Dt At + 5) m
= : )
‘Swen | ap ae Sie (Qi+ 3) + 5)
—2Ha? sinus ne. | ...(39),
Pra age o[EREED ne EGLO 6 9)
ae = a eo = ~ Qi 1) at
0-59 araaes* 2 rears +? |
ar eam ol |
(2¢+ 1)(27+ 3) a8
where £;,, is given by (12), and H by (14).
The other surface-conditions are to be obtained from (38) by cyclical interchanges
of the letters (A, B, C) and (a, y, 2).
From these equations we are to find Aj,..., Wis--., is. in terms of W,,, and
the other harmonics occurring in the disturbing potential.
8. We may find the solution for each term of the disturbing potential by sup-
posing all the other terms to vanish. We shall therefore suppose that W,,, is the
expression of the disturbing potential and proceed to determine the unknowns so far as
they depend on it.
Now in (38) the function on the left is finite continuous and one-valued within the
region containing the origin, satisfies Laplace’s equation, and vanishes at the surface r=c,
It is therefore identically zero. Take then the equations such as (38) and differentiate
them with respect to #, y, 2 respectively and add, we thus obtain the equation
— (20 +5)(0 + 2) (Q: Wis + Qin + Q2r* p+ 4)] +2 (641) Pin =0......... (40),
where we have picked out the terms containing surface harmonics of order (7+ 1).
ESTIMATE OF THE RIGIDITY OF THE EARTH. 115
Again multiply equation (38) and the like equations by «, y, z add and use (30) and
we get
{P; (+1) —Q; (+ 2) 2} Wirt [Pi (6 +1) — Qi (0 + 2) 27} Wins + {Pi” (6 +1) — Q/" (64-2) 17} 8h _ 5g
n(t+1)r* 5 oe
es ie Wis ie (a = 1) a*4 (Qi +1) b= Olessseeesemee Orestes (41),
where as before we have picked out the terms containing surface harmonics of order (i +1).
Using (40) to simplify (41) we have
prits
a (+1) An LA resect oor
Kquations (40) and (42) determine y;,, and ¢ ;. in terms of W;,, and they shew that
each of these functions is simply proportional to W,,,.
(@ a 1) {P; Wes =e Pitti Fr att peat =n (i = (42).
To find the A, B, C observe that all the terms of (38) except the Se yee
a
contain spherical surface harmonics of order 7 or else of order 1+2 so that the only A,
B, C that can occur are A;, Aj, and the like B and C@. Thus picking out the terms
containing’ surface harmonics of orders 7 and 7+ 2 oy we have the equations
=F) (i a 1) Vie = P; es ne iBs a « pee oe - (773; 9)
; .. (43).
Ae + W; , i PG —i—2
ney ditraree[ ad (Hie Beare |)
And the ena a is given by the equations
a=A;— “+ Age qe oe (a? — 1°) Mix. —
iz : ot (1 + a) Wi tid; Winton b_ eeeshoee (Ee)
m+n 2(2i+5) a
and in like manner the other displacements can be written down.
The amount of the harmonic inequality ¢;Q., is given by the equation (37), in which
as we now see Wi4, and ¢_;. are proportional to W;,, so that to each term in the dis-
turbing potential there corresponds one term in the equation of the surface
r=at+eQi,
and these terms contain the same surface harmonic.
9. We proceed to reduce the question to one of arithmetical calculation in two
special cases. These will agree in that we shall take W to consist of a single term W,
which is a spherical solid harmonic of order 2, ie. we shall take i=1. They will also
agree in that we shall assume p=5'6 or that the density is about the same as the Earth’s
mean density. They will differ in that in the first we shall suppose the solid incompressible,
ie. we shall take m great compared with nm and great compared with myp*a? which will be
taken of the same order as n, while in the second we shall suppose m and n connected by
the relation m=2n which is nearly verified for most solids that have been tested by
experiment.
15—2
116 Mr LOVE, ON SIR WILLIAM THOMSON’S
Let us write @ for the number {7yp*a?/(m+n) and g for the value of gravity at the
surface, Le. for 4arypa. Then
2 = gpal nein) lec $s Ute Foe eee neha eomagame needs sateen ee (45).
In the first case @=0 but @m is finite and = gpa.
In the second case @=1gpa/n so that @ has in this case the meaning given to the symbol
S% in the introduction. The two symbols are distinct in the first case.
We shall have for both cases
2n
(0. — St) vet Q'r-2=- QW,
PY Wr. + Py? h_s =—P,W,
by (40) and (42); and the equation giving the amount of the harmonic mequality is the
surface-value of
70 (Wy _s\_ W, 150 :
soos ams oe 4 =)- TF aiSb at kee eae (47).
Also the values of the P’s and Q’s are
P= ad 70 (" +2n | 36 5m = Tn
—9 70+90\ 5 +10 |
6 TO /im—n fae 5m — -21 |
= ag 70+ 98 ( 35 35 |
pes ad Gag (m+ 2n 8 5m + Lies i Oa? 5m a Tn |
' g 70+ 90 ( 5 35 |
.. (48).
Q, = S Gag (7m—n ie s 5m — = ca Py 5m — 2 _ ndm—5n
a) 2 70+ = 35 LOWS 5 835 72m+ 5n
Ps a|¢ l4ag_ (™ + 2n zs 36 5m + im) _ Oa? 5m+7n _ 7 |
ri g 70+90\ 5 10 35 5 15 3
i [ @ l4ag (3 —n 305m— = 05m — =] |
Q'= oe ~ag70+90\ 35°10 35
Denna
10. Taking up now the first case putting @=0 but m= gpa and substituting in (46)
we have as: is easily verified
D)
ee a 7900) ‘oe 2gpa 7 ,=1 pW: |
175 75 a ee = (49)
Oana ls ai) = 3 eS asta ease at '
ps Ye (34 45) b= BPM
from which by solving and substituting in (47) where @ is put =0 we find
apW, E
6 Q. TEE LL aaa ge Ps Ea (50)
Se tape
5
and this may be written
158 e
A). By +19 See e eee e ewer ec ceesesessecsssrecesescase (51),
where 3 = }9pa/n.
ESTIMATE OF THE RIGIDITY OF THE EARTH. 117
Hence if Q, be taken to be W,/g and if we write ¢ for «, € will give a measure
of the amount of the inequality and we have
5S
== 52
SSE lg OO eo ee secant eghea sea A ec uucace (92).
11. Again taking up the second case, putting everywhere m=2n, @=1gpa/n, we find
20 2s 5
pees 80 +510
15 70+96 ’
9, <P, 130+ 218
515 HOS e
pra? gpa’
3.70490’
ao Bae 770 + 1850
* 21° «189 (70+ 96) 6”
* ~~ 9a? 6(70 + 98)’
Ne 49p 1
* 9 at 70+ 96°
Substituting in (46) we have for the surface-values at r= a,
Hise AYE egy vile
9 ia 5ag (53)
26yr,— (70 + 236) _ = (280 + 516) ay |
fiers F a :
and thence in (47) taking as before Q, = ri we shall find «, or € given by
SS 3356500 + 8631008 4+ 5548538? =
62 SS SS SS mE OD OU ONONBOAGSOOODODNE (54),
70+9S3 53900 + 271609 + 26019?
where S$ is written for 6 the two numbers being in this case identical.
12. Now taking the data furnished in Thomson and Tait, Arts. 837, 838 as to the
rigidity of steel and glass we shall find that 3=% nearly for the rigidity of steel, and
%=5 nearly for that of glass, the density p being taken equal to the Karth’s mean
density 56. To see therefore how the inequality e« depends on % or on the rigidity it
is only necessary to trace the curves (52) and (54) with e for ordinate and 3 for
abscissa. The curve (52) is a rectangular hyperbola passing through the origin and the
part S positive of the branch through the origin is the part to be considered. It can
be easily seen by calculation that the corresponding part of the curve (54) hes always
very near to (52). The tangent lines at the origin to (52) and (54) start out at in-
clinations of tan“'4 and tan,8, nearly so that the points of (54) begin by being slightly
above those of (52) which ia the same abscisse. This state of things goes on until
S>5 but the difference is diminishing all the way from S=3 to S=5. When ‘% is
infinite the hyperbola touches the asymptote e=§ and the curve (54) touches the asymp-
tote © = 55485 + 23409 which is slightly less than 3. It is difficult without taking a
118 Mr LOVE, ON THE EARTH’S RIGIDITY.
very large number of points to draw both curves. I have therefore contented myself
with a drawing of the hyperbola (52). On the scale to which the figure is drawn it
would not be easy to distinguish the two curves.
| B
aa
—_t
A pe
ee
ee
O 1 2 3 1 5
Of the points A and B, A eorresponds to the rigidity of steel and B to that of
glass, ie. A to S=% and B to $=5. The ordinate of A is about ‘803 or nearly 4,
that of B is about 1°53 or slightly greater than $.
To determine the “tidal effective rigidity” we may with sufficient exactness compare
the value of € as given by (52) with that which would obtain in a homogeneous liquid
sphere of the same mass and diameter. The latter will be found from (50) by making
n=0, Le. it gives e=8.
We have seen that for rigidity equal to that of steel e« is nearly 4 it follows that
the ratio of the elastic solid yielding in this case to the fluid yielding is nearly & or
about 4. Consequently the height of the ocean-tide will be reduced to about 3% of the
true equilibrium amount by the elastic yielding of the nucleus when the “tidal effective
rigidity” is that of steel. In lke manner it will be reduced to about 2 of the true
equilibrium amount when the “tidal effective rigidity” is that of glass.
V. On Solution and Crystallization. No. III. By G. D. Liverne, M.A.,
Professor of Chemistry in the University of Cambridge.
[Read May 26, 1890.]
In my last communication on this subject I made the supposition that all the
molecules of the same substance have, on the average, under similar conditions of tem-
perature, pressure, and other external circumstances affecting their mechanical state, similar
motions; and that the excursions of the parts of any molecule from the centre of mass
of the molecule are, under given conditions, comprised with a certain ellipsoid. This
ellipsoid I called for convenience the molecular volume, and assumed it to be of the
same average dimensions for all molecules of the same substance under the same circum-
stances. In passing from the fluid to the crystalline state the molecules will pack
themselves as closely in the solid state as is consistent with their molecular volumes,
and then, as I shewed, each ellipsoid will be touched by twelve others, and the orien-
tation of the axes will be the same for all of them. It is on this arrangement that
I conceive the ordinary properties of crystals to depend.
If the ellipsoids have all their axes equal, that is be spheres, the crystal will
belong to the cubic system with the principal cleavage octahedral: if the ellipsoids be
oblate spheroids with longest and shortest diameters in the ratio /2:1 and the axes
of revolution perpendicular to one of the planes in which the points of contact of each
spheroid with its neighbours are four in number (Part II. fig. 2), the crystal will belong
to the cubic system but the principal cleavage will be dodecahedral: if the ratio of
the greatest and least diameters of the spheroids be 2: 1 and the axes of revolution
perpendicular to the plane of fig. 1, the crystal will still belong to the cubic system,
but the principal cleavage will be cubic. Now if we conceive the spheres and spheroids
to be material, instead of being merely the geometric boundaries of the excursions of
the parts of the molecules, and to be subject to a uniform stress perpendicular to one
plane of the fundamental cube, those originally spheres will be strained to spheroids,
and those originally spheroids with axes of revolution perpendicular to the plane of four
contacts (fig. 2) will have the ratio of their greatest and least diameters altered, and
those with their axes perpendicular to the plane of fig. 1 will become ellipsoids. By
any of such changes the arrangement of molecules will lose symmetry in consequence
of the strain and the crystal will become pyramidal instead of cubic.
If the stress be in the direction of one diagonal of the cube, the effect will be
to convert the crystal from cubic to rhombohedral. In the arrangement indicated in
fig. 1, one diagonal of the cube is perpendicular to the plane of the figure and if the
120 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION.
stress be in that direction the original spheres will be strained into spheroids with axes
of revolution perpendicular to the plane of the figure, and in the case of the spheroids
with greatest and least diameters in the ratio 2:1, this ratio will be altered; and
in both these cases the arrangement of the molecules will be the same as if we sup-
posed space divided into equal and similar rhombohedrons and a molecule placed with
its centre in each angular point of the rhombohedrons. In the unstrained system of
spheres the arrangement is that which would ensue if space were divided into equal
cubes, and spheres were placed so that there should be the centre of one in each corner
of the cubes, and also the centre of one in the centre of each face of each cube. The
strain which converts the cube into a rhombohedron will leave the spheroids similarly
arranged one at each corner of the rhombohedron and one at the centre of each face;
but this arrangement can be represented more simply since the planes which pass through
one extremity of the axis of a rhombohedron and through the centres of two adjacent
faces, will cut up space into rhombohedrons all similar and equal to one another, which
will have one spheroid at each angular point and none in any other position. The
new rhombohedrons will be more acute than the old. In fact if the unbroken lines
in fig. 9 represent the original rhombohedron viewed in the direction of its axis of sym-
metry, the dotted lines will represent the new rhombohedron, which will have the same axis
as the original one and will be placed transversely. There will be four times as many of
the new rhombohedrons in a given space as there were rhombohedrons of the original form.
In the remaining case, in which the original cube had the centre of a spheroid in
each corner and one in its centre, the spheroids will become strained into ellipsoids, the
cube will become a rhombohedron with the centre of an ellipsoid in each angular point
and one in its centre. Figure 10 will represent the ellipsoids of one rhombohedron pro-
jected on the plane of four contacts (fig. 2, Part II.), the ellipses with unbroken outline
representing the ellipsoids with centres a, b, c, d in that plane, the ellipse with dotted
outline representing the ellipsoid, with centre e, lying next above them, and those with
broken outline representing the ellipsoids with centres A, B, C, D lying above that
with dotted outline. Figure 11 represents a section through ab, perpendicular to the
plane of fig. 10. The ellipsoids with centres in the plane of fig. 10, or in planes
parallel to it, will touch each other at the extremities of the equal conjugate diameters,
and the diameters through aA, bB and so on will be conjugate to the plane of fig. 10.
These data will suffice to determine the ratio of the axes of the ellipsoids and their
orientation when the angle between the axis of the rhombohedron and the normal to
one of the faces is given’.
* For if A, B, C be the points where three adjacent angles AC, CB, BA be a, and AD (which is the angle dal
edges of the rhombohedron (100), 4 in fig. 11) be ¢, we have
passing through the axis and in triangle OCD, the angle at O=60°,
through the centre of the sphere
f St t th f f and cos 60°=cot D cot (¢+D-90°),
of projection, mee e suriace 0: ara ovel . 1
that sphere, O be the pole of (111), or tan (¢+D—90°)=2 cot D...........ccee (1),
and the angular element, that is, Ait cos D=cot 60° tan 5
the angle between the normals to
100 and 111, be D, OD will be or, tan 2 =,/3 COB) pacetnsn:ssaestecimertny (2).
90°-D. If further, each of the 2
Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 121
It does not appear that there is any other form and arrangement of the ellipsoids,
when packed as closely as possible, which will give rise to the structure of a crystal of
the rhombohedral system. At first sight it seems as if these would not suffice to explain
the occurrence of what are called hexagonal crystals; but this difficulty vanishes when
the following considerations are taken into account. Let us confine our attention for
the present to crystals built up of spheroids having their axes of revolution perpendicular
to the plane of fig. 1, and let the circles with unbroken outline in that figure, centres
a, b, c, d, e, f, g, represent spheroids with axes of revolution perpendicular to the plane of
the figure and centres in one plane, then the next layer of spheroids may either take the
positions indicated by the circles with dotted outline, centres h, k, l, or those indicated
by the circles with broken outline, centres m, n, 0. Either of these arrangements equally
well fulfils the condition of maximum concentration of the spheroids, and so far either
is equally probable. Now in the first case the three planes of the fundamental rhombo-
hedron (100) will be parallel to hel, Ick and kch, and in the other case they will be
parallel to mco, ocn and nem. The second rhombohedron will be transverse to the first ;
or will be in the position of a twin to the first, the twin axis being the axis of the
rhombohedron. The crystal may therefore, so far as concentration of molecules is con-
cerned, be built up of alternating layers, of indefinite thicknesses, of such twin crystals.
Now what are called hexagonal forms, that is the forms for which the poles lie in great
circles bisecting the angles between the three planes which pass each through the axis
of symmetry and through one of the three poles of the fundamental rhombohedron, are
not in any way affected by this sort of twimning. In fact the forms hkl, when h+k+l=0,
and when A—2k+/=0, are identical with the twin forms when the twin axis is the
axis of symmetry of the crystal. None of these forms therefore will be at all affected
by the alternations of twin layers referred to. It will be otherwise with rhombohedral
forms. Any face of such a form which grows when the deposition consists of alternating
layers of twins, must either be formed of alternating layers of transverse rhombohedrons,
or the face will be ridged and irregular. In the former case the average condensation
Since the plane of fig. 11 is parallel to the stress, it will ft Rie
oF : ae 5 or, aR ndeaeeyvedes
be a plane of principal section of the ellipsoids and contain Paes So t+ ag oe 2)
two of the axes of the ellipsoids, which will be the axes of , aq
the ellipses in that figure and may be called 2x and 2z. In like manner, zr N?
The third axis, 2y, will be perpendicular to that plane and But by fig. 10,
will be the axis parallel to cd of the ellipses in fig. 10. ~,
= A aq =ac coSs-=2r cos we
In fig. 11, ab’ is conjugate to the plane of fig. 10, and 2 2°
ab’, aa’ are conjugate semidiameters of the ellipse with a
centre a in fig. 11. Let ab’ be z', aa’ be x’ and ao whichis “° that, V/2rcos DO cross ect eer eee eee eee eeeeeeeeeees (4).
half of Aa be r. oe
AN\2 /NE\2 Also —=,/2,
Then (>) + (=) =i, joe
and since the inscribed parallelogram is half the cireum- and /2r sino =y fi eae el ine Decrees (5).
scribed parallelogram, 24N . NP=z'x’, and therefore i
AN\2 1/2! \2 Wiesiavelal soni gato 22a lt een (6)
=) + i (air) =1; and AIAN SENCA. ss etc 9 eee (7).
; AN\2 Vrom these equations the ratios x: y : z may be found
aBerce 2( i ) Fens; when D is known, and vice versa.
Vou. XV. Parr I. 16
122 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION.
of the molecules will be the mean of that im the two rhombohedrons transverse to one
another: but this will not be a true measure of the surface tension which, for these
rhombohedral faces, will change with each alternation of growth. If the alternations took
place with perfect regularity, so as to produce alternate layers of each rhombohedron of
uniform very small thickness, the effect might be the same as that of a form having
the mean condensation. But in fact the alternations will not in general be regular, but
determined by causes which depend on the mechanical conditions of the fluid at the
poimts where crystallization occurs; causes which, so far as the forms developed are con-
cerned, may be called accidental. The growth of such faces will therefore be impeded
in comparison with the growth of hexagonal forms.
It is obvious that in those cubic crystals in which the molecular volumes are
spherical, there will also be the same tendency to grow in alternate layers of twin
erystals with the twin axis perpendicular to the octahedral faces. And such alternations
have not infrequently been observed. But in the cubic crystal the twinning may take
place equally well about any one of the four axes perpendicular to the faces of the
octahedron, and in general the only indication of such twinning would be a roughness
of the faces. Neither in the hexagonal nor the cubic crystals would the optical and
other physical characters be affected, unless the crystal were grown under some stress
which gave a peculiar character to those properties.
It is also plain that if the system of spheroids arranged with their axes perpen-
dicular to the plane of fig. 1 be strained in a direction lying in that plane, the spheroids
will become ellipsoids and that plane will be a plane of principal section. In this case
also alternations of twins will be probable as before.
Similar alternations of growth may also occur when the plane of fig. 1 is not a
plane of principal section, because the ellipsoids which represent the molecular volumes
may assume in an irregular manner sometimes the positions indicated by the dotted
lines and sometimes those of the broken outlines in fig. 1. In these cases the crystals
will belong to the less symmetrical systems, and the alternations, though definitely related
to one another, will not have the relation of ordinary twins.
Returning to hexagonal forms, if a face has been developed parallel to the plane of
fig. 1, that is, a face of the form 111, and the other faces developed be also hexagonal,
there will be no cause to interfere with the alternation of twin layers as the crystal
grows. But if besides 111 a rhombohedral form, as for example 100, has been developed
and the crystal grows by an addition to the face 111, the twinning will cause a dis-
continuity of the surfaces of 100 at the edges where the forms 111 and 100 intersect.
If the transverse form 122 be developed as well as 100, there will be no discontinuity
of surface at these edges but some discontinuity of surface tension, which is not the same
in the faces of the two forms. This will be a force tending to prevent the twinning
or else to prevent the growth of the rhombohedral forms. In most cases the rule that
the crystal will grow in such a way that the surface-tension shall, on the whole, be a
minimum will, unless the condensation in the rhombohedral form is much greater than
Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 123
in any hexagonal form, ensure the preponderance of the hexagonal forms. These hexa-
gonal forms likewise lend themselves more readily to the formation of nearly globular
crystals, that is to crystals with a minimum of total surface.
The cleavages of the hexagonal forms will not be at all affected by the alternations
of twins, but cleavages in rhombohedral forms will be rendered difficult and, if they occur
at all, will be interrupted. In general the average condensation in a di-rhombohedral pair
of forms will be the mean of what it would be in those two forms if there were no
twinning. With this consideration we may calculate the relative condensation in the
faces of different forms. For this purpose, if p be the perpendicular distance between
successive sets of molecules parallel to a face of the form Akl, P the point where the
normal to that face meets the sphere of projection, O the corresponding point for the
face of the form 111, and X, Y, Z, the traces on that sphere of the crystallographic axes
we have, as shewn in Part [, poe BP ;
and cos PX = cos PO cos OX + sin PO sin OX cos POX.
Also if D be the angle between the normals to the faces 111, 100
y k—l
tan POX = /3 Fea
AL Fay KO _ hry SAT
tan PO= Ja {E—lP + C= hy + hk} tan D,
h+k+l
tan OX = 2 cot D,
and similar equations with reference to the axes Y and Z.
The hexagonal forms are those for which either POX or PO is 90°, and for these
the condensation in the faces is p.
For the other forms it will be $(p+p’) where p’ is the value of p for a face of
the transverse form.
For shortness we may designate the form 011 as a, the form 100 as r, and so on,
and the corresponding values of p as pq, p,, and so on.
Then taking first hexagonal forms, we have for a or 011, PO=90°, POY =30°,
V/3
J(tan Dy! +4’
which increases as D diminishes, or as the fundamental rhombchedron is flatter, that is
Pa =Ssin OY cos 30° =
more obtuse.
= : 2 ne sO Xa
li) = 2, JOIN SHO. a 5 = Fg Pa:
For «=210, POY =90°, POX =30°, tan PO= <5 tan D,
eS GE J/3 tan D + V3 tan D .
/{3 + (tan D)*} {(tan D)? + 4}
con Px /3 tan D
eu Po 2 > Vien DYES} (Gan Dy + 4
16—2
124 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION,
9
| no
For z= 311, POX =30°, tan PO=—5 tan D,
meat V3 tan D =
si /{4 (tan Dj? + 3} {(tan D)? + 4}
i=231, POX =90°, tan PO = ae
ee V3tanD _
Pi [i(tan Dy + 12} ((tan D+ 4)
- 1
=32 — ? La See
h=321, PO=90°, tan POX 33"
= v3
Pa Ji (Gan DP +4)
ae WI ee po= ens OX =
an D/
Next for rhombohedral forms.
For r= 100, PO=D; POX=0°,;
3 sin D
Pr [ten DY +4"
which increases as D increases up to 45°, and diminishes as D increases from 45° to 90°.
For 7,= 122, the rhombohedron transverse to 7, POX = 180°,
wed We er ‘
ve J(tan DP +47
whence if we put p,/=4(p,+ pr),
7f = —2sin D
Pr ~ Jian DY 34
For e= 011, tan PO =4 tan D=cot OX, POX =180°, POY=60°,
_ 3 tan D -
Pe= ‘(tan Dy? + 4}’
which increases as D diminishes.
For ¢,= 411, tan PO =} tan D=cot OX, POX =0°,
_ __tanD Fete, 2tan D
Pe, = (tan Dp + 4} °™° Pe = (tan Dy +4’
for s=111, tan PO=2 tan D, POX = 60°,
ee 4 ited, oy’
* /{4 (tan DP +1) {(tan Dy + 4}
Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 125
For s,=511, POX =0°, tan PO = 2 tan D,
ae tan) oe
- J [4 (tan D? +1} {(tan D+ 4}
os 2 tan D
J (4 (tan DP + 1} {(tan DY + 4}
whence Ps
tan D
4 >
Hor 7= 211, tan POX =0°, tan PO=
fp ie 3 tan D i
= J {(tan Dy + 4} {(tan D)? + 16} ;
paeaaeres. $n) i
"J i(tan DY + 4} {(tan DY + 16}"
and for n,= 255,
These formulae will help us to compare the relative probability of the occurrence of
the several hexagonal forms. For the reasons given above they are not applicable for
the comparison of rhombohedral forms with hexagonal; for we cannot say that p,’,
which is the average condensation in a plane parallel to a twin face of the form r and
of the transverse form 7,, is a measure of the smallness of the surface-tension on such a
face, though it indicates a minimum below which that tension will not on the average fall.
From these formulae we get
Pa : Po=3 cot D, which is greater than unity if D be less than 60°;
Pa > Pxr= J1 +3 (cot Dy, always greater than unity:
Po : Pr=J/1+3(cot D) : /3, which is greater than unity if cot D be greater than ./2
or D less than 39° 13’;
Px: pz =V4 (tan DP +3: J(tan D+3, which is always greater than unity;
pz : pi =J (tan DP +12 : /4(tan D)?+3, which is greater than unity if D be less
than 60°;
Pa : Pr : Po=V3: 2sin D : tan D, and p,’ is always intermediate between p, and p,.
In crystals having for their molecular volumes spheroids arranged with their axes
perpendicular to the plane of fig. 1, we should therefore expect the faces a and o to
predominate, and faces to occur in the same zones with the faces of those forms, but
the rhombohedral forms to occur rarely. And in fact we find that the distinct cleavages
of hexagonal crystals are parallel to either o or a.
If we examine particular cases we find in Apatite, D=55°40’, and if A be the
radius of the principal section of the molecular volume, B the semi-axis,
126 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION.
And for hexagonal faces the values of p, which are proportional to the condensation, are for
a, O11, ‘69877,
o, 111, °59070,
wv, 210, 45106,
b, 211, *40344,
z, 311, ‘30078,
z, 321, 27206,
and the mean values of p for pairs of transverse rhombohedra are for
rr,, 100, 122, 49974,
ss, 111, 511, ‘47724,
ee,, O11, 411, °35746.
The cleavages are parallel to a and o, the former being the more easily obtained.
In the (nearly) isomorphous crystals of Mimetite and Pyromorphite, the most frequent
forms are a, 0 and x; and they have an imperfect cleavage parallel to 2, In Vanadinite a
and 0 occur, and Des Cloizeaux gives a figure of a crystal which is exactly like a crystal
of Apatite.
In Greenockite, D=58° 47’, the condensations in a and o differ but little, the faces
most frequent are all hexagonal, a, 0, , 2, i, and the cleavages parallel to a and o.
In Molybdenite the faces occurring are a, 0, and there is a very perfect cleavage
parallel to o.
In Polybasite, D=71° 31’, the condensation in o is therefore greater than in a, the
cleavage is parallel to 0, and the forms which occur are 0, @, @.
In Covelline, forms 0, a occur and the cleavage is very perfect parallel to faces of o.
In Pyrrhotine, D=60°7’ so that the condensation in 0 is slightly greater than in a,
and we find that it has a perfect cleavage parallel to 0, a less distinct one parallel to
hb: and the forms which occur most frequently are 0, a, b, #, z and the pair 7, 7.
In Graphite the forms developed are hexagonal, the usual forms 0, a, and the cleavage
parallel to 0, but the striation seems to indicate an unsuccessful struggle for the develop-
ment of rhombohedral forms.
In Ice the usual forms are o and a, and the cleavage parallel to o.
In Brucite forms o and a occur, and the cleavage is very perfect parallel to 0,
traces parallel to a.
In Hydrargillite, 0, a, 6 occur and there is perfect cleavage parallel to o.
In Emerald, Miller gives D= 44°56’. The most common forms are a and o, then
b, « and the pair r, 7,, cleavages o and a, the latter interrupted. With D=44° 56’ we
Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 127
find pq : pp=17360, and we should therefore expect that the cleavage parallel to a
would be more perfect than that parallel to o. If however we take the form which
Miller assumes to be 100, to be O11, as we are perfectly at liberty to do, we shall get
a different value for D, namely 63°15’, and pg : p,= ‘87302, and the facts then cor-
respond closely with theory.
In Nepheline D=59° 10’, the most frequent forms are 0, a, 2, z and the cleavages
o anda. As D is nearly 60° pz and p, are nearly equal.
In Pyrosmalite, D = 46° 42’, the forms 0, a, 2, z occur, and the cleavages are o perfect,
a less perfect.
In Davyne, D=59°15’ according to Miller, who assumes the most common six-sided
pyramid to be the form 231. It seems more reasonable to assume this form to be 120,
the other six-sided pyramid which occurs will then be 311, and D=40°2’. The forms
occurring will then be o, a, b, x, z, and the cleavage is perfect parallel to a.
The varieties of Chlorite known as Pennine and Ripidolite appear to me to be
hexagonal, or rather to have their molecular volumes spheroids with their axes perpen-
dicular to the plane of fig. 1. Des Cloizeaux taking the acute rhombohedron, which is
developed in crystals found on the Rimpfischwange near Zermatt, as the form 100 finds
--/
D=76° 15’. Miller makes the corresponding angle 79°55’. The former angle gives
jo 2 JO 2 fOr) 2 PAE S ARV SME
the latter gives 1°732 : 2954 : 5623. o is the plane of perfect cleavage, a is rarely
developed but there are traces of cleavage parallel to it. The rhombohedral faces are
usually striated and ridged or undulated parallel to their intersection with o. In large
erystals the face o is so dominant that the crystals become six-sided tables. These
characters correspond well with theory. The condensation in planes parallel to o is much
greater than in any other plane, and it is so large in r that there must be a strong
tendency to the development of that form. At the same time the unevenness of the
faces 7 betrays the peculiar growth of hexagonal crystals. Specimens from localities other
than Zermatt are much more hexagonal in their appearance, the form 311 and its trans-
verse form occurring frequently, and striated parallel to their intersections with o. The
molecular volume will be a prolate spheroid with greatest and least semi-diameters in
the ratio 1:444 if we take Des Cloizeaux’s measure, or 1°988 if we take Miller's measure,
of the angular element. As an illustration of the application of the theory to the facts
it does not matter which we take.
Tamarite may very likely have a similar molecular grouping. D=71° 16’, and it
has a very perfect cleavage parallel to the faces of 0, with traces parallel to the faces
of r, and the crystals are very thin in a direction perpendicular to o.
In Coquimbite D=43°50', the forms developed are a, 0, «; and it has imperfect
cleavages parallel to a and z.
In Parisite the forms which occur are 0 and z, D=81° 20’, and it has a very perfect
cleavage parallel to 0, and a very imperfect cleavage parallel to r. With so large a
128 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION.
value for D the concentration in planes parallel to r is much greater than in planes parallel
to a.
Although the twinning which produces hexagonal forms is very likely to occur, yet
its occurrence is mainly determined by the more or less accidental circumstances under
which the growth of the crystal takes place. The chief obstructive cause to such twin-
ning will be, as stated above, the variations of surface-tension which will occur at the
junction of the twin layers where adjacent faces do not belong to faces in the zone oa
or the zone ab. In cases in which the condensation in planes parallel to 7 is much
greater than in planes parallel to a, the obstruction to the twinning may suffice to
prevent its occurrence. This will be the case when the value of D is large, as in the
case of Pennine. And it is probable that those crystals which have a very perfect
cleavage parallel to 0, but are usually classed as rhombohedral, really have their molecular
volumes spheroids and arranged with their axes perpendicular to the plane of fig. 1.
In Bismuth if we take the rhombohedron which in natural crystals is most common,
namely that to which Miller assigns the symbol 111, to be the form 100, we get for D
71°37’, which differs very little from a cubic form. The forms occurring in natural
crystals will then be 111, 100 and 211. There is a very perfect cleavage parallel to
111 or 0, less perfect parallel to the faces of the other two forms. The form developed
in crystallizing bismuth from fusion will be 011, but there is no cleavage parallel to its
faces. The anomalous expansion of bismuth in solidifying indicates a change in the
dimensions of the molecular volumes at that temperature, and this circumstance may
affect the form assumed by the metal in crystallizing at that temperature.
Antimony is very nearly isomorphous with bismuth, and if we take the form to
which Miller assigns the symbol 111 to be 100, D becomes 71°40’, and the forms ob-
served are 111, 382 and O11. The cleavages are o very perfect, n distinct, r less
distinct, @ traces.
Arsenic also is nearly isomorphous with bismuth. Making a similar assumption as to
the symbol of the most common rhombohedron namely that it is 011, we find D=72° 33’,
the cleavages are parallel to the faces of 0, perfect, and parallel to the faces of 211
imperfect; while the faces observed are 111, 011, and 977. The crystals are of course
laboratory preparations.
Spartalite is most probably hexagonal. It has distinct cleavages parallel to o and a,
and if we take the form to which Miller assigns the symbol 513 to be 210 we find
for D 71°57’. If however we take that form to be 311 we get for D 56°56’. The
latter is perhaps more probable, as it makes the condensation in planes parallel to a
and o more nearly equal. We get in that case, pg : po=1:023, which agrees well with
observation. The natural mineral gives only cleavage faces, as far as I am aware.
Of the isomorphous minerals Haematite, Ilmenite, and Corundum, the last shews a
decided tendency to hexagonal forms. The cleavages are parallel to the faces of o and
and r, D=57°34 and we find p, : p, : po>=1 : 1462 : ‘908. There is a great difference
between these values, and they seem inconsistent with the cleavages. But the cleavages
Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION, 129
are very variable in these minerals, in some specimens seemingly perfect, in others in-
distinct ; the apparently perfect cleavages are sometimes only faces of union of aggregated
crystals, so that after all the inconsistency may be more apparent than real.
In specimens of Willemite from Vieille-Montagne near Moresnet there is an easy
cleavage parallel to the faces of 0, a difficult one parallel to the faces of a, while in
specimens from Franklin in New Jersey, the cleavage is easy parallel to the faces of
a, according to Des Cloizeaux; and D=37° 43’. Miller gives a different value for D,
but Dana agrees with Des Cloizeaux. Dana says the rhombohedral faces are seldom
smooth, while the prismatic are smooth. It seems therefore probable that in this case
also the molecular volumes are spheroids with their axes perpendicular to the plane of
fig
Susannite has an easy cleavage parallel to the faces of 0, and D=68° 38’,
In Tellurium if we take the form which Miller puts as b to be a, and those which
he puts as rr, to be z, we find D=53° 46’, and the faces which occur are 0, a, z, with
a very distinct cleavage parallel to the faces of a, and an imperfect one parallel to the
faces of 0.
In Osmiridium, Miller gives the faces which occur as 0, a, z, and D=58°27’. There
is a tolerably perfect cleavage parallel to the faces of o. If we take the form to which
Miller assigns the symbol 311 to be 210 we shall have D=72°56’, the forms occurring
will be 0, a, 2, and the condensation greatest in the planes of cleavage.
Breithauptite exhibits forms 0, a, 7, and 251, and Kupfernickel the forms 0, «.
Amongst laboratory crystals of hexagonal development we find
Lithium sulphate, with forms a, «, 0, with cleavage parallel to 0, and angular element
73° 26’.
Barium perchlorate, with forms a, x if crystallized from alcohol and a, z if crystallized
from water, and angular element 52° 57’.
Ethyl-ammonium chloroplatinate, with forms r, 0, b hemihedral, with perfect cleavage
parallel to o and angular element 54°6’. More probably the forms are x, a, 0 and angular
element 67°19’, x and a being hemihedral.
Todoform, with forms w and o and angular element 53° 32’.
Ceroso-ceric sulphate, with forms rr,, b, z, 0 and angular element 69°45’; or if we
assume the hexagonal prism to be a, and the di-rhombohedron rr, to be a, the forms
will be a, 2, 0, 144, 522, and angular element 77° 58’.
Basic ferric-potassium sulphate, with forms a, o.
All these agree well with theory if we assume (as I have done) that the six-sided
prism is the form 011 and the six-sided pyramid 012.
There are yet two natural crystals which are commonly classed as rhombohedral but
to me appear rather to be hexagonal. These are quartz and cinnabar. Both are remark-
able for exhibiting asymmetric hemihedry (trapezoidal tetartohedry of some crystallographers)
and for their rotation of the plane of polarization of plane polarized light.
Vou. XV. Parr I, 17
130 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION.
To begin with quartz. The most common, I believe the invariable, form is a six-
sided prism terminated by a six-sided pyramid with or without other forms. This gene-
rally hexagonal appearance is modified frequently by unequal development, and unequal
smoothness, of the alternate faces of the terminal pyramids, which is thought to mark
them as di-rhombohedral combinations. The cleavages are so difficult to obtain and so
interrupted that they hardly help us, but as far as they go they confirm the hexagonal
character of the crystal. They are given by Miller, and by Des Cloizeaux, as perpendi-
cular to the axis of the six-sided prism, and parallel to the faces of both rhombohedrons of
the di-rhombohedral combination, and there is no indication that the cleavage parallel to
the faces of one rhombohedron differs in character or facility from that parallel to the
faces of the transverse rhombohedron, I know no other case of equal cleavages parallel
to the faces of a di-rhombohedral combination, and it appears to me essentially an
hexagonal character. Twins are common, almost universal, with the twin axis the axis
of the prism. This is very frequent amongst hexagonal crystals, but is not confined to
them. If we regard the crystal as hexagonal the difference in size and roughness of
the alternate faces of the terminal pyramids will be indications of hemihedral develop-
ment, or growth under stress, as is the case in many hemihedral crystals when the
hemihedry does not extend to the complete suppression of half the faces. The asym-
metric hemihedry of quartz is an indication of the formation of the crystal under stress,
and there is no reason why both kinds of hemihedry should not coexist. If the crystal
be taken as hexagonal the prism will be the form (a) or 011 and the terminal pyramids
the form (#) or 012. We shall then have for the angular element 65°33"2, and if
wv'w be the symbol of a face referred to the new axes and uwvw the symbol of the
same face referred to the axes assumed by Miller,
w=w+t2u v=u+2v, w=v+ 2w.
The abundance of quartz in nature, and the great variety of circumstances in which
it has erystallized, have caused a great many combinations of forms to be recorded.
The symbols of some of the most frequent forms as referred to the old and new axes
are given in the following table:
Miller’s Hexagonal | Miller’s Hexagonal
Symbol Symbol Symbol Symbol
211 101 101 112
100 and 122 210 722 412
142 010 | 221 and 814 324
O11 123 412 625
511 and 111 113 1a, 22.°7 v, 10) ae
13; 5."5 618 452 223
The symbol of the form 111 remains unchanged and though it never occurs except
as a cleavage face it is the regular twin-face. This form and the first three forms in
the left-hand column have the greatest condensation in their faces, and the supposition
that quartz is hexagonal agrees sufficiently well with my molecular theory.
Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 131
Cinnabar has quite a rhombohedral appearance so far as external form goes, but it
has a perfect cleavage parallel to the faces of a hexagonal prism. There is no truly
rhombohedral crystal which has such a cleavage, and I infer that the apparently rhombo-
hedral development is due to hemihedry. This inference is confirmed by the fact that
cinnabar sometimes shews in its external form an asymmetric hemihedry, and shews by
its powerful twisting of the plane of polarization of light that it has this asymmetry in
its internal structure. In this respect it presents a striking analogy to the hyposulphates
of lead, strontium and calcium, described further on. These three substances are iso-
morphous, and the strontium hyposulphate has decided hexagonal symmetry, while the
erystals of lead hyposulphate resemble those of cinnabar. If we take cinnabar to be
hexagonal we must take the cleavage prism to be the form (a) 011. The most common
forms besides the hexagonal prism, are those to which Miller assigns the symbols 111,
100, 522. If we take the last of the three to be the hemihedral development of 012, we
get for the form 100 the new symbol 412, the form 111 retains its symbol, and the
less frequent forms become 125, 741, and 13, 5,1. The angular element becomes 56° 47’.
The asymmetric hemihedral forms observed by Des Cloizeaux seem to be the alternate
faces of 211 and of a scalenohedron. They are however rare.
We might assume the form 100 of Miller to be 012. We should then get for 522
the new symbol 432, and for the less frequent forms the symbols 123, 543, 753. The
numerical values of the indices become a trifle more simple on this assumption, but the
angular element, 70° 43’, would give a smaller value for the condensation in planes parallel
to the faces of the hexagonal prism than in planes at right angles to them, and the
facility of cleavage in the former planes seems to negative this. Again it might be
assumed that the form given as 011 by Miller should be 012. This would give still
more simple indices for the forms observed but would still give a greater condensation
in planes parallel to 111 than in planes parallel to the faces of the hexagonal prism.
On the whole the first supposition corresponds very well with the facts and entirely
with my theory. In twin crystals of cinnabar the twin face is 111, as in most hexagonal
crystals.
In lead hyposulphate, mentioned above, the forms observed, if we take the crystals
as rhombohedral, are r, e, 0, a, b, s, and 155, the first three being most common, and the
angular element 60°. If we change the axes and take the form r to be 012 (a), we
get the hexagonal forms 2, 7, 0, b, a, 2 and 137, and the angular element 71° 34. There
is no cleavage, and the facts agree well with theory.
Calcium hyposulphate and strontium hyposulphate are isomorphous with the lead
salt, but the forms of the strontium hyposulphate are o and 2, o being largely developed,
and « holohedral but with uneven faces. There is also an imperfect cleavage parallel
to 0, as we should expect because the maximum concentration (on the hypothesis that
the angular element is 71° 34’) is greatest in the planes parallel to o.
Crystals of sodium periodate with three molecules of water have a very unusual
appearance from unequal development of the faces. The forms commonly developed,
17—2
132 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION.
considered as rhombohedral, are 7, e, s, b, 0, 0 being hemimorphic and b sometimes hemi-
hedral, and the angular element 51°38’. They rotate the plane of polarization of light,
and besides the hemihedral character of 6, sometimes shew the alternate edges formed by
the intersection of r and e truncated by a hemihedral scalenohedron. If we assume the
crystal to be hexagonal and hemihedral and make the forms 7, b, to be 012, 101, respec-
tively, we get for e, s, the symbols 123, 113, respectively, and for the angular element
65° 26’, which makes the facts and theory agree. The corresponding silver salt appears
to be isomorphous with it, or very nearly so, and it exhibits quite as irregular an
appearance. It is very likely endowed with the power of rotating the plane of polari-
zation of light, but I am not aware that any one has actually observed this fact. In
a few other crystals similar characters have been observed, but they hardly call for a
detailed discussion.
Next referring to fig. 2 of Part IL. let us consider that the circles with dotted out-
line eee represent spheres with their centres in the plane of the paper, while those with
unbroken outline bed, &ec. represent the projections on that plane of the outlines of a
set of spheres which touch the former set and have their centres in a plane below the
plane of the figure. We may suppose that there is another set of spheres also touching
the first set, but lying above them. The projections of their outlines on the plane of
the paper will correspond with the circles of unbroken outline, and to distinguish the
set lying above the first set we may designate their centres as B, C, D &c., b and B,
ce and C, d and D, &e. having the same projections, respectively. Then the points
c, C, c’, C’, d’, D’, d, D, lie in the corners of a rectangular parallelopiped with the centre of
a sphere e in its centre, and the whole space may be cut up into similar and equal
parallelopipeds, each having the centre of a sphere at each corner and one in its centre.
If the spheres become oblate spheroids with axes perpendicular to the plane of the
figure, these parallelopipeds will be cubes if the ratio of the greatest to the least
diameter be 2. If further we suppose the spheroids to be all strained in the direction
of one of the diagonals of the cube the spheroids will become ellipsoids and the cubes
will become rhombohedrons. The axes of these rhombohedrons will not be perpendicular
to the plane of fig. 1. In fact if the circles with unbroken outline are supposed to have
their centres in the plane of the paper, those with dotted outline below, and those with
broken outline above, that plane, and c be the central sphere, the eight centres which
form the corners of the parallelopiped may be abmnlkfe, and two of the diagonals ae,
bf lie in the plane of the paper, the others mk, dn lie in an inclined position. If the
parallelopiped become a cube by changing the spheres into spheroids their axes of revo-
lution will be perpendicular to the plane amnb. If further the system be subject to a
uniform stress in the direction of one of the diagonals of the cube, the spheroids will
become strained into ellipsoids and the cube into a rhombohedron with its axis in the
direction of the strain. The arrangement of the ellipsoids will be the same as if space
were divided into equal rhombohedrons with the centre of an ellipsoid in each angular
point and one in the centre of each rhombohedron, This is the same as if two sets of
rhombohedrons were superposed, all being equal, similar, and similarly situated, and each
Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 133
having the centre of an ellipsoid at each corner but none in its centre, but one set
having its angular points at the centres of the other set. The planes of a set of parallel
planes which pass through the corners of one set of rhombohedrons will not in general
pass through the corners of the other set, so that, if the arrangement represent the
structure of a crystal, the relative condensation of molecules in the direction of the
sets of planes will in general be the same as if there were but one set of rhombohedrons
with molecules at their corners only. But there are certain cases in which the same
plane will pass through the corners of both sets of rhombohedrons, and in such a_ plane
the condensation will be double of what it would otherwise be.
To see what planes have this property, let figure 12 represent the traces on three
planes of reference of the planes forming one set of rhombohedrons. Then a plane which
passes through z, and y, and is parallel to the axis OX, will pass through the centres
of the rhombohedrons as well as through their corners, This will be the face 011. Also
any plane parallel to OX, which passes through z,ym, where m and n are odd, will
also pass through the centres of some of the rhombohedrons. The symbol of the face
in this case will be Ohk where h and k are both odd numbers. Next if the plane pass
through «,, where J is odd, and also through the intersection of the lines in the plane
ZOY drawn parallel to OY and OZ through z,4¥,, where m and n are odd numbers, it
will pass through the centres of some of the rhombohedrons. That is for such a plane
the reciprocals of the indices (reduced to whole numbers) must be one of them an odd
number, and the others equimultiples by a power of 2 of some odd numbers; or the
indices, without regard to sign, must be of the form
2'(Qm+1)(2n+1), (2m+1)(2r+1), (2n4+1)(2r4 1),
where & is an integer, and m, n, r are integers or zero.
Such will be 211, 433, 631, &e.
How to find the relation between the axes of the ellipsoids, and their orientation,
when the angular element of the crystal is known, has been already explained. Taking
the same notation as before we get in the faces of certaim forms double the concentrations
which were obtained when there was no molecule in the centre of the rhombohedron,
= 24/3
For a, O11" Ee Gan D tea:
_ 2
ij, PALL. =
V (tan D) + 4
6 tan D
ot EE tan De
2 tan D
éy ATL, Ps = Gan DY +4’
fi, PD Gita,
Po Jican Dy + 4}{(tan Dy +16)’
134 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION.
way i 2 tan D F
may 29% Bu, ]i(tan Dy + 4} (tan Dy + 16) '
2./3 tan D
G2h Bea Ga DRE
‘ Pi [tan Dy + # (tan Dy + 13}
rel 2/3
h, 321, ee >
eee Ti \(tan D)? + 4}
and so on; while those forms of which the indices do not satisfy one of the conditions
above enunciated, will have the same concentration as if there were no molecule at the
centre of the rhombohedron.
Comparing the concentration in some of the forms we find
Pa _ J (tan DY +4
pe AdtanD ”
which is greater than unity if tan D is less than /2 or D less than 54° 45".
Also 2 = 2. We which gid always greater than unity; and hence, with
Pr cosDJ(tanDP+4°
this arrangement of molecules, the rhombohedron with the easiest cleavage will be O11
and not 100.
Pe _< eee 1
Ds (tan D+ 4 ’
which is always greater than unity.
pats aia = P :
Again po tat DY and p,q is greater than p, if tan D is less than 24/3, or D less than
73° 54’; and i and p, is greater than p, if tan D is less than 4,/2, or D
Po (tan DP +4
less than 79° 59’.
Now if Calcite have the molecular arrangement now under consideration, the cleavage
form must be 011, not 100, and we must change the axes. If we make a change of
axes so that form 100 becomes 011, we shall have the new axes parallel to the inter-
sections of every two of the faces of the form 111, and for a face wow referred to the
original axes we shall have the symbol w’v'w’ referred to the new axes, where w=v+w,
v=ut+w and w’=u+v.
In the case of any face for which u+v+w=0 the symbol will remain unchanged.
Also for any face for which 2u—v—w=0 we shall have 2u’—v’—w’=0.
Form 100 (r) becomes 011 (e’), Form 210 becomes 211,
a EEO. Gio 211 4m, g (BLL We) ohn. « LOK
« Bite = 5 238 » S210) A> eh
> Laas, 00); ai) SIL (a): > Ge,
& 129 (Ry Se ell ey 8 Millie) ee 265 Ca);
5 210ay Pt Pw), 3 old ee TTT:
Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 135
Forms 111, 011, 211, 321, retain their symbols.
Also for the new angular element of the crystal, we have tan D’ = 2 tan D.
Hence for Calcite D’=63°7"3, and the relative condensation in the planes of faces
of the most common forms are given in the following table:
Symbol Symbol
referred to referred to Condensation.
old axes new axes
100 O11 100000
101 101 82211
111 100 "63505
O11 211 62991
211 211 47465
111 111 46823
210 321 “40687
311 111 34508
122 411 33333
511 122 ‘21168
Caleulating the ratios of the axes of the ellipsoids representing the molecular volumes
we find them as 1] : "76159 : 57216.
A similar change of axes will be needed in the case of other crystals which have
a perfect rhombohedral cleavage. Most of these are isomorphous, or nearly so, with
ealeite, and it may be assumed that the anhydrous carbonates of rhombohedral form are
all similarly constituted. Nitratine follows them. Pyrargyrite and Proustite both have
rhombohedral cleavage, and if we assume the symbol of the cleavage face to be 011 we
find the angular element for the former 61°12"6 and for the latter 61° 40°5. The cha-
racters of Chabasie, which has a tolerably perfect rhombohedral cleavage, are satisfied by
a similar supposition.
Phenakite has a not very distinct rhombohedral cleavage, and also a similar cleavage
parallel to the six-sided prism 101. If we assume the symbol of the cleavage rhombo-
hedron to be O11, as before, the angular element will be 56°44’, and the condensations
in planes parallel to the two faces named will have the ratio 9525, or nearly one of
equality, which agrees with the facts of the case.
Dioptase has a perfect rhombohedral cleavage parallel to the face 011, but as the
angular element is 50°39’ the ratio of the condensations in planes parallel to the faces
of 101 and O11 respectively is 1:109, and we should have expected a cleavage parallel
to the faces of 101 as well as of 011. No such cleavage has been observed, though the
form 101 is almost always developed. The faces of that form are however striated in
such a way as to lead to the supposition of some sort of alternations having occurred
in the growth of the crystals, which may possibly interfere with the cleavages parallel to
those faces.
136 Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION.
Millerite has perfect rhombohedral cleavages parallel to the faces 011 and 100. The
angular element is however only 20°51’, which should give the condensation in planes
parallel to the faces of the form 011 much greater than in either of. the cleavage forms.
This form is that which is chiefly developed and the crystals are usually capillary so
that it would be hardly possible to observe whether they had a cleavage parallel to the
faces of O11.
The cleavages in tourmaline are imperfect parallel to faces of the forms 100 and
111. If we change the axes as before the symbols for these faces become 011 and 100,
and the angular element 45°57’, which makes the condensation greatest in planes parallel
to the form 011. If however we take the form to which Miller assigns the symbol 111
to be O11, the form 100 becomes 211 and the angular element 76°24. The concen-
tration in the faces of the most common forms then become
Symbol Symbol
referred to referred to Concentration
old axes new axes
(LL O11 100000
(r) 100 211 ‘79839
(0) 111 111 76553
(a) 101 101 64134
(y) 311 100 53984
(b) 211 oni ‘37028
These figures agree sufficiently with the observed facts. The tendency to the develop-
ment of the form (b) 211, for which the concentration is much less than for some other
forms, seems to be connected with the stress producing hemihedrism (as explained in
Part I.), since the form (b) 211 is almost always hemihedral.
Of laboratory crystals not many of rhombohedral character require special mention.
In magnesium sulphite the forms observed are 7, €, a, 0, and the angular element is
50° 29’.
The double ferro-cyanide of barium and potassium has forms 7, 0 and angular ele-
ment 61°7’, and cleavage parallel to the faces of r, If we take the cleavage form to
be e or O11, the angular element becomes 74° 35’, and theory will agree with the facts.
Aldehyd-ammonia has 7, e, a, 0, with cleavage r, and angular element 58°10. If
we take the cleavage form to be e or 011, 7 becomes n or 211, and the angular element
72°45’, which agrees well with theory, since with that angular element the condensation
is greatest in the faces of e, next in n, a, o, in order.
In crystals of sodium chloride with grape sugar and two molecules of water, the
faces of a, rr,, e and more rarely b, 0 have been observed, and the angular element is
63°15’. This agrees with theory, but the forms rr, might be taken as ee,, when the
other forms observed would be a, n, b, 0 and the angular element 75° 51’,
Pror. LIVEING, ON SOLUTION AND CRYSTALLIZATION. 137
Some may think, in the light of Reusch’s experiment in producing the rotation of
the plane of polarized light by a pile of plates of mica successively twisted through 60’,
to which the twinning of quartz and other hexagonal crystals bears a close resemblance,
that such twinning would account for the effect of quartz on plane polarized light. This
cause is however, as it seems to me, inadequate. The rotation can hardly be accounted
for by any static arrangement of molecules. It is a phenomenon more nearly related to
the rotation of the apsides of a planetary orbit, and seems to imply a stress. This
view is borne out by the fact that it is produced by some liquids, and that these liquids
appear, so far as it is possible to judge of such a fact in a biaxal crystal, to
lose their rotatory power when crystallized in asymmetric hemihedral forms; while the
asymmetric crystals which have the power of rotation lose that power when liquified.
The stress reacts, as it should do, on the external form, because the tendency must
always be for the molecules, so far as they are free, to arrange themselves in such a
way as to counteract the stress.
On the whole the molecular arrangement for which the principles of mechanics give
adequate reason accounts remarkably well for the main features of hexagonal and rhombo-
hedral crystallization. I say the main features, because surface-tension, though the primary
and principal cause of crystalline form, is not the only cause which affects the growth
of crystals. The other causes mentioned in Part I. have a secondary influence, and
produce in some cases disturbing effects, but they are only disturbing not overpowering.
WOin, SONG IAN Ie 18
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3§ 37—41.
wr
On some Compound Vibrating Systems.
By C. Cures, M.A.,
Fellow of King’s College.
INDEX.
Section I, General Principles.
Surface conditions.
Definition of layer, shell, core, altered layer,
change of pitch, ete.
Type of displacement.
Frequency equations in simple shell.
Frequency equations in compound
general formulae.
Result of differentiating frequency functions.
General Remarks.
shells,
Radial Vibrations in Solid Sphere.
Types of displacement and stress,
Determination of ratios between arbitrary con-
stants.
Deduction of expression for change of fre-
quency.
Definition of central
general formula.
Remarks on form of change of frequency.
Change of type of vibration.
Type of vibration in simple sphere, node, loop,
and no-stress surfaces.
General results as to relations between radii of
various surfaces.
On quantities Q and Q’. Table of their values.
Central layer and core. Terms stiffness, elas-
ticity considered.
On layer at surface.
Formulae obtained for special forms of altera-
tion of material.
Consideration of form taken by expressions for
change of pitch,
Explanation of mode of construction and
application of curves, showing graphically
the law of variation of the magnitude of the
change of pitch with the position of the
altered layer.
Table of functions employed in drawing curves
and applications.
Layers of altered density.
Layers whose elastic constant m differs from
that of remainder.
Layers whose elastic constant n differs from
that of remainder.
layer, restriction of
Worn <V. PART IL
§§ 42—7.
§ 48.
Section IIT,
§ 49.
§ 64.
§§ 73—4.
Case when both elastic constants are altered
in same proportion, or m,/m=n,/n=1+p.
On nature of elastie quantity on whose altera-
tion change of pitch depends.
Transverse Vibrations in Solid Sphere.
Types of displacement; explanation of term
rotatory vibration.
Surface conditions.
for change of pitch.
Change of type of vibration.
Type of vibration in simple sphere; node, loop,
no-stress surfaces, ete.
Cases of central layer, core and surface layer.
Expressions for change of pitch, 1° when den-
sity alone altered, 2° when rigidity alone
altered; curves showing law of variation of
magnitude of change of pitch with position
of altered layer.
Case when density alone altered.
p rigidity ,, ee
Type of displacement in rotatory vibrations.
Frequencies of several notes. Table of node,
loop and no-stress surfaces.
Quantities Q and Q’ for rotatory vibrations.
’otatory vibrations, density alone altered.
elasticity ,, i
Deduction of expression
” ”
IV. Radial Vibrations in Solid Cylinder.
Type of displacement, surface conditions,
deduction of change of frequency.
Change of type of vibration.
Type of vibration in simple cylinder.
of node, loop and no-stress surfaces.
Quantities Q and Q’, with table.
Axial layer, and core.
Surface layer.
Formulae for change of pitch in several cases
considered.
Equations to curves representing law of varia-
tion of magnitude of change of pitch with
position of altered layer, and table of func-
tions employed.
Case when density alone altered.
Table
19
140 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
$$ 75—6. Case when elastic constant m alone altered. S$ 98—9. Surface layers.
S$ 77—8. el ee se - Rs ae § 100. Thin compound shell of three or more layers,
$$ 79—81. ey » both elastic constants are altered in
same proportion, or m,/m=n,/n=1+p. Section VII. Transverse Vibrations in Spherical Shell.
§ 82 On nature of elastic quantity on whose altera- § 101. Forms of frequency functions and results of
tion change of pitch depends. differentiating them.
Note to Section IV. On relations between maxima of § 102. Deduction of expression for change of fre-
(Jo(x)}3, {Jy (x)}2 and {J, (x)}?. quency.
§ 103. Case of solid sphere.
Section V. Transverse Vibrations in Solid Cylinder. § 104. Discussion of general formula for change of
§ 83. Type of displacement, surface conditions, | . ess
a §§ 105—6. Surface layers.
general formula for change of pitch. s 107 Thi alahalllarthn
§ 84. hace ativan ce parecer § i in compound shell of three or more layers.
§ 85. Type of vibration in simple cylinder. Table Section VIII. Radial Vibrations in Cylindrical Shell.
: of node, loop and no-stress surfaces. | § 108. Forms of frequency equations; deduction of
§ 86. Axial layer and core. sonitostah
os eke aera expression for change of frequency.
8 TST § 109. Case of solid cylinder.
§$ 88—9. Case when density alone altered. P -
es 2 § 110. Discussion of general formula for change of
§$§ 90—91. a » elasticity ,, 5 fi
requency.
Section VI. Radial Vibrations in Spherical Shell. sane pane layers.
§ 113. Thin compound shell of three or more layers.
on
92. Forms of frequency functions and results of
differentiating them. Section IX. Transverse Vibrations in Cylindrical Shell.
§s 93—4. Deduction of expression for change of fre- | § 114. General expression for change of frequency.
quency. § 115. Case of solid cylinder,
§ 95. Case of solid sphere. § 116. Discussion of general formula for change of
§§ 96—7. Discussion of general formula for change of frequency.
frequency. § 117. Surface layers, and thin compound shell.
SECTION I.
GENERAL PRINCIPLES.
§ 1. In the most general type of vibrations of an isotropic elastic solid there have to
be considered three displacement and three stress components at every point of a surface
along three mutually orthogonal directions. In the general case at a common surface of two
media there are six necessary conditions, arising from the equality of the displacements and
stresses at adjacent points on opposite sides of the surface. In the types of vibration
discussed in the present memoir the surfaces limiting the several media are either
concentric spheres or coaxial right cylinders, and the displacements are either entirely
radial or entirely transversal. In all the cases considered the number of independent
conditions to be satisfied at the common surface of two media reduces to two, one
arising from the equality of the stresses, the other from the equality of the displacements
at adjacent points on opposite sides of the surface. If a surface where no stress exists
be termed free, and one where the displacement is zero be termed fied, then in the
types of vibration treated here, there is at a free surface a single condition expressing
the vanishing of the stress, and at a fixed surface a single condition expressing the
vanishing of the displacement. The centre of a solid sphere and the axis of a solid
cylinder may be regarded as fixed surfaces.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 14]
§ 2. For the sake of brevity we shall frequently have occasion to apply the term
layer to a portion of homogeneous isotropic material limited by two concentric spherical
or two coaxial cylindrical surfaces. When one such layer exists alone it will be termed
a simple shell, while a series of layers one above another will be termed a compound shell,
provided there be no material at the centre of the sphere or at the axis of the cylinder.
When the material extends to the centre of the sphere or the axis of the cylinder, the
system will be termed compound when more than one medium exists. The inmost
material, whose outer surface is of course spherical or cylindrical, will be spoken of as
the core.
The principal object of this memoir is to determine how the pitch of the several
notes of a simple shell or core would be altered by the existence in it of a thin layer
differmg from the rest of the material. Now the elasticity of a layer can doubtless be
altered without altering its volume, but of course the density cannot. For the sake of
brevity, however, the term altered layer will be applied here whatever be the difference
between the structure of the layer and that of the rest of the material. The term merely
indicates the existence of a certain definite want of homogeneity, and does not imply
that the vibrating system ever was homogeneous. By the change of pitch due to an
altered layer is meant the difference between the pitches of corresponding notes in two
vibrating systems, the only difference between which is the existence in one of them of
a layer differing in an assigned way from the rest of the material.
§ 3. A vibrating system is in general capable of producing a large—theoretically
an infinite—number of different notes, answering to each of which there appears a separate
term in the expression for the displacement. The expression for the representative dis-
placement at any point in a layer may be regarded as a product of two factors. One
of these is coskt, where k/27 is the frequency of the representative note and ¢ the
time. This factor is the same for every point in all the media of a compound system.
The other factor is the sum of two functions each multiplied by an arbitrary constant.
These functions have for their variable the radial or axial distance r, and contain, in
addition to k, the density and one or both of the elastic constants of the medium; they
thus vary from layer to layer. In a core one of the above two functions of r must be
omitted, as it would become infinite when r vanishes.
In the case of the transverse vibrations of a sphere there exists in general a third
factor in the representative displacement. It is, however, a function solely of the angular
coordinates. It does not in fact enter into the surface conditions and may for our
present purpose be left out of account.
The following remarks apply equally to the radial and to the purely transversal
vibrations of spherical and cylindrical systems.
If a compound shell consist of n layers the expressions for the representative dis-
placement contain 2n arbitrary constants. At each of the n—1 surfaces separating the
layers there are two surface conditions, and at each of the bounding surfaces of the
shell—whether fixed or free—there is 1 surface condition. There are thus 2n equations,
19—2
142 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
ef which 2n—1 suffice to determine the ratios of the 2n arbitrary constants. Thus we
are left with a single equation from which all the arbitrary constants have been elimi-
nated, and this supplies the frequencies of the vibrations of the given type which can
occur in the compound system.
If there be a core and n—1 layers there are 2n—1 arbitrary constants and 2n—1
equations connecting them, so that the result is exactly the same. In general it will
be unnecessary to consider separately the case when a core exists.
§ 4 At the common surface, r=a,, of two media the two surface conditions may
be put in the form
Aig (gesagt Beg (Gigs span) Ag Ei (Ga vine teal (Cig teisye) neisosmisew ces tee (1),
ASG (Gee Yea) Ben Gude yen) = ALG (ay 98) BG (Ggeye) heer eee (2),
the first representing the equality of stress, the second of displacement on the two sides
of the surface. Here the A’s and B’s are arbitrary constants whose absolute magnitudes
depend on the amplitude of the vibration. The F’s and G’s represent certain functions
of a,, of the density and of the elastic properties of the media. For brevity the letter y
is employed to represent all the material properties of the medium, Le. its density and
elastic constants m and n combined. F'(a,.¥s-) is of course the same function of ps4,
Ms and n,_, that F(a,.ys) is of ps, ms and n,.
The right-hand side of (1) is proportional to the stress and the right-hand side of (2)
to the displacement at the surface »=a, in the medium y;. It must, however, be clearly
understood that the expressions in (1) and (2), multiplied by coskt, need not be the
exact stresses and displacements themselves.
If +=a, were the outer bounding surface of a compound shell then the surface
condition would be got by equating to zero the left-hand side of (1) or the left-hand side
of (2), according as the surface was free or fixed. Similarly, if »=a, were the inner
bounding surface, we should equate to zero the right-hand side of (1) or the right-hand
side of (2) according to circumstances.
In a shell, whether simple or compound, there are four fundamental types of vibration,
the free-free, the fixed-free, the free-fiwed, the fixed-fixed,
where the first term applies to the inner surface.
In what follows it is necessary to adopt some one notation free from ambiguity.
Thus a compound shell of, say, three layers, the inmost of material (p,, m,, n,)—repre-
sented by y,—bounded by the surfaces r=e and r=c, the middle of material (p,, m,, n,),
and the outmost of material (p,, m3, 7;) bounded by r=b and r=a, will be spoken of as
the shell (e.%,.¢.72-b. 4s. a).
The letter f will be invariably employed for the function which equated to zero
gives the frequency equation, and inside the accompanying bracket will be given the
letters necessary to define the system. If a bounding surface be fixed, then the radius
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 143
of that surface will appear in the bracket with a horizontal line over it. Thus, for
instance,
FkEs Wie va Olitys em) 0
represents the frequency equation of the three-layer compound shell specified above, the
imer bounding surface, r=e, being free, the outer, =a, being fixed.
From the remarks made on the forms assumed by (1) and (2) at a bounding surface,
we find at once for the frequency equations of the four fundamental types in the simple
shell (b. y.a) the following—
S(b.y.a)=F(a.y) Fi(b.y)—Fy(a.y) Fb. y)=0 once ecceecec eee eee (3),
f(6.y7- =F (a. 4) Gib.) — F(a. 9) E(B.) HO ceeeeseccencccescceee (4),
f.7-a)=Ga.y7) Fi6.y)—Gi@.y) FO.y)=0 «22.2 ..c0ceceee eee (5),
f(6.y.@ =G4(a.y) Gb.) — Gia. y) @(b. 7) HO weeeeeeeeeeeeeeeeees (6).
The terms in these functions will always be supposed to present themselves in the same
order as above.
§ 5. Suppose now we proceed to find the frequency equations for the two-layer shell
(a,.%-42-%2-4). For the free-free vibrations we have to eliminate the arbitrary constants
from
Zell Od (ase) aa Oh/ J (CARY A eA ois 2s id. seo aes no np BRODER AROSE ree eee (7),
AB (as). yi) + By Py (ds. 91) = Alani” (Ga; Ya)\- Bala Gaietys)) c2scse-00-ceno-eee (8),
A,G (aq. y:) + BG, (de . 71) = AaG (de - Y2) + BiG, (Ge. Yo) -.c2ceecceseeceeee (9),
vile (Ge Gp) $2 JEW (@aeGP)) ccocosseadoonnanee (10).
The result of elimination is easily found to be
T(G, Yi - Ay. Yo. Az) = {F (az : )F, (a. ) —Fy(a,. 1) F(a, -)} {F(a3.72) G (ds.Y2) —F, (as. Y2) G(a..7y2)}
=, iG (Qs. %1) F, (a -%) a G, (ds. 1) F (a,. %)} {F (as. 2) F(a... Yo) — Fi (a3. Yo) F (ay. y2)} =())
Comparing this with equations (3)—(5) we obviously have
FG .%1- e- Yo» Az) =f (Gy . My « As) f (Ge. Yo. Az) — f (Ay. Yr - Go) f (Aa. Ya+ Ag)-e2ee+0e- (11);
similarly we may easily prove
SF (Gi «r= Ge» Yo» As) = f (A, . Yr As) f (Go. Yo - As) — f (Gy - Yr- Ge) f (Ae Y2- Gz)......(12),
F(G «+ As. Yo. Gz) = f (dy. Yr. de) f (Ba. Yo. Us) — f (dy - Yr - Ee) f (Ae. Yo. Hy)a-+0-. (13),
S(G «Ya « A «Yo» As) =f(G “N- ay) f (Ge - Yo - As) —f(% “N- Gy) f (As. Yo - Ag)... (14).
In each of these identities there is a very obvious physical meaning. For instance,
we see from (11) that f(a,.4,.d..9.a;)=0 will be satisfied by any value of k which
satisfies simultaneously either
J(G-91-%)=0, and f(a,.7,.a;)—= 0,
or F(a." -d&)=0, and f(a. y2-a;)=0.
144 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
This merely signifies that if there be a common frequency of vibration for the two
layers existing separately with their common surface either a free or a fixed surface, then
this too is the frequency of a vibration which the compound shell can execute.
At first sight it might appear that in (11) we had also the two alternatives
F(G-m- ds) =0=f(a,.%. de),
FT (de. Yo. As) = 9 =f (Gy. Yo - G3).
Neither of these alternatives is, however, possible in any case, as might easily be foreseen
from the physical meaning of the functions.
§ 6. The relations (11)—(14) are particular cases of a general law which will now
be proved.
It will be sufficient to limit our proof to the cases when both surfaces of the com-
pound shell are free or when the outer only is fixed. The method of proof in any other
case is practically identical.
Let us assume that for a compound shell (a,.%.@)...Gn-Yn-Gnii) of n layers the
frequency equations take the forms
F (Gy. Yr Qe ++ On Yn» Ansa) =f (hy «++ An) f (Gn - Yn» Anti) —f («++ On) f (Gn - Yn» Oni) = 0...(15),
F (Qe N- Ge --- On» Yn» Onaga) = f (Gy «-- On) fF (Gn Yn - Ans) —f (hr --- On) f (Qn- Yn - Gnas) = 0...(16),
where f(a,...@,)=0, and f(a,...¢,)=0 are the frequency equations in the compound
shell (a,...@n) of n—1 layers.
Now the difference between the frequency equations
St (a ws Any) =0, and f (q ... Qn4r- Yn4i + An42) = 9,
is that whereas two arbitrary constants A,, B, have in the first case their ratio deter-
mined by the single equation
AnF (anti - Yn) + BrP; (ants. Yn) = 9,
this ratio is in the second case determined by means of the three equations
AnF (An4i- Yn) + BaF (nti + Yn) = Ant (Ants Ynts) + BnasFi (Gna - Yn+i)s
AnG (Qnsi- in) + BaG (Ans - Yn) = AntG Ans» Yass) + Baa Gh (Ans + Yntr);
0 = Any (Anse - Ynt) + Bn Fi (Gnse- Yn)
Eliminating A,,, and B,., from these three equations we find
An —F (Ans Ya - Mn+) G, (Anis - Yn) +f (Gn his Ynti + An+2) F, (Qn41- Yn)
= Ba —Fk Ona» Yns1. Onis) G (Ona la) +f Gan “asian pa) tn)
Thus we obtain f(4,...dn4.) by replacing in (15) the ratio Fy(dni1-%n): F(dn4i-Yn) by
the ratio given by (17) for A, :—B,.
Mr ©. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 145
The only factors in (15) in which F, (dy4:.yn) and F'(ay4,.%) oceur are
SF (Gn + Yn + Uns1) = F (ns. Yn) Gi (Gn « Yn) — Fi (Ania. Yn) G(Gn- yn),
F (Qn © Yn + Anss) = F (dinsr Ym) Fy Gn» Yn) — Fy (Gntr- Yn) F (dns Yn):
These factors are thus to be replaced, the first by
—Ff (Anta - Ynti + Anta) {@ (Ansa + Yn) Gh (An - Yn) — Gi (Ani Yn) G (Gn. Yn)}
+f (Gna + Ynt1 + ngs) (F (Ania - Yn) Gi (An - Yn) — Fy (Gnsr- Yn) G (dn - Yn}
the second by
—F (Anta » Ynta + Ing2) (G (Ants - Ym) Fi (An « Yn) — Gi (nga - Yn) F (Gn. yn)}
+f (Gn+1 + Inti» Ante) LF (Anas - Yn) Fy (Gn » Yn) — Fy (Gnas - Yn) F (Gn - Yn)}-
In other words, we obtain f(a,...dn4.) from (15) by substituting
—F (Ansa Ynsa Ant) F (Gn Yn» Ens) +f Gn -Yntr » Uns2) f (Gn - Yn» Ons1)
for f(Gn.Yn+Gn4), and
SF (Ants Yaa + Ants) F (An Yn « Ings) + (Fata + Yn « Ings) F (dn Yn + Ins)
ROMA (On Yr» Onqs):
Thus we find
SF (dy---On+2) =
= {Ff (dy. --On) f (Gn Yn - Unsi) —f (Gys--Gin) f (Gn- Yn- Gnta)} F Gnsr » Yn41 > nse)
+ {f(Gr.--On)f (Gn - Yn + Ins) —f (dh---En)f (An Yn = Onss)} fF (Anta Yn Inte) = 0 ...(18).
Hence we find from the assumptions (15) and (16)
SI (Gi: «Onis - Yat - Inga) =f(dh-- Ong) f(Gn41- Yat - Ings) —f(Ar- Onin) fi Opens Maree laces) = Oren 19):
Similarly we may prove that if (15) and (16) be the proper forms for the frequency
equations of an n-layer shell, then
F(a Ansa - Yt» Ing) =f(G-+ Ont) (Anta na» Ents) ~f i: +-Gnga)f (Ansa « Ynta + ngs) = 0...(20),
Thus if (15) and (16) be correct types of the frequency equations for the free-free
and free-fixed vibrations of a compound shell of » layers they are likewise correct types
for a compound shell of n+1 layers. But they agree with the forms (11) and (13)
which we obtained for a shell of two layers, and so their universal truth is established.
We can easily establish in like manner the formulae
F Gr - r+ Caen Yn» Onagr) = f (G+ An) F(Gin - Yn» Ontr) —f (Gh. +n) f (An Yn + Ons) = 0...(21),
FAG: hr - Ga.- On. Yn - Onis) =f (Ay... .n) f (Gn Yn» nga) —f (G---Gn) f (Gn - Yn - Engr) = 0...(22).
§ 7. We can obviously by means of these results obtain very simply the frequency
equations of any compound shell in terms of the functions which when equated to zero
are the frequency equations of the individual layers. Thus in the case of (15) our next
146 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
step would be to express f(q...a@,) and f(a@...dn) in terms of f(@...dn4), f(Gi-+Gr—),
F (Qn—1 + Yn—1+On)s f (Gn-a+ Yn—1- An), (Gnas Yn Gn) and f(Gn1- Yn-1- Gn), and so on,
The final form so obtained for the function which when equated to zero constitutes
the frequency equation of a compound shell of n layers is a series of terms each composed
of n factors. Each of these factors when equated to zero constitutes a frequency equation
of one of the four fundamental types for one of the layers of which the shell is com-
posed, and each layer contributes one factor to each term.
For instance, the frequency equation for the free-free vibrations of the three-layer
shell (a, . 1. @o-Y2+ As» Ys- Ms) 1S
F (Gi. r+ Ga Yo Us- Ys» Us) =f (Ga « Yr - Ae) f (Aa - Yo « Az) f(s « Ys - As)
—f (a). Yr - Gs) f (a+ Y2+ As) f(s» s+ s) +f (Gi - Ce) F (Ge « Yr As) F (Us + Ya» Us)
— F (dans ia) Fieri Yasha) Ji (Gas Og) 10 ws «cee, «ake Oden cate sass opiate Sette re (23).
§ 8. There is a considerable resemblance between the functions we are here dealing
with and the sines and cosines of multiple angles. An illustration of this, which is also
of importance in itself, is the following:
Instead of converting (18) into (19) we can write it as
vA Gy. ints) =f(a,. =-Cn) { f(Gn ~Yn- Onis) f (Onis »Ynu- Gn42) —f(dn -Yn- Ginta)f (Gna *Ynu-: An+2)}
— f(a. «.@n) { flan *Yn- Onis) f(Gn4a Yn: Gn+2) —f(an -Yn- Gnta) f(@n4a *Ynti- Ants)} =0,
or
tT ,..-An+2) = f(a. : hn) (Gn =Yn+ Gn Yn An+2) — f(a. . In) f (An Yn + Ont» Ynti + nts) = 0...(24),
by (11) and (12).
This can easily be extended so as to lead to the result
F Ga--dn) =f (Gh. «.5) f (Gigs. Un) — f (Gy. ig) f (es. Gin) = Oo... cceeeces coven (25),
where a, is the boundary surface separating any two of the m layers.
The corresponding results for the other three types of vibration are
F (Ga-e-Gn) =f (Gy. «g) f (Gg. «<Oin) =f (a-s <Og) f (Mg. -By) =O. 0. c 0. ceeccncenes (26),
Ff Gi.» Gn) =F (Gi --0,) f Ge: On) — fF Gs Gy) F (Gg: -.On) =O. vice ece eens (27),
FG a) =F Gea FG.) = fo (Ga te) =O oe ee (28),
§ 9. As the results we have obtained for the frequency equations arise from the
elimination of arbitrary constants, different methods of elimination may lead to results
which can be reduced to our standard forms only through multiplication by some factor,
which ought of course to be incapable of vanishing. The existence of factors which can
not vanish, and therefore supply no additional roots to the frequency equation, is obviously
of no importance,
As this point is a little obscure without an example, let us consider the following
case. Let us suppose ¢ to be any length intermediate between a and b. We can regard
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS, 147
the shell (b.y.a) as composed of two layers of the same material whose common surface
is of radius c. Thus
F(b.y.c.y.a)s=f(b.y.c)f(e.y.a)—f(b.y.af(c.y.a)=0
ought to supply all the roots of f(b.y.a)=0 and no additional roots, but the two
functions f(b.y.c.y.a) and f(b.y.a) are not identical.
It is in fact easily proved that
fb.y.c.y.a={F(c.y) G(c.y)—Filc.y) Gc. a} f(b. y d)e-vecveereee (29).
Now referring to (4) we see that
F(c.y) G(c.y)-Ki(e.y) G(c.y)=0
would be the frequency equation for the vibrations of an infinitely thin simple shell of
radius c, one of whose surfaces is fixed. But it is subsequently proved in the case of all
the forms of vibration treated here that the free-free is the only possible form of vibration
in a very thin shell. Thus f(b.y.c.y.a) is the product of f(b.y.a) and a factor which
cannot vanish.
The result (29) can easily be extended so as to lead to
FG Ya Yee Use Y - Ug yr--An) = [F (ayy) Gs (Qe y-) — Fi (Qe y-) G (Gey) } X +
x {F (as. y) Gi (ds. ¥) — Fi (as. ¥) G (as. y)} x. Xf (GY An)... (30),
where the number of factors such as F'(a,.y) G(as.y) — F, (as. 7) G (as. y) 1s equal to the
number of intermediate surfaces whose radii are a...ds,.... These same factors will also
present themselves though one or both of the bounding surfaces r=a,, and r=a, be fixed.
§ 10. There is another class of general results which regarded as independent facts
seem very curious. They present themselves repeatedly, so their explanation at an early
stage is advisable.
Suppose we have a simple shell (b.y.a+0a), where da is so small that (@a/a)* is
negligible. We may write the frequency equation for the free-free vibrations of this
shell in the form
fb.y.a.y.a+0a)=f(b.y.a)f(@.y.at+0a)—f(b.y.a)f(a.y.a+ 0a) =0,
or, since f (@.y.a+ 0a) cannot vanish,
‘(b.y.a
f(b.y.4a) = fae nO Foi a: Gli) <0) sogocenarscessncsenee (31).
This must be equivalent to f(b. y.a+ 0a) =0 and so, as (da/a)? is negligible, to
fb.y.a)+ea © f(b.y.4)=0 Ce ae Pat ee ene ae (32).
Since the equations (31) and (32) are equivalent we must have
d S(b.y.a)
$f 6-9-0) == Naa Ot FO 4 00) onssnene(8B)
Worn XV. Parr II. 20
148 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
But f(b.y.@)=0 is the frequency equation for the free-fixed vibrations of the simple
shell (b. y.a), and f(a.y.a+ 0a)=0 is the frequency equation for the free-free vibrations
of a very thin shell of radius a. Thus if we take the function f(b.y.a) which when
equated to zero gives the frequency of free-free vibrations in a simple shell (b.y.a), and
differentiate it with respect to the radius a of the outer surface, this differential co-
eticient equated to zero must supply us with the frequency of the free-fixed vibrations
of the shell (b.y.a) and with the frequency of the free-free vibrations of an infinitely
thin shell of radius a, when we modify it in a suitable way by introducing the facts
that
and that (da/a)? is negligible.
as follows from (32)—f(b.y.a) differs from zero only by a term of the order da/a
Examples of this result will be found in § 14; Sect. IL, § 50, Sect. IIL, § 64, Sect. IV.,
§ 92, Sect. VI. ete.
pe eae, ,
A similar treatment of ap! (o-7-% when the result is equated to zero, leads to
the equation
f(b.y.a) f(b —0b.y.b) =0.
Such a result as this last, in which it is tacitly assumed that 6 does not vanish,
cannot of course be applied to any case in which a core exists, but all the results such
as (21) or (22) where no such assumption is latent apply immediately in the case of
a core. The result (33) also applies to a core when 0 is replaced by 0.
§ 11. In so far as the results of the present section are mathematical they may
doubtless be deduced from the properties of the determinant which would result from the
elimination of the arbitrary constants in the surface conditions treated as simultaneous
equations.
The methods of this section are probably the simplest for obtaining the change of
pitch due to a thin altered layer in an otherwise homogeneous system, and their application
to this object will be found in Sections VI. to IX. which deal with spherical and cylindrical
shells. In Sections II. to V., however, a different procedure is adopted in dealing with
solid spheres and cylinders in order to determine how the type of vibration changes.
SECTION II.
RADIAL VIBRATIONS IN SOLID SPHERE.
§ 12. In a simple spherical shell of material (p, m, n) vibrating radially the repre-
sentative displacement may be taken as
= cos k f (= pea * (see ra t .
u = cos kt ag: cos kar ) gash (Gay ea kao Pelle selene (ae
where CET Gi Eh) ado non anesadgo dos tonn cen satodsnsensanrcsed: (2).
* Cambridge Philosophical Transactions, Vol. x1v., equation (60), p 320.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 149
The corresponding radial stress is
U= - cos kt [4 fom +n) kar sin kar — 4n ss — cos har)!
)
+B fom +n) kar cos kar — 4n ae + sin kar) eee (a
Suppose now we have the compound solid sphere (0.a.c.a,.b.%.a), where b—c is
so small its square is negligible. Here we denote V p,/(m, + m) by a, supposing p,, m, 7,
to be respectively the density and the elastic constants of the thin layer.
The presence of the thin layer will produce only a corresponding small change in
the type of vibration throughout the rest of the sphere. We may thus assume for the
type of vibration answering to a note of frequency k/2r,
in the core u/cos kt = 4 (= bee cos kar) SOS EBREEHE CEORCE eer nn AS oROecaee (4):
r \ kar
in the layer u/cos kt = ay (andar — cos Koy ) te 2 conkm + sin kar) esesosctoese (5);
’ r \ kar kar
in the material outside the layer
u/cos kt =
A+0A /sinkar ) 0B cos kar
- — cos kar} + — (
( ae re +sin kar mieidaisrecacriectas (6).
The several quantities A, A,, etc. are constants to be connected presently through
the surface conditions.
If the layer did not exist the expression (4) would apply to the whole sphere. Thus
oA/A and 0B/A must be of the order b—c of small quantities at least.
§ 13. We shall confine our attention entirely to the case when the surface of the
sphere is free. The relations connecting the constants of the solution may then be
written in the form
A Gs —cos hae | =A, Gen — cos ka.) + B, (eee + sin kaa ae (7),
A jm +n) kac sin kac — 4n (awe — cos kae |
7 k ;
=A, jem +m) ka,e sin ka,e — 4n, | (Tacs — cos hae)!
Ble ta are ae Le, Bae | s
+ B, ym + n,) ka,e cos kaye — +n, ( Tie + sin me) SP eae =e tod HARE ee (8)
sin kab coskab \
+0A) Cir — cos kab) +0B ( kab + SD kab )
m sin ka,b : \ coskab
=A; an — cos kab) +B, as + sin ka) benacee sctace (9),
20—2
150 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
4 +04) {(m +n) hab sin kab — 4n ("4% _ cos kad)
(A +04 Miia ab sin kab — nas os ka
coskab .
+0B 4(m+n) kab cos kab — 4n ( kab + SD kab)
=A, {om +7) kab sin kab — 4n, Cae kad — cos ia)
+ B, jm +) kab cos kab — 4n, (a4 ua sin ka »| Bais dieu eee ae (10),
/ 0A = sin kaa )
\J = A {om +n) kaa sin kaa — 4n ( tie wan kaa |
+ “= \(m +n) kaa cos kaa — 4n (== © + sin kaa) = Drasepou (lh)):
In equation (7) put c=b—(b—c) and neglect terms in (6—c)?; then subtract the
equation from (9) and we find
0A (aa — cos kab) +0B (a + sin kab)
=—A a {hap sin kab — ae — cos kab)}
veil = {bab fae a 555 kab)!
1
+B, = {ib cos ka,b — (e La +sinka »)| Jase RS ee tee (12).
Treating (8) and (10) similarly, we deduce
A {im +n) kab sin kab — 4n a — cos kab)| +0B {(m +n) kab cos kab — 4n (me = + sin kab)
=-—A = = om +n) kab (sin kab + kab cos kab) — 4n (ab sin kab — ae _ + cos kab)
+ A, b 7 2 \ m, + n,) kab (sin ka,b + kab cos kab) — 4n, (iad sin ka,b — so + cos ka)
1
+B, b } S fom +7) ka,b (cos ka,b — ka,b sin kab) — 4n, (ib cos ka,b — oC — sin kab)
Now as terms in (b—c) are negligible we are to determine 0A and 0B from (12)
and (13) by substituting in these equations the approximate values for A,/A and B,/A
deduced from (7) and (8) by putting c=b, or from (9) and (10) by neglecting 0A and
0B. These approximate values are
= (m, + 7,) kab = (‘o i + sin ka b) jim +n) kab sin kab — 4n (= pil cos kab)
TN tiga ag i kab ; kab
cos ka,b
-(32 — cos kab) \ m, +m) ka,b cos kab — 4n, Gas —-+ sin in)! tiene (14),
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 151
B, in kab . i
oT (m, +m) kab = — a -— cos kasd) \(m +n) kab sin kab — 4n = — cos kab)
sin kab ; sin kab
at a — cos kab) {im +m) ka,b sin kab — 4n, ( So cos kab) Seren (15).
Substituting these values of 4A,/A and B,/A in (12), reducing and arranging the terms
we get
2
0A /sin kab OB/coskab
=| ab °° kab sag ( kab t S12 kab
_ b—c (4(m—7n) /sin kab m+n ;
ay { ey ( Fab 298 kab - (1 - ate) kab sin ka} Spade (16).
The same substitutions enable us in like manner to reduce (13) to
0A : sin kab 0B . coskab . |
a {im +n) kab sin kab — 4n ( ab 7°08 lab} cee \(m +n) kab cos kab — 4n ( ab SB kab)
_b—-c aa 2 ajo, £(ta — n)(3m, — n,)) /sin kab
a [ {om + n) kea2b? — (m, +m) kea2b? + Pe ( kab 7 ©°8 kab)
n ny ea cs
+4(m +n) (. ae =) kab sin kab] Rasta (17).
Solving (16) and (17) we obtain
0A _b-—c
- {im +n) kath? — (m, +m) kayrb? — 4n (38m — n) & 4n, (8m, — “al
m+n m +N,
x es — cos kab) (= cee + sin kal
kab kab
Bud. aa Green ieab ein cab = (4g (ee kab)
(— aah m+n in n( ah — cosh
x jm +n) kab cos kab — 4n (a + sin kab)
+ 4 (- = =e = ) (a — cos kab) {im +n) kab cos kab — 4n (a + sin kab)!
1 1
coskab : sin kab ;
( Tai + sin kab) {(m +n) kab sin kab — 4n ae — cos kab | \eaicaupeemeceetsbaie/i (18),
oB _b-c
a oe, yo 42 (38m—n) , 4m, (38m,—7)) (sin kab =
=— {im +n) k?a?b? — (m, + n,) k*a,2b? — =e, eee —— ae are — cos kab)
1 1 heen aes sin kab PNG
- ( a ae {im +n) kab sin kab — 4n ae — cos kab)!
a nL a (sin kab _ sab) ne sinkab _ ap
8 (— ata) hah cos kab) +(m-+n) kab sin kab An ( Tab cos kab) sort (II).
152 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
§ 14 If the thin layer did not exist the frequency equation would be got by
putting 0A =0=0B in (11), which would give
F(0.a.a)=(m-+n) kaa sin kaa — 4n ee
— cos kaa) =i! Bapanoneceennce (20)*.
In consequence of the existence of the thin layer, f(0.a.a) is no longer zero but is
of the order b—c. We may thus neglect 0A/A in (11). Further as 0B/A is by (19)
of order b—c, we may introduce into its coefficient in (11) any modification consistent
with the supposition that (20) is exactly true. We thus reduce (11) to
(m+n) kaa sin kaa — 4n oie! cos kaa) ts (rant ay eae Oven ecsee (2a):
kaa }) sin kaw
fae oO haa
Now in this equation 4/27 is the frequency of the vibration of the compound system.
Thus if the presence of the layer has raised the frequency by 0k/27, then (k—0k)/2a
was the frequency of the corresponding note of the simple sphere, and so k—0k must be
a root of (20).
As ok is of order b—c we are thus to substitute /—0k in (20) and neglect terms
in (ok). We thus find
J (0.a.a) ke Le qf 9-4) Wpepaorcodon sno swosacosce sanee.101( (2%)
iNOW ip O.a.a =haa ax SES (0 4.0
=hkaa (( m+n) (sin kaa + kaa cos kaw) — 4n ee + sin haa So :
{ kaa (kaa)?
As this occurs in (22) in the coefficient of 0k we may modify it by any transformation
consistent with the hypothesis that (20) is exactly true. We thus easily transform it into
ds sin kaa 4n (3m —n))
a yy Ss & (Shaan Ae alc dig = a ee 23).
k apd (0.4.4) ( Tan cos kaa) (m +n) kata eae (23)
We may thus replace (22) by
(m+n) kaa sin kaa — 4n (= ae cos kaa)
kaa
ok /sin kaa ‘ pe ea Ae om= 2) >
al rant — cos kaa) |(m-+ n) Kata hie =o.
This equation being necessarily identical with (21), we obtain
Ok _ —(m+n) kaaoB/A :
5 7 ish ee ge a (24).
( kaa er ae) 1 ) m+n |
As 0B/A and so odk/k is of order b—c, we may in this equation regard k/27 as the
frequency in the simple sphere (0.a.a). Thus the ratio of the small change in the
frequency of a typical note to the value it possesses in the simple sphere is found by
substituting in (24) the value obtained for 6B/A in (19).
* Cf. Transactions, Vol. x1v., equation (55), p, 318.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 153
§ 15. Some preliminary considerations will enable us to give for dk/k a compara-
tively short symbolical expression.
I os 2 ‘5 2 . E 4
Let 55 Kaa) denote the frequency of the free-free radial vibrations in an infinitely
ane
“
thin spherical shell of material a and radius a. Then it is known that
CO ee (25)
a? (m+n)? p(m+n)
This result may also be obtained by equating our expression (23) to zero in accordance
with the general result established in Sect. I.+
Also let U, = 1 & kaw — cos kar’) “SHEER ee ORCL ware eT ERS aT aCe eee (26),
r\ kar
ent (9 yaa sin kar als) a
Disgaea SEM S27) ae — 808 kar); SR eee emnae (27).
These represent respectively the amplitude of a displacement and the corresponding
greatest radial stress at a distance r from the centre of a simple sphere of material
(p, m, n). Whatever be the magnitude of the displacement or the instant considered,
the simultaneous displacements at radial distances r and r’ are in the ratio u, : u,, and
the ratio of the radial stress at 7 to the simultaneous displacement at ris always
ea us.
Employing these several abbreviations in (19), and then substituting for 0B/A in
(24), we finally obtain
ok _b—c (p(*— K*a.») — p(k — Ka.) G .
k ~~ a p(k? — K*,.4)) )
1 1 i U;\? fy in my Is bd =) 5
rf las mn m,+ a) ap (k? — K* 4.4) a Ko ies +n m+n,/ ap (2 — K*%q.a) (eos if eed (ie)
§ 16. In establishing (28) certain assumptions have been made which limit its
applicability.
The primary assumption is made in § 12 where 04/A and 0B/A are supposed to
be small quantities of the order b—c. In the proof this is interpreted as meaning that
(b—c)/b is small. The form of the expressions (18) and (19) constitute a complete
justification of the primary assumption and of the mathematical treatment provided kab
be not very small.
If however we were in (18) and (19) to suppose kab very small, we should find
that while 6b/A varies as (b—c)b*, 0A/A varies as (b—c)/b. Now in (11) we are
* Transactions, Vol. xtv., equation (67), p. 321.
F ut
f(0.a -ay=a J(0.a.a), and that SEA coskaa=0 is by (4) identical with
d
S 10, ici hat k —
+ See § noticing that ae
dk
f(0.a.a)=0.
154 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
justified in neglecting @4/A only if it be of the same order of small quantities as 0B/A.
Thus our method and assumptions are legitimate only when (b—c) b?/b® as well as
(6—c)b/a® is a quantity whose square is negligible. In other words the volume of the
layer must be small compared to the volume of the mass inside it.
It would thus be unjustifiable to apply (28) to the case when the material (p,, mm, ™)
forms a core, but by supposing (b—c) sufficiently small it may be applied to any true
layer however small its radius may be. When the layer is of infinitely small radius, its
thickness being supposed of course of a still higher order of small quantities, it will be
designated the central layer.
The results obtained for the central layer are practically useful, because as will
presently appear, the effect of a given alteration of material is for the central layer
either zero or else a numerical maximum. Thus the values obtained for the central layer
are asymptotic limits, and they supply very close approximations for practical cases in
which the layer has a finite though small radius.
Further discussion of the central layer and core is reserved for § 22.
§ 17. We notice in (28) the separation of the expression for the change of pitch
into three distinct terms, the first depending on the square of the displacement at the
altered layer, the second on the square of the radial stress, and the third on the product
of the displacement and radial stress.
If the layer differ from the remainder of the sphere only in density then the first
term alone exists. This is also the case when the position of the layer coincides with
the surface of the sphere, or more generally with any no-stress surface—i.e. a surface
over which the radial stress U vanishes.
If on the other hand the layer occur at a node surface—or surface where the dis-
placement u is always zero—then the second term alone exists.
If the material of the layer remain the same, then however its distance from the
centre may vary the signs of these two terms remain unchanged.
The third term vanishes when the layer coincides either with a node or with a no-
stress surface. It differs from the other terms in the important respect that its sign
varies with the position of the layer. Another important feature of this term is that
it vanishes if m,/n,=m/n, a relation which on the uniconstant theory of isotropy is
necessarily true.
§ 18. Before entering on a discussion of (28) it will be convenient to consider shortly
the type of vibration throughout the sphere. In the core there is no pronounced change
of type because (4), with of course a different value for k, would apply equally to a simple
sphere. The only consequence of the existence of the layer is that every node, no-stress
and loop surface—or surface where the displacement is a maximum—alters its radius r
according to the law
= Orr = Ole Mein sc smscisunonucsaragertrssass. stese sete (29).
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 155
Substituting in (6) the values of dA/A and 0B/A from (18) and (19) and reducing,
we find outside the layer
u/A coskt = eed cos kar’)
r\ kar
aK.) Ce On Oe r)+( tie se ) UFO a.7)
(m+n) kar |‘? nd : eae oe : ee eer) ee es
au - = ia {R= ma i ye "
4 (— ae a) (b-u f (b.a.7) + Up f (0.0. r| POR a Rey WHA. & (30);
where i Gaaen) = @ + aa) sin ka (r — b) — = ( -- *) cos ka (7 — b),
M aiaere (31)
S(b.a.r) =(m+n) ( sin ka (r — b) — kab cos ka (7 — »)| — 4nf (b.a. |
The functions f have the same significations in reality as in Sect. I.
This is easily proved if we notice that
F (b.a)=(m+n) kab sin kab — 4n (ean — cos kab 5
F, (b.a) =(m+n) kab cos kab — 4n — + sin hab) ;
; 32).
ji pee we lf Peo R REESE (52)
G (b.a)= ep kab,
Cu(Ou.G= ee + sin kab
It will be noticed that f(b.a.7) vanishes and changes sign as r passes through any
value answering to a node surface of a simple shell of material (p, m, n) performing
radial vibrations of frequency k/2a, whose inner surface is of radius b and is fixed.
Similarly /(b.a.7) vanishes and changes sign as 7 passes through any value answering
to a node surface of a simple shell of material (p, m, n) whose inner surface is of radius
b and is free, the frequency of vibration being also k/27.
The formula (30) differs from that for the displacement in the core by the addition
of the long expression which has b—e for its factor. This expression we shall here call
the change of type. It consists of three terms corresponding to the three terms in (28).
If the difference between the material of the layer and that of the remainder be
such that one or more terms in the expression for the change of frequency vanish, then
the corresponding term or terms in the expression for the change of type also vanish.
Again if the position of the layer is such that either of the first two terms in the
expression for the change of frequency vanishes, then too the corresponding term in the
change of type vanishes.
While, however, the third term in the expression for the change of frequency vanishes
when the layer occurs either at a node or at a no-stress surface, the third term in the
change of type cannot vanish except for a chance value of r, for w and U, cannot be
simultaneously zero. It thus appears that, except on the uniconstant theory of isotropy, an
alteration of elasticity occurring throughout a thin layer coincident either with a node or
Wot, OY, IPrate IDE oil
156 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
with a no-stress surface may produce a change of type to which there is no corresponding
change of frequency.
It is also worth noticing that while the first two terms in the expression for dk/k
depend respectively on the squares of wm and U,, the first two terms in the change of
type depend for their sign on the position of the layer.
A special interest attaches to the displacement just outside the layer. As f(b.a.b)
vanishes and f(b.a.b)=—(m-+n)kab by (31), the displacement in question is
1 n ny x =
u=A coskt E —(b _ c) (ee rT = -——) U, + 4 as. _ os ) b vf. soee(O3).
Now if in crossing the layer the type of vibration existing in the core were maintained,
the displacement just outside would be simply
u=A coskt.up.
Thus the coefficient of b—c in (33) is the measure of the change of type met with in
crossing the layer.
The displacement in the layer itself may be got very simply from the consideration
that it must have the value (33) when 7=b, and the value
A cos kt — (ae — cos kac)
kac
when r=c, terms in (b—c)* being ee It is thus given by
1 /sin kab b— sin kab
u/A cos kt = b ( 7 kab) — = 5 {ka sin kab — 2 ( Tob °° kab)!
it n Ny Le :
rail ¢) ee an Mm, + ) Uy as (= + na m+n ) b rush. (34).
The term in r—c in (34) represents the progressive change of type, due to alteration
of material alone, met with as we cross the layer from within outwards, and it reaches
the value represented by the term in b—c im (33) when the layer is completely crossed.
If the layer differ from the remainder only in density no change of type is met
with in crossing it. In other words the layer vibrates as if it formed a portion of the
included core.
Any alteration of elasticity will in general produce a progressive change of type in
the layer, but this will not be the case when the layer coincides with a no-stress surface
if the uniconstant theory be true, or if both constants in the biconstant theory be altered
in the same proportion.
§ 19. As continual references to the properties of a simple vibrating sphere are
essential for a discussion of (28), and as a good many of these properties have not, so
far as I know, been fully discussed elsewhere I shall briefly notice them.
The frequency equation for the simple sphere (0.a.a) is (20).
The roots of this equation answering to the six notes of lowest pitch have been
calculated by Professor Lamb* for the values 0, ‘25, ‘3 and 3 of Poisson’s ratio
a =(m—n)/2m.
* Proceedings of the London Mathematical Society, Vol. x11. p. 202.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 157
Answering to o=1/2 the frequency equation is
itn Ac HE (Dssasuabecaetose cocosaS couse RAC AO ACD eCOnae (35) ;
whence kaa =i, where 7 is any positive integer.
The following table incorporates some of Professor Lamb’s results.
TABLE I,
Values of kaa/r.
ee lee 2 See
(1) 6626) /-8i60)) Giberas Wh
(2) 18909 19285 19470 2
(3) 29303 29539 29656 3
(4) 39485 39658 39744 4
(5) 49590 49728 49796 5
(6) 59660 59774 59830 6
It will be noticed that except in the lowest note or two the frequencies are nearly
independent of the value of o, and that the case o="5 supplies asymptotic values to
which the results in the other cases tend.
As (4) is the type of vibration in the simple sphere the node surfaces are the con-
centric spheres whose radii are given by
PEAR VEY TN PSE, coconooondoacnouseedoboonocenboasas (36).
The following are the first six roots, taken from p. 266 of Verdet’s Lecons d’Optique
Physique, Tome L.,
= =0, 1:4303, 24590, 34709, 44774, 5:4818.
The higher roots are approximately odd multiples of 7/2.
The no-stress surfaces are likewise concentric spheres, and their radii are supplied
by (20) for the note of frequency k/27 when the @ in that equation is replaced by +.
Thus for a given note and a given value of a, the ratios of the radii of the no-stress surfaces
to the radius of the sphere are obtained by dividing the values of kaa/a in Table I.
for all the notes of less frequency, and for the note itself by the value of kaa/m for the
note in question, all being taken for the assigned value of o.
This method of determining the positions of these surfaces is given by Professor
Lamb in his p. 197. The surfaces so determimed he, however, speaks of as loop surfaces.
I have here ventured to employ the term in a different sense, defining a loop surface
as one over which the displacement is a maximum.
I employ the term no-stress surface only in default of a better. It must be borne
in mind that over a surface so named it is only the radial stress that vanishes.
21—2
158 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
1 (= kar
As defined above Joop surfaces are the loci where — P — cos kar’) numerically
ih rar 7
considered is a maximum. They are thus concentric spheres whose radii are given by
: 2 /sin x
r=ax/ka, where sina —= (age — cos 2) = Vi iities. danvicnenens sitet (37).
Now if we write z for kaa, and 1 for m/n in (20) we transform it into (37). Thus
the radii of the loop surfaces are found by equating kar to the values ascribed to kaa
in Table I. for the value 0 of o The loop surfaces accordingly coincide with the no-
stress surfaces only when Poisson’s ratio is zero. For all other values of Poisson’s ratio
each loop surface les inside the corresponding no-stress surface.
The following table gives the positions of the node, loop and no-stress surfaces for
the first six notes for the limiting values 0 and ‘5 of o, and the value ‘25 of the uni-
constant theory.
Taste II.
Values of v/a over node, loop, and no-stress surfaces.
a=0 can) =
Number Node Tenet Node Loop No-stress Node Loop No-stress
of note surfaces ireae surfaces surfaces surfaces surfaces surfaces surfaces
(1) 0 10 0 8120 10 0 ‘6626 10
(2) (0 3504 i) 3436 ‘4231 0 3313 35)
| “7564 1-0 ‘T7417 "9805 10 Lot “‘O454 10
0 ‘2261 0 2243 2762 0 ‘2209 3
(3) 4581 6453 "4842 6401 “6529 4768 6303 6
8392 10 8325 ‘9920 10 8197 ‘9768 10
| 0 1678 +380 1671 2058 0 1656 ‘25
(4 | 3622 ‘4789 3607 “4768 ‘4863 3576 ‘A727 5s
) 6228 ‘7421 6201 ‘7389 7448 6147 7326 diss
| 8790 10 8752 ‘9956 10 ‘8677 ‘9871 10
{? ‘1336 0 1332 1641 0 13825 ‘2
"2884 3813 ‘2876 3802 ‘3878 2861 3782 “4
(5) = °4959 5909 “4945 5893 5940 “4918 5861 6
: | ‘6999 ‘7962 ‘6980 ‘7940 ‘7975 6942 ‘7897 $8
9029 1-0 ‘9004 ‘9972 10 8955 9918 10
0 ‘1111 i) 1109 1365 0 1104 ‘16
| 2397 3169 2393 3163 3226 2384 “olbl ‘3
(6) ] “4122 4912 4114 “4902 4942 4098 “4884 35)
\ 5818 ‘6618 ‘5807 ‘6606 6635 ‘5785 6581 6
| "7505 8312 ‘7491 8296 8319 ‘7462 8265 ‘83
‘9155 10 ‘9171 ‘9981 10 ‘9136 9943 10
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 159
§ 20. Counting the centre as a node surface and the outer surface as a no-stress
surface, the number of the node, loop, or no-stress surfaces is always equal to the number
of the note. We shall refer to any such surface by its number, supposing the surface
of the same kind of least radius to be number (1).
For the node surfaces kar is equated to certain numerical quantities independent of
@, viz. the roots of (36). Thus the ratio of the radii of the node surfaces of numbers
(¢) and (2’) in a given sphere, when ¢ and @ are given integers, is the same whatever be
the value of o for the material of the sphere or the number of the note. In like
manner for the loop surfaces kar is equated to certain numerical quantities. Thus the
ratio of the radii of the loop surfaces of numbers (¢) and (7’) in a given sphere is inde-
pendent of the value of o or of the number of the note.
For the no-stress surfaces, however, kar is equated to the values obtained for kaw from
the frequency equation, and these vary with the value of oc. It thus appears that while
in a sphere of given material the ratio of the radii of the no-stress surfaces of numbers
(i) and (7v’) is the same for all the notes, this ratio is different for materials which differ in
the value of Poisson’s ratio.
It will be seen from the table that unless o be small there is a marked difference in
the positions of the corresponding loop and no-stress surfaces of least number. Between the
loop and no-stress surfaces of high number the difference is obviously very small. Their
radii, as well as those of the node surfaces of large number, are but little dependent
on o. As the number of the node surface increases it tends continually to become equi-
distant from two successive loop or no-stress surfaces.
§ 21. Im all the expressions we are about to deal with for the change of frequency
there occurs one or other of two factors. The first is
= kea\* = §p8n272 y =;
Q= is + {k®ata? — 4n (3m — n) (m + n)-}, | cree eke eB
the second 1Q' = theaa?Q |
As the expressions (38) occur in the coefficient of b—c in the expressions for
ok/k we may, to the present degree of approximation, simplify them by any transformation
which regards kaa as a root of (20) or the quantity tabulated in Table I.
Thus we may take
m+n
4n
3
= [heart ee + ( 4n i 1 }
(m+n)? m+n) kaa
whence we get the following alternative formulae
(ka sin kaa,
ae n= ay (3 kaa
“~ kaa\ kaa
— cos kaa) =
Q = {4n(m +n)“ cosec kaa}? + (hata? — 4n (Bm — n) (m+ n)7} veeveeevees Feotne acres (39),
8n (m — n) ( 4n
= k°ag? — ——
m+n
(m+ ny ) (taa)*| + {keea? — 4n (3m —n)(m+n)} 0... (40).
160 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
In the higher notes (40) is much the safer formula to use, because with it any
small error in the value attributed to kaa in Table I. has a wholly insignificant
effect.
The method by which (40) was deduced requires modification when o="5. It is
easy however independently to prove for this case
Q=1,
a result consistent with (40); whence we also get
Q = en
where i is an integer equal to the number of the note.
Employing the results in Table I, I find the following values for Q and Q’—
TaBLE III.
Value of Q Value of Q’
Number - a ns on
of note o=0 o="25 o='5 c=0 o='25 o="5
(1) 2:253 1369 9-762 8995 98696
(2) 10635 1:0401 37°53 38:18 39:48
(3) 10247 ~=-1-0161 87-51 88:83
(4) 10133 1:0088
(5) 1:0084 1:0055
(6) 1:0058 10088
155°93 156°59 157-91
24.4°74, 245-415 246-74
35331 35398 355°31
ee |
(0.2)
fon)
(o'2)
-_
We may regard the case o='5 as supplying an inferior asymptotic value, viz. 1,
for Q, and a superior asymptotic value, viz. 7? where i denotes the number of the
note, for Q. Except in the case of note (1) we may in rough calculations treat Q as
unity, and regard Q’ as varying as the square of the number of the note whatever be the
value of o.
§ 22. We shall first discuss some special cases of (28).
By supposing b/a very small we pass to the case of the central layer mentioned in
§ 16. Supposing V the volume of the whole sphere, dV that of the layer, we have
0V/V=3(6—c) B’/a’.
Retaining in (28) only the lowest powers of b, and treating the function of kaa in the
manner just discussed, we easily find for this case
Ok, ty Vay 3m, —n, — (8m —n) {1 4n, =n}
ie Ve 9 (m+) 3 m+n)
where the suffix / signifies that the material (p,, m, ,) forms a true layer.
As already explained, the case when the material (p,, m,, ™) forms a core cannot be
derived from (28). I have, therefore, worked out this case by a rigid method inde-
pendently. Supposing 6 the radius of the core and dV its volume, so that
0V/V=1%/a*,
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 161
and retaining only the lowest power of b/a, so that the result assumes the core of
very small volume compared to the sphere, I find
Ti Ng ist (Ey
avs 3 (38m, — m+ 4n)
The suffix ¢ signifies that the material (p,, m,, m) actually forms a core.
The physical conditions under which (41;) and (41,) apply are totally different, so
there is no reason to expect an identity between the two results. It will be noticed,
however, that when the difference between the material of the layer or core and that of
the rest of the sphere is small (41;) and (41,) lead to the same result, viz.
dk OV .,38m,—1, — (3m —7n)
Fea ye SIGE ai sauces aaa
Since p;—p appears neither in (41,) nor (41,) we see that an alteration of density
alone throughout either a central layer or a small core has to the present degree of
approximation no effect on the pitch of any note.
In investigating the effects of alteration of elasticity we shall mainly consider the three
following special cases :—
*1° when the elastic constant m alone is altered,
2° when the rigidity » alone is altered,
+3° when both elastic constants are altered in the same proportion so that
MG |= Ny IU — cbs Dewecneeess eens cccercesoces vsiscl eee oes (43)
where p must of course be algebraically greater than — 1.
The relation (43) is on the uniconstant hypothesis necessarily true, but on the bi-
constant hypothesis of isotropy there is no @ priori reason to expect it to hold.
Employing the suffixes / and ¢ as in (41;) and (41,), we find for the changes of pitch
in the above three cases :—
1° when m alone is altered
chit pte m, —™m x,
Berm: Gasca aa a
2° when 7» alone is altered
Ok, yi Ons m—N | 4n,— “} }
k V * 9(m+n) 3m+n)’ | (44”)
Oke __ OV Q’ Ny —N + aise 03 (44).
ko V © 8(8m—m+4+ 4n) |
3° when the relation (43) holds |
ok; =p ai 3m —n ab 7 ha) |
kk 1+pV “© 9(m+n) 3? int n\” (44’”) |
Oke _ oV Q 3m —n fhe, 3m—n ) |
EL Y¥. 9(m +n) v 3(m+n)) J J
* This gives the most general alteration of the com- + This is the most general alteration consistent with
pressibility, or of Young’s modulus, which is accompanied _ the constancy of Poisson’s ratio.
by no change in rigidity.
162 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
We see that an increase in m alone throughout a small volume at or close to the
centre raises the pitch and a diminution of m lowers it; also for a given numerical
alteration of m the fall of pitch when m is diminished is greater than the rise of pitch
when m is increased.
Since 3m—n is essentially positive we see that in both forms of (44”) the sign of
ok is opposite to that of m,—mn. Thus when the rigidity at or close to the centre is
altered the pitch is raised or lowered according as the rigidity is diminished or increased.
The fall of pitch due to a small increase of rigidity at or close to the centre is greater
than the rise of pitch due to an equal small diminution of rigidity.
In the case of the core this is obviously the case whatever be the magnitude of
the alteration of rigidity. In the case of the central layer we may regard 0k, as com-
posed of two terms, the first varying as m,—n and indicating a change of pitch opposite
in sign to the alteration of rigidity, the second varying as (m,—7)? and always indicating
a fall of pitch.
If the alteration of elasticity satisfy (43), then the pitch is raised or lowered accord-
ing as the elastic constants are increased or diminished. In the case of the core the
rise of pitch due to a given numerical increase in the elastic constants is obviously
always less than the fall of pitch due to an equal diminution in the constants. The
same is easily proved true for the case of the central layer when the alteration in elasticity
is small.
For any alteration of elasticity other than those above considered occurring at or close
to the centre, we obtain from inspection of (41,) and (41,) the general law that the
pitch of all the notes is raised or lowered according as the elastic quantity m—n/3—
i.e. the bulk-modulus—is increased or diminished.
§ 23. When, as necessarily happens on the uniconstant theory of isotropy, only one
elastic quantity is involved, the meaning to be attached to the terms stiffness and elasticity
is in general free from ambiguity, and the statement that a local increase in stiffness raises
the pitch may be im all cases sufficiently definite to admit of its truth being tested.
As applied to the case (43) it is strictly true, and so when proceeding from supporters
of uniconstant isotropy is in accordance with the facts here arrived at.
When, however, the statement is made by supporters of the biconstant theory it fails
in the present case to have any exact meaning. This is obvious if we consider that
the terms stiffness and elasticity might be interpreted to mean the rigidity, the bulk-
modulus, Young’s modulus, or any other modulus.
Now an increase in the rigidity produces an increase in Young’s modulus and a fall
in the bulk-modulus, while an increase in m increases both Young’s modulus and the
bulk-modulus. Thus a given increase in Young’s modulus may be accompanied by a rise
or by a fall in the bulk-modulus.
Our recent investigation shows that if the term stiffness is limited to mean the bulk-
modulus the general statement is here in accordance with the facts; whereas if it be
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 163
supposed equivalent to Young’s modulus it may be true or false according to circum-
stances.
§ 24. As concerns the numerical magnitude of the change of pitch we may regard
in the case of the central layer
OV 38m, —n,—(38m — of i: 4n,—n)
V 3 (m, +7) 3m+ nf Se
and in the case of the core
OV 3m, — m —(8m—N) _
Vo 3m, —n, + 4n OL,
as measuring the magnitude of the alteration of elasticity.
The expressions (41;) and (41,) may then be written
1 1 r
dh + Oly = 5 the + BB. = 30.
Thus if in Table III. we divide the values given for Q by 3, and alter the heading
from Q’ to 1 Oh, + 0B; =} Ok, = OE, we obtain at once a numerical measure of the changes
k
in the pitch of all the notes considered in that table. The forms taken by @#, and
eH, in the special cases when m alone is altered, or n alone is altered, or (43) holds
are obvious from equations (44).
The forms given above are convenient when we examine the effect on the pitch
due to a given alteration of material occurring throughout a given volume.
We shall also have occasion to deal with layers of given thickness, for which 6—c
is constant. The square of the thickness is supposed in every case negligible, thus the
effect on the pitch of any note due to any alteration of material throughout a central
layer of given thickness or throughout a core of equal small radius, being at least of order
(kab)?, must be held to be zero.
§ 25. A second special case arises when the alteration of material occurs at the
surface.
As the proof on which (28) rests assumes that the material (p,, ™,, n,) has material
(p, m, n) outside it, its application without further proof to the case when (p,, m,, 7)
forms a surface layer might be objected to. I have thus worked out independently the
case of the two-material compound sphere (0.a.b.a,.a), and proceeding to the limit
when {(a—b)/a}* is negligible I obtained a result identical with that derived from (28)
by supposing b=a.
Denoting the thickness of the layer by ¢, and remembering that in virtue of the
surface condition in a simple sphere U, is zero, we easily obtain from (28)
Vou. XV. Parr II. Dy
164
ee el 4n (38m —n) (nr (8m, — %) (M+ %)7 fi
“ae (m+n) n(38m—n)(m+ny>
hi Gm=n)
a
k°ata? ae
. (m +n)?
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
The value of 0k/k when the density at the surface alone is altered is shown in the
following table for the first six notes answering to the values 0, °25 and °5 of oa.
TABLE IV.
Talue of -F.( a—e) for a surface layer.
a p
Number
of note (1) (2) (3) (4) ()
0 1857 1-060 1:024 1-013 1-008
o=1'25 1511 1:064 1:0265 10145 1009
15) ] 1 1 l u
Noticing that if we suppose in (45)
My (3m = M%) (M+) _ 4 _ Pr
n(3m—n) (m+n) Pp
it reduces to the wonderfully simple form
we deduce at once from Table IV. the following results for the change of pitch due to a
surface alteration of elasticity alone—
TABLE V.
Value of x a E {7 (Bm, — %) (M+) _
a| n(Bm—n) (m+n) >
Number
of note (1) (2) (3) (4) (5)
0 ‘857 ‘060 024, 0138 008
ao =<'25 531 FE 064 0265 0145 009
13) 0 0 0 0 0
it| jor a surface layer.
(6)
‘006
‘006
0
From Table IV. we see that in every case of a surface alteration of density the pitch
is raised or lowered according as the density is
diminished or increased.
The effect of a surface alteration of elasticity whatever be the value of o is very
small in the case of the higher notes, and continually diminishes, as measured by the
percentage change of pitch, as the number of the note increases.
For the limiting value
‘5 of o the effect of a surface alteration of elasticity alone is always zero.
From Tables IV. and V. we see that if a thin surface layer of an isotropic sphere
be altered in any manner consistent with its remaining isotropic, the ratios of the
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 165
frequencies of all the higher notes can only be very slightly affected; but, unless the
value of o for the unaltered material approach the limiting value *5, or else both density
and elasticity be altered in such a way as approximately to satisfy (46), the ratio of the
frequency of the fundamental note to that of any of the higher notes may be sensibly
disturbed.
If we suppose the relation (43) to hold, then (46) takes the form
Dir
or the percentage alterations in the density and in the elastic constants are to be numeri-
cally equal and of the same sign.
§ 26. An exhaustive analysis of (28) being out of the question, I propose limiting
the investigation to the following cases:
1°. Suppose the layer to differ from the remainder only in density, then remember-
ing (38) and (26) we have
ok tp — sin — oM QY (1 /sin kab a: 2 ‘
Ee sellers cos kab) = M3 = kab 08 kab)! Sees (48),
where t=b—c, M=4a'p/3, 0M = 4b? (b—c)(p, —
and dM/M is supposed small.
The form of (48) to be used is the first or second according as the layer is of
given thickness or given volume.
2°. Suppose m alone altered, or the layer to differ from the remainder in all its
elastic properties except the rigidity. For this case there are the two alternative
formulae
Ok _tm—™p «377 _ dV m, — m YY /sin kab\? ,
Se eR Q sin? kab = Vicatens ( Ta ) SOOM URGE ao Hae Hee (49)
where V =47ra*/3, oV =47b?(b —c),
and 0V/V is supposed small.
*3°. Suppose m constant and n alone altered. The following seems the most con-
venient way of representing the expression for the change of pitch—
= ON ae
pa! iP Q | {sin kab — —— (a — cos kab) sn kab — oe ee — cos kab)
ko am+n, TET) Nw Hea Ta
_ 4-7) fl sin kab s 2 i
m+n (kab ( Tay cog kab soon Ceeae (50).
We can obtain an alternative form in 0V by putting
igen (3) Pein Moowboaite els. hein. cul. (51),
employing 0V under the same restriction as in (49).
* For the case where the compressibility is constant while the rigidity is altered, see the note at the end
of this Section.
22—2
166 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
4+. Suppose the relation (43) to hold. The formula for the change of pitch is
4n (3m —n){ 1 /sin kab ‘este
EO & pQ | CEs) a {ab Tap UO kab))
1 fos pgp, 4H (m +n) /sin kab salt ae
+(1+ p) eee kab ih e Tab 0° kab) woeaclee Dens
The substitution (51) gives the equivalent form in 0V, applicable under the usual
restriction.
§ 27. Comparing the several expressions (48), (49), (50) and (52) for the change of
pitch we see that each is a product of three factors.
The first factor is such as
tpi—p _ 0Vm—m
-—— or =- ——__,
Wis, Vi m+n
and may be regarded as measuring the magnitude of the alteration in the material. For
a given alteration of material the first factor is the same for all notes, and for all positions
of the layer. The second factor is either Q or @Q/3. These quantities vary with the
number of the note and the value of o, as shown by Table III, but are independent
of b. The third factors are such as sin?kzb, They determine how the effect on the
pitch of a given note of a given alteration of material varies with the position of the
altered layer.
In the case of (48) and (49) these third factors do not contain m or n explicitly,
and depend on o only in so far as ka does. They may thus be regarded as functions
solely of the variable kab. We thence arrive at a comparatively simple way of treating
the subject.
§ 28. We shall first examine the case of (48) and (49).
As an example let us take the first form of (48) and draw a curve B, fig. 1, viz.
sin « 2 me
y= ( = R08 “) Soacsobsgndaabeppacnaspovnenc tt Pes cecetrdaee (53),
whose abscissae are the values of 2=kab. Then the ordinates of this curve indicate
the variation in the magnitude of
with the radius of the layer of altered density, supposed of given thickness, whatever
be the number of the note or the value of o The only effect of a variation in the
number of the note or in the value of o is to vary the value of the factor, viz.
(kaa), by which the abscissae must be multiplied to get the corresponding values of b/a,
and the factor, viz. Q, by which the ordinates must be multiplied so as to give the
numerical values of
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 167
In the fundamental note for instance the position of the layer to which the abscissa
« refers answers to b/a=2/('66267) when o=0, and to b/a=2/('81607) when o =°235.
In the first case the portion of the curve which applies is limited by the abscissae 0
and ‘66267, whereas in the second case the limiting abscissae are 0 and ‘81607. In
the first case to find the numerical value of
Um é — |)
ip Nae fay JP”
we must multiply the ordinates by 2°253, whereas in the second case the factor of multi-
plication is 1°369.
Suppose again we consider one of the higher notes, for instance note (4) when
o='25. Here the position of the layer to which the abscissa x refers answers to
b/a = x/(3'965877),
and the whole of the curve between the origin and the point whose abscissa is 3:96587
applies. To get the numerical value of
we must in this case multiply the ordinates by 1:0088.
Still employing the same curve we shall illustrate its application to the determi-
nation of relations between the successive positions of the layer when the change of
pitch vanishes or is a maximum. Since 0k vanishes when the ordinate of (53) vanishes, the
several positions of the layer when its existence has no effect on the pitch are found by
equating kab to the successive roots of equation (36), which are absolute constants inde-
pendent of & or a.
In like manner the several positions of the layer when its effect on the pitch is
a maximum are found by equating kab to those abscissae which supply the maxima
ordinates of (53), i.e. to the successive roots greater than zero of the equation
sin 7 — -(="- cos x) = Qe cratsssatcon set aerate teaemstre esses (54).
The roots of this equation are likewise numerical quantities. We thus conclude that as
ka is constant for a given sphere performing a vibration of given frequency, the radu
of the several positions of the layer where its existence has no effect or a maximum
effect on the frequency of a given note are to one another in certain constant ratios
wholly independent of the number of the note, of the value of o, or of the magnitude of
the sphere.
If we denote the 1 positive root in ascending order of (54) by 2;, and the radius
of the corresponding position of the layer for the note of frequency k/2m by 6;, then
b;/a = x;/kaa Devlacecisecccovenccecseccescccscserscccccccee (55).
168 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
Thus the ratio to the radius of the sphere of the radius of the layer when in the
position answering to the maximum change of frequency of given number (7), in the note
of frequency &/27,—the position nearest the centre being held number (1)—varies inversely
as the value of kaa for the note and material considered. The same is obviously true of
the radii of those positions of the layer where its effect on the pitch vanishes.
Again since the numerical value of ok/k for a given note in a given sphere is
obtained by multiplying the ordinate of (53) by a constant factor, we find between the
maxima changes of pitch of numbers (¢) and (j) in a note of frequency k/27 and the
maxima ordinates of numbers (7) and (j) in the curve (53) the simple relation
C)CRIRYG/ OER VO AE DRO RRR. sonencep ae Ron oe esonnousnga. coh (56). °
Now y; and y; are certain numerical quantities, thus, whatever be the number of
the note or the value of o, the ratio of the maxima changes of frequency of numbers |
(i) and (j) is the same. Thus if we desire to compare the relative magnitudes of the
successive maxima changes of frequency in the pitch of a note of given number in a given
sphere, due to an assigned alteration of density throughout a layer of given small thick-
ness, all we have to do is to compare the lengths of the successive maxima ordinates of
the curve B, fig. (1).
Conclusions of the same general character obviously apply to the three following curves—
5 _ {lysing _ ) a 4
A, fig. 1, viz. y= ie ( Second | (57),
does th ilar ie ==]. ooderanpAaBpEehsdda op puoanapandonscusoéaedec! (58),
Ay Gg Os 45 aie Ria oo: wach osm oon ee coe omer (59),
which represent the variation of 0k/k with the value of kab in the second form of (48),
and in the first and second forms of (49) respectively. In the case of (57) and (59),
where the layer is supposed of given volume, the restriction of the formula im the case
when the radius of the layer becomes very small must be remembered. The ordinates
however at the origin give correctly the change of pitch due to a central layer.
§ 29. There are various other general conclusions which are easily derived from (48)
and (49), in the elucidation of which the curves (538), (57), (58) and (59) are useful.
If we suppose the curves drawn on the same scale, then the value of b/a which
answers to a given value of @ is, for a given note in a given material, the same in all
the curves.
Again if we are considering the effect of an altered layer of given thickness, the
second factor, which determines the variation of 0k/k with the value of o or with the
number of the note, is Q in the first forms of both (48) and (49).
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 169
We thus conclude that if the same scale be adopted in the curves, then the quantities
t py— ok t m,—™m
a ae an k Ee G 7 +n
ratio of those ordinates of the curves B, fig. 1, and B, fig. 2, whose abscissae are found
by multiplying the values of b/a for the assigned positions of the layer by that value of
kaa which applies to the note and material under consideration.
) for any given note and material are simply in the
If we suppose the thicknesses of the layer of altered density and the layer whose
elastic constant m is altered the same, and further suppose
pPi—p_m,—m
p m, +n
then the numerical magnitudes of the changes of pitch in the two cases in a given
note and material are simply as the lengths of the ordinates of the curves.
Similarly if we are considering the effect of altered layers of given volume, we see
from the second forms of (48) and (49) that the second factors are the same, viz. Q’/3,
whether the alteration be in the density alone, or in the elastic constant m alone. We
thus conclude that the magnitudes of the quantities -F+(Gr) 2 : (F 7)
for apy given note and material are simply in the ratio of the ordinates of the curves
A, fig. 1, and A, fig. 2, supplied by the abscissae which correspond to the assigned positions
of the layer.
nd
§ 30. The expressions (50) and (52) do not admit of so simple a treatment.
We may, however, regard (50) as composed of two terms, each of which may have
its dependence on 6 represented by a curve whose ordinate is a function solely of
a =kab.
When the layer is of given thickness, these curves are
ah (a) Ya a (09) pace noseaudoconebeeaaaneenodbenoscsonpena (61),
where VW, (@) = sin 2 — 24 (a sin « — cos a 62
‘p. (a) = sin 2 — 6a\(a-1sin @ cos) “nen (62),
and (57).
We may then suppose a compound curve drawn whose ordinate is the ordinate of
(61) diminished by the product of the ordinate of (57) into the quantity
4 (nm, — n)/(m+7n).
Since this quantity depends on the value of o and on the magnitude of (n,—n)/n
the compound curve varies with the value of o in the material and with the magnitude
of the alteration of rigidity.
170 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
When the layer is of given volume the two curves are
Y SAAN (&) Wes (@) b conevscan wert paces se see se aeeseeen ose (63),
and U)—)aa8 (ae Sint as— COS a:) ae cuneaaecennncha oceania eet (64).
A compound curve may be derived from (63) and (64) precisely as one was derived
from (61) and (57).
The expression (52) may likewise be regarded as composed of two terms. The first
of these may have its dependence on 6 shown by a curve whose form is independent
of o. This curve is (57) or (64) according as the layer is of given thickness or of
given volume. The second term has its dependence on } shown, according as the thickness
or volume of the layer is given, by the curves
y = {sin 2 — 4n(m + 2) ao (2 sin & — COS Z)}® 2.02.2. .eeneceenene (65),
y = a~ {sin w— 4n (m +2) ae (a7 sin £ — COS @)}*... eee eee eee (66)
respectively.
Compound curves may as before be constructed showing the variation with b of the
complete expression (52). These compound curves vary with the value of o and with
the magnitude of the alteration of elasticity.
If we suppose a compound curve drawn in the case either of (50) or (52) answering
to a given alteration of elasticity and a given value of o, then it applies to all possible
notes. There are thus for a given alteration of elasticity and a given value of o the same
species of relations between the relative positions of the layer when its effect on the pitch
is a maximum, and between the magnitudes of the several maxima of dk/k, as there
were in the case of (48) and (49).
§ 31. When the alteration of elasticity and the value of o remain unchanged then
in (50) and (52), precisely as in (48) and (49), the variation of the several maxima of
dk/k with the number of the note depends only on the factor @ when the layer is of
given thickness, and on the factor Q’/3 when the layer is of given volume.
Now as appears from Table III, Q differs but little from unity except for note
(1); whereas in the higher notes Q’ increases at least very approximately as the
square of the number of the note. Thus for any one of the four types of alteration
of material treated here, the maxima percentage changes of any given number in the
frequencies of the several notes above the first are all nearly equal when the layer is
of given thickness, but vary approximately as the squares of the numbers of the notes
when the layer is of given volume.
§ 32. The evaluation of some of the functions of # represented by the curves being
a very laborious process, I have carried none of the calculations beyond the value 37
of «. The results are given in Table VII. This supplies most necessary data for the
first three notes in any material, but in the case of the higher notes its scope is
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 171
limited to positions of the layer which, roughly speaking, lie inside the third loop
surface.
The unit abscissa adopted in the table is 7/18. For shortness the functions are
represented by f((z)...fis(#). Full information as to the first eleven of these headings
is supplied in the following table. The entry “p” in the column headed “ Property of
material altered” means that both elastic constants are supposed altered in the same
proportion, as in (43):
TABLE VI.
3 S 3 m = Letter Iv alues
= Pit: so whore ed to | | 2G A
| Ee | (m |thickness| 2 | B | all
x) = sin? x | 4 |
J1 lip | : el aos) 85
ir! | (m | volume eal eeAas|nalll
fe (2) = Si (x) | \p 55 x | 5
|
Fi (@) = (a7 sin # — cos @) | p | thickness} 1 B | all
p volume 1 A | all
fi(@) = 27 f, (2) n | thickness | _,, » |
P | » »” » | ”
: | volume Ul C i} all
fala) =a-*f,(2) Eto spe eels
Js (#) = {sin # — 2a (a sin & — cos x) }? | p ‘thickness | 4 Bea\enO
a2) — if (a) p volume 4 | A, 0
Js (2) = {sin @— 4 a4 (a sin w — cos 2)}* p |thickness| 4 | By | -25
So (@) = 4 fa (2) Pp | volume AW Zékece || 253
eet |
age) — {sina my (= *—cos z)t {sin “L— 6 (= * —cos “)} | 2 A thickness | 3 Binal
DNs J PN te /) |
FL O=E AO) | nm | volume | 3 FAualmalll
As the first nine functions cannot be negative no signs are attached to their values.
In the case of f(z) and f,(#) signs are attached to those entries which occur next
the zero value. Any number without a sign attached has the sign last entered in the
column.
The functions f,(#) and f,,(©) which appear in the table apply both to the radial
and the transverse vibrations of a sphere. Their use in radial vibrations is stated at the
end of this section; their form is more fully discussed in Sect. III.
Vou. XV. Part II. 23
CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
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Mr ©. CHREE, ON SOME COMPOUND VIBRATING SYSTE
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2
\74 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
§ 33. We shall now discuss in some detail the effects of the several alterations
of material.
When the layer differs from the remainder only in density the change of pitch is
given by (48). The positions of the layer when the pitch of a given note is unaffected
coincide with the node surfaces for that note.
When the layer is in any other position the pitch is raised or lowered according
as the density is diminished or increased.
When the layer of altered density is of given volume 0k/k varies simply as wj*, as
may be seen by comparing (26) with the function of # occurring in curve (57), Le. A,
fig. 1.
The points of this curve whose ordinates vanish answer of course to the node
surfaces including the centre. The successive maxima ordinates answer to positions of
the layer coincident with the successive loop surfaces.
The number of maxima is always equal to the number of the note. When o=0
the surface of the sphere is always a position supplying a maximum.
We see at once from the curve that the first maximum is far the most important.
Thus the effect on the pitch of any note of an alteration of density throughout a layer
of given small volume whose radius exceeds that of the first, or at all events the second,
node surface is comparatively insignificant. The calculation of the lengths of the maxima
ordinates may be simplified by the consideration that since the corresponding abscissae
are the roots of (37) we may put
1 /sin Ys ‘ae 1 4\7)
—(——— cos 2} =( sin 7 =3(1+5)
al z J 2 e Fis |
where «/m has the values ascribed in Table I. to the case o=0. For the ratios of
the first to the successive maxima ordinates, and so of the first to the successive maxima
ok oM reas
eat et, *
1 : 1485 : 0620 : 0342 : 0217 : :0150....
As already explained the absolute magnitudes of the maxima vary as Q’ and so
depend on the value of o and on the number of the note. The following table gives
the first and so the largest maximum for the first six notes.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 175
Taste VIII.
ok By oM
First maximum of = on
Value Number
of o | of note (1) (2) (3) (4) (5) (6)
0 619-238) 551 989 15:52 22-41
"25 570 2°42 5°55 9:93 15°56 22°45
3) 626 2°50 5°63 10°01 15°65 22°58
When o='5 the first maximum for note (2) is given by
Ok JOM) ae 20),
are fee (cy) cubaanaocdadnouconeencad cagabooueedcor (68):
and for all values of 2 above 6, this equation will give a close approximation to the
first maximum whatever be the value of oc.
§ 34. When the layer of altered density is of given thickness ¢, the mode of varia-
tion of 0k/k with kab is given by curve (53), 1e. B, fig. 1. The successive maxima
ordinates diminish slightly as the values of 2 to which they correspond increase.
The exact values of the abscissae supplying the maxima ordinates are the positive
roots of
eve 1 (= x
a\ 2
excluding zero. It will be noticed that (20) may be made identical with (69) by writing
x for kaa and taking m=3n. Thus the roots of (69) are the values assigned to kaa
in Table I. in the column for o=°3. The corresponding positions of the layer thus
coincide with the no-stress surfaces when o=°3, and lie outside or inside these surfaces
according as o is less or greater than this value. It follows that provided o be not less
than ‘3 the number of true maxima of 0k/k is equal to the number of the note. If, how-
ever, « be less than 3 the number of true maxima is less by unity than the number of
the note.
This point requires special attention in note (1), as there is here no true maximum
if o be less than ‘3. This simply means that when o is less than ‘3 the portion of
curve B, fig. 1, which applies to this note does not extend as far as the first maximum
ordinate. The value of 0k/k in such a case increases continually as the layer moves out
from the centre. The value arising when the layer is at the surface may be called a
maximum, but it must be carefully distinguished from the true maxima which answer to
the maxima ordinates of curve B.
All the data necessary for calculating the positions of the layer answering to the
true maxima in the case of those notes and materials considered here exist in Table I.
I have, however, thought it worth while to record the results in the following table.
The blanks indicate the absence of true maxima.
176
Values of b/a when
Number Value Number of
of note of ¢ maximum (1)
(0 sae
(1) 4-25 —
(5 8733
(0 “4618
(2) 4-25 4528
(5 4366
(0 2930
(3) 495 2956
(-5 2911
(0 2212
(4) 4°25 2202
5 2183
| 0 1761
(5) 4°25 1756
(-5 ‘1747
(0 1464
(6) 4-25 1461
(5 1455
TABLE IX.
9735
6644
6591
6490
4931
4909
4867
3926
3915
3894
3263
3257
3245
a
tpi—ep
a Pp
(3)
"9885
7511
‘T7478
‘7414
5980
“5964.
5931
‘4971
“4961
4945
) is a maximum.
(4)
9936
S015
“7992
“7949
6662
6649
6624
(5)
8299
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
(6)
“9972
In calculating the lengths of the successive maxima ordinates of curve B, fig. 1,
we may, since z is a root of (69), replace
an — cos x) by qd = > | n= Vy
For the ratios of the first to the successive maxima ordinates I find
12 “908 =) <395"= -S90M osee- Sole...
These are thus the ratios of the first to the successive maxima of —dk/k due to a
given alteration of density occurring throughout a layer of given thickness in a given sphere,
whatever be the density or elasticity of the sphere or the number of the note con-
sidered.
The absolute values of the maxima vary as Q. In the following table are given
the absolute values of the first and so largest maxima for the cases considered here.
TABLE X.
First maximum of ee (5 Pu °) :
k a p
Value Number
of | of note (1) (2) (3) (4) (5) (6)
0 1°857* 1:202 1158 1145 1140 ey;
25 159 Wh lie Vi75 1148 1140 1136 1135
oy 1130 1130 1130 1:130 1130 1130
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 177
In the case of note (1) there are no true maxima for the values 0 and ‘25 of o.
I have, however, given the greatest values which the quantity tabulated can have im
these two cases. They answer to positions of the layer coincident with the surface,
and are distinguished by asterisks.
As in the case of all quantities varying as Q, it is only in the first few notes that
the percentage change of pitch depends to any marked extent on o. For any note
above the sixth in any isotropic material the formula for the limiting case o=°5, viz.
-#-(¢ PP) - 1130 eset ee tes, OAL Gt Dts (70)
p
k a
supplies a very fair approximation to the first maximum,
§ 35. In the second case we are to consider m alone is altered. Mathematically con-
sidered this change is very important, as the expressions which occur in the formula for
dk/k ave of extraordinary simplicity.
The change of pitch in this case is given by (49). The positions of the layer when
the change of pitch in a note of frequency k/2 vanishes are given by the equation
where 7 is any positive integer. For all other positions the pitch is raised or lowered
according as m is increased or diminished throughout the layer.
Employing (71) we can easily calculate from Table I. the values of b/a for those
positions in which the layer does not affect the pitch of the several notes. When o ='5
these positions coincide with the no-stress surfaces. For other values of o it seems
unnecessary to tabulate them, because they lie exactly midway between the successive
positions given in Table XI. where the layer when of given thickness has most effect
on the pitch.
Supposing first that the layer is of given volume, we have the variation of 0k/k
with the value of kab given by the curve (59), 1e. A, fig. 2.
Of the maxima ordinates that at the origin is very much the largest. Thus the
maximum change of pitch which arises when the altered material forms a central layer
is extremely large compared to the other maxima.
In the present case to obtain the change of pitch due to a central layer, we have
only to divide by 3 the values given for Q’ in Table IIL, and to alter the heading from
: ok (oVi,—m
Q fe ie a a :
From (59) we see that the several maxima ordinates have for their abscissae the
roots of tanw=. The corresponding positions of the layer are thus coincident with the
node surfaces.
178 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
In comparing the lengths of the maxima ordinates it is convenient to notice that
since tan 2=a2, :
(a sin #)? = (1 + 2°)-.
Employing this relation, I find for the ratios of the first to the subsequent maxima
ordinates, and so for the ratios of the first maximum change of pitch—answering to a
change of m throughout a central layer—to the subsequent maxima
1 : 04719 : 01648 : -00834 : -00503 : -00336....
For notes above the sixth a close approximation to the first maximum in any material
is supplied by the equation
ok . (OV m—m\— Pr
+> = oar -secgsinreect ree eee eee (72),
k Ve m+n 3
where 7 is the number of the note. This is the exact equation for the value ‘5 of o.
§ 36. Suppose next that the layer whose m differs from that of the remainder is of
given thickness. The corresponding curve is (58), i.e. B, fig. 2, which is merely a special
form of the curve of sines.
The zero ordinates coincide of course with those of curve A, fig. 2. The abscissae
supplying the maxima ordinates are found by ascribing positive integral values to 7 in the
equation
xe = (20 + 1) 7/2.
The corresponding values of b/a for the notes and materials treated here are given in
the following table:
TABLE XI,
, ok . (tf m—m) . :
Values of b/a when —~++4- —*——-> is a maximum.
k am+n
Number Value Number of
of note ofc Maximum (1) (2) (3) (4) (5) (6)
(0 7546
(1) 725 6127
( 0 2044 ‘7933
(2) 4°25 "2593 ‘7778
[-5 25 75
(0 ‘1706 5119 "8532
(8) » 425 1693 5078 ~~ 8463
(-5 16 5 ‘83
(0 1266 3799 6332 8864
(4) 4°25 1261 3782 6304 $826
(5 125 375 625 875
(0 ‘1008 3025 5041 ‘7058 9074
(5) 4°25 1005 +3016. «= 5027'S 7088 ~— 9049
(-5 ‘J 3 5 7 9
: {0 0838 2514 4190 9867 7543 9219
(6) 4°25 0836 2509 4182 5855 7528 ‘9201
(:5 083 25 416 583 75 ‘916
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. A?)
Comparing the preceding table with Table II. it will be seen that the positions of
the layer of given thickness when an alteration in m has most effect on the pitch are,
with the exception of the first, only a very small distance outside of the corresponding
node surfaces. The distances separating the two sets of surfaces become less and less
the higher the note.
The maxima ordinates are all exactly equal. The exact expression for the maxima
changes of pitch is
ok | é M,—m\ _
kk \am+tn/] —
Their numerical values are thus given explicitly in Table III. by altering the heading in
ah e
that table from Q to (- —).
k am,+n
§ 37. In the next case we are to consider when the layer differs from the re-
mainder only in rigidity the change of frequency is given by (50).
This may be regarded as composed of two separate terms, one varying as the first,
the other as the second power of n,—n. When the difference between the rigidities of
the layer and the remainder is small the second term may be neglected, except for such
values of b as make the first term nearly vanish. By supposing the difference of the
rigidities sufficiently small we can indefinitely reduce the limits wherein the second term
is comparable with the first. We shall thus for the sake of simplicity commence by
supposing that ,—mn is very small and that the term in (n,—n)* is negligible.
The law of variation of ok/k with the value of kab is in this case given by (61)
or (63) according as the layer is of given thickness or of given volume. The sign of
ok/(m,—n) is thus the same as that of the product of the functions (x) and w, (2)
defined in (62).
The ordinate of curve A, fig. 3, is the quantity 2, (x). (e), or fu («) of Table VIL;
while the ordinate of curve B, fig. 3, is the quantity W,(w)y.(«), or f(a). Thus the
ordinates of these curves are proportional to the changes of pitch when a small alteration
in rigidity occurs throughout (1) a given volume, (2) a given thickness.
The sign of ok is the same as that of m,—n or the opposite according as the
ordinates of the curves are positive or negative. The zero ordinates have for their
abscissae the roots of the two equations
As « increases through a root of (74) the curves cross from the negative to the
positive side of the axis of #, while as # increases through a root of (75) they cross
from the positive to the negative side.
Comparing (62) with (20) we see that the roots of (74) are the values ascribed to
kaa in Table I. for «=0, the corresponding positions of the layer being coincident with
Vou. XV. Parr II. 24
180 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
the loop surfaces. For the first two roots of (75), excluding zero, I find approximately
16947 and 2°7977.
If we denote by ,#; and .2; the 7 roots excluding zero of (74) and (75) re-
spectively, then it is easily proved that as 7 increases the roots ,7; and ,#;, both con-
tinually approach im. Also ,#;—,t;. remains positive but continually diminishes as 7
increases. Thus the breadth of the segments which lie on the negative side of the axis
becomes less and less the further they are from the origin, while the breadth of the
positive segments approaches 7.
For further information as to details the reader may consult the following table,
remembering that the term in (n,— 2)? is neglected in its conclusions.
TABLE XII.
Sign of 0k/(nm,—n), and values of b/a for which its sign changes.
Number
of note Gk/(n, —n) = — tO) = @ SF Oi 0
1
812 1
663 1
b/a
ll
uwno
Or
350 896 1
B44 ‘878 ‘980 1
331 ‘S47 945 1
+ b/a =
/
Il
wNo
Or
226 578 645 955 1
b/
—
9 9 9 9 9
ll
Ci CK)
Or
—_—— aa ——— ee 0 ee eet
bla = 224 574 640 ‘947 "992
0 ‘168 “429 479 ‘708 “TA2
(4) = 25 a= ‘167 4.27 ATT ‘705 ‘739
i) ‘166 “4.24 473 699 ‘733
0 134 342 381 “564 591
(5) = 25 +b/a = 133 341 380 562 589
5) 132 339 ‘378 ‘559 586
0 SWE 284 ‘B17 “469 “491
(6) \e = 25 |e = gl 283 316 468 “490
0) 0 110 ‘282 B15 466 “488
0
0)
0
0
0
0
0
0
0
0 221 565 630 932 977
0
0
0
i)
0
0
0
0
For the fourth and higher notes the table is complete only for positions of the layer
inside the third loop surface. The other positions of the layer in which dk/(n, —n)
vanishes in changing from negative to positive, being the same as the loop surfaces above
the third, are given for notes (4)—(6) in Table II.
§ 38. For the numerical magnitudes of the changes of pitch we must separately con-
sider the cases when the layer is of given volume and of given thickness. In the former
case the curve A of fig. 3 applies. This curve has its largest maximum ordinate at the
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 18]
origin, The numerical magnitude of the first maximum change of pitch may be obtained
from §§ 22 and 24. As explained there its values for the several notes and materials treated
here may be found by dividing by 9 the values assigned to Q in Table III. and equating the
resulte too = = s or
k \m+n V
Thus the change in pitch due to a given small alteration in n throughout a central
layer is numerically equal to one-third the change in pitch due to an equal alteration in
m throughout the same central layer. The fact that dk is opposite in sign to m,—n is
thus important practically as well as theoretically.
The abscissae answering to the subsequent maxima ordinates are the roots of a
complicated equation. The approximate values of the first few roots can be seen from
the figure or from Table VII As regards the higher roots it is comparatively easy to
prove that they split up into two sets, one set approaching the values (2i + 1)7/2, the
other set approaching i, where 7 is an integer. Answering to the first set are those
maxima for which ¢@k/(n,—m) is positive, to the second those maxima for which @k/(n,—n)
is negative. The number of negative maxima, including that for the central layer, is
equal to the number of the note and exceeds by 1 the number of positive maxima.
Tt is not difficult to prove that the successive positive maxima ordinates vary
approximately as the inverse squares of the corresponding abscissae, while the negative
maxima ordinates after the first vary approximately as the inverse fourth powers of the
abscissae. No great interest thus attaches to the numerical magnitudes of any but the
first positive and negative maxima ordinates which can be approximately derived from the
figure or from Table VII.
§ 39. When the layer whose rigidity suffers a given small alteration is of given
thickness the variation of ok/k with the value of kab is shown by curve B of fig. 3.
The equation determining the abscissae corresponding to the maxima ordinates is very
complicated. It is, however, easily proved that there are two sets of roots, the higher
roots of the first set being approximately odd multiples, and the higher roots of the
second set approximately even multiples of 7/2.
The first set supply the positive, the second the negative maxima ordinates. It is
easily proved that the positive maxima changes of pitch which answer to those of the
maxima ordinates which are most remote from the origin in the case of the higher notes
are all approximately given by
ok (b-cm—n\ _
T+ ( - man) =Q ee et ee (76).
They thus approach to equality amongst themselves and likewise to equality with the
; ok —cim—MmM\.
maxima of dls & me ) in the same notes.
k a m+n
The positions of the layer answering to the (t—1) positive maximum in the case
of n altered, and to the 7 maximum in the case of m altered are also when 7 is
large nearly identical.
24—2
182 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
The abscissa supplying the first and largest positive maximum ordinate is greater
than mw; thus the corresponding maximum change of pitch cannot apply to note (1).
This ordinate is greater than the maxima ordinates of curve B, fig. 2, by fully 50 per
cent. Thus the greatest possible change in the pitch of any note, except the first, due
to a given small alteration of throughout a layer of given thickness is fully 50 per
cent. greater than the maximum change of pitch in the same note due to an equal
alteration of m throughout a layer of equal thickness.
The abscissa answering to the first and largest maximum negative ordinate is
approximately ‘447, and the corresponding value of
Ok ary
UG at; m+n
This is a far from insignificant change of pitch, and it applies to all the notes in
slightly exceeds ‘29Q.
every material. In the case of note (1) it is the only trwe maximum there is, and
when o is small it is the numerically largest change of pitch which the given alteration
of rigidity can produce, If, however, « approach *5 an equal alteration of rigidity through-
out a layer at or near the surface of the sphere is more effective in altering the pitch,
and in this position the sign of 0k is the same as that of n,—n.
The subsequent maxima negative ordinates rapidly diminish as the corresponding
abscissae increase,
§ 40. We must next take into consideration the term im (n,—7)? in (50). Its con-
tribution to the change of pitch is given, writing # for kab, by
ok Ut Am NP %
ko a(m+m) (m+n) Qf.(2) | i
ak ail) Lee Does GORA Ea ae (77),
ne kV (m+n)(m+n)3 Jt)
according as the layer is of given thickness or of given volume. The term in (m—n)
indicates a fall in pitch whether the rigidity of the layer be increased or diminished.
The curves
y=f,(a), and y=f,(a)
are A and C' of fig. 1 respectively, the former of which was discussed in § 33. The zero
ordinates of both curves answer to positions of the layer comeident with the node sur-
faces. Of the maxima ordinates of curve A the first is much the largest. The corre-
sponding contribution to the change of pitch in the present case may easily be calculated
approximately from the curve and Table III. It is far from being insignificant com-
pared to the contribution of the term in m — when the alteration in rigidity is large.
As the subsequent maxima ordinates of curve A, fig. 1, rapidly diminish as their abscissae
increase, while the several maxima ordinates of curve B, fig. 3, remain large, it follows
that for an alteration of rigidity throughout a layer of given thickness the relative im-
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS, 183
portance of the term in (n,— 7)? rapidly diminishes as the layer moves outwards from
the first loop surface.
Exactly similar conclusions for the case when the layer is of given volume follow
from a comparison of curve C, fig. 1, and curve 4d, fig. 3. Of the maxima ordinates of
curve C, fig, 1, that at the origin is much the largest. In fact the second maximum
is so small that I have not attempted to draw the curve further than the first zero
ordinate.
§ 41. Our investigations show that for positions of the layer inside the first loop
surface the term in (n,—7) is in general far from negligible unless the alteration in
rigidity be small; but that in the case of the higher notes for positions of the layer
outside the first loop surface this term is in general comparatively insignificant even
when the alteration im rigidity is large.
It must, however, be remembered that the term in n,—n vanishes when the layer
coincides with a loop surface, whereas the term in (n,—7)? has its maxima when the
layer is at or very close to the loop surfaces, Thus, however small the alteration in rigidity
may be, when it occurs in a layer immediately adjacent to a loop surface the term in
(n,—n) is the larger of the two.
We thus arrive at the following conclusions.
There are certain volumes within a sphere performing any given note where any
alteration in rigidity’ throughout a thin layer lowers the pitch. As the term in (n,— 7)?
varies as (m+) the corresponding fall of pitch is greater when the rigidity is diminished
than when it is increased.
The principal volumes of this kind are in the immediate neighbourhood of the loop
surfaces L,, L,.... There are, however, similar volumes in the neighbourhood of the
surfaces S,, S,, ete. which answer to the roots of (75). The volumes surrounding two
adjacent surfaces S;, and L; may possibly in some cases when m—n is large become
coterminous, but when »,—7 is small they are certainly separate. An alteration of
rigidity throughout a layer within one of these volumes acts to some extent as what is
frequently termed a constraint.
In general terms it may be said that the existence of the term in (n,—7)? extends
the regions wherein an increase of rigidity lowers the pitch, and increases numerically
this lowering of pitch. On the other hand it restricts the limits of the regions -wherein
a diminution of rigidity raises the pitch and reduces numerically this rise of pitch.
§ 42. In our last special case the change of pitch is given by (52). For the limiting
value 5 of o this assumes the simple form
o : I a Ofsinekabmreecste terete etecehercrcnee <tc: (78).
Now the coefficient of p/(1+p) m (78) is the same as that of (m,—m)/(m,+n) in
(49). Thus the curves of fig. 2 and the conclusions already come to in the case when
m alone varies apply at once with merely a change in phraseology.
1 i.e. any alteration of elasticity which leaves m unaltered.
184 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
Except in this extreme case the coefficient of p on the right-hand side of (52) is
the sum of two squares.
Further as the equations sinw=0 and tan#=0 have no common root other than
zero, the two squares cannot simultaneously vanish unless b=0. Thus an alteration of
both elastic constants in the same proportion necessarily affects the pitch unless it occur
at the centre, and the pitch is raised or lowered according as the constants of the layer
are increased or diminished.
It will also be seen from § 22 that when such an alteration of elasticity occurs through-
out a core of given volume there is a change of pitch whose sign agrees with that of p.
Thus the statement that the change of pitch is of the same sign as the alteration of
elasticity is on the uniconstant theory universally correct as well as unambiguous.
§ 43. It will be convenient to suppose
Ok = 0k, + Ok,
where
ok, t 4n(8m—n)(1 /smkab _,.\)?
7 = a Pp ~(m+ny = ( rah — Cos kab ) m/ulpinelaimieteisiersia(a-cfeiminieta plate etqdsteteisiarsiei (79),
ok, _t p 1 ) ee cs Ly fT a { ) 2
k ai+tp Q = a {hab ae ara sees cos kab f Bcc dccaceee (80).
The numerical magnitude of @k, is independent of the sign of p, whereas dk, is
numerically greater for a given negative value of p than for an equal positive value.
Again ¢k, depends on the square of the displacement. It thus vanishes when the
altered layer is at a node surface, and when the layer is of given thickness it has its
maxima when the layer coincides with the loop surfaces. On the other hand 0k, depends
on the square of the radial stress. It thus vanishes when the altered layer is at a no-
stress surface, and when the layer is of given volume it has its maxima when the layer
coincides with those surfaces over which the radial stress is a maximum.
Again the law of variation of @k,/k with kab is wholly independent of the value
of ¢, but the absolute values of 0h,/k diminish rapidly and become inconsiderable as o
approaches near the limiting value ‘5. On the other hand the law of variation of 0k,/h
with kab varies with the value of ¢, and this is very conspicuous in the case of the
fundamental note, or so long as b/a is small in the case of the higher notes.
Perhaps the most important difference of all is that in the case of the higher notes
when the layer, supposed of given thickness, travels outwards from the third node surface
ék,/k becomes rapidly insignificant, whereas 0k./k has a succession of important maxima
of nearly uniform magnitude and nearly independent of o. By supposing the layer of
given volume we should come to precisely the same conclusion as to the relative pre-
ponderance of ok, when the layer is outside the third node surface. An exception must
of course be made of positions of the layer immediately adjacent to the no-stress surfaces
where dk, vanishes.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 185
§ 44. To obtain some idea of the numerical magnitude of the change of pitch we
must consider separately the cases when the layer is of given volume and when it is of
given thickness.
In the former case, with the usual limitation as to the centre, writing « for kab,
Le AG) - ‘= * — cos o\F sooponcenconGanoBasouDDE (81).
a
pe VAS (m+ ny?
The variation of 0k,/k with kab is thus shown by /f,(v) as tabulated in Table VIL,
and by curve C, fig. 1, for values of w less than 32/2. This curve has by far its largest
maximum ordinate at the origin. This ordinate is by no means insignificant. It has also
in the present case to be multiplied by 4n(38m—n)(m+n)~%, a quantity which varies
between 2 and 9/4 for values of o less than °3. ‘Thus the corresponding change of pitch
is of considerable importance in ordinary isotropic materials. So long in fact as « is less
than 7 the ordinates of curve C, fig. 1, are fairly comparable with the ordinates of the
other curves which apply when the layer is of constant volume.
For positions of the layer, however, answering to points beyond the first zero ordinate
of curve C, fig. 1, 0k, is always extremely small. It is in fact easily proved that the
second maximum ordinate is less than 1/134 of that at the origin.
Still supposing the layer of given volume, we have with the usual limitation, writing
« for kab,
Cheep OV OL fee An, Aaa NP 5
ea as T 3 E Ran IEET Al = — cos.x) Melcicraratetoreseeiclerereiete (82).
The function of « inside the square bracket reduces when ¢='5 to a sin2z, the
quantity appearing as /f,(#) in Table VII., and represented by curve 4, fig. 2. This curve
has been already exhaustively considered. The function is also tabulated for the values
0 and ‘25 of o in Table VII. under the headings f,(#) and f,(#) respectively. The cor-
responding curves are A, and A.., of fig. +.
The differences between the three curves last mentioned are very conspicuous near
the origin.
For small values of « the ordinates of curve C, fig. 1, are comparable with the ordinates
of the curves mentioned above. Thus in comparing the changes of pitch due to a given
percentage alteration of elasticity for different values of o we must, at least when the
altered layer is inside the second node surface, construct compound curves of the kind
mentioned in § 30,
The compound curves showing the variation with kab of
Glee (i Duat Oe _
k =(j ip Vir
are found as follows :—
186 Mr C, CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
when ¢=0, multiply the ordinate of curve C, fig. 1, by 2(1+p), and add it to the
ordinate of curve Ay, fig. 4
when ¢=°25, multiply the ordinate of curve C, fig. 1, by 20(1+~p)/9, and add it to the
ordinate of curve A..;, fig. 4
when o=°5 there is the simple curve 4, fig. 2.
When the alteration in elasticity is small we may neglect p in forming the compound
curves, i.e. replace 1+p by 1 simply.
In deducing the numerical value of 0k/k for a given value of p the ordinate of the
corresponding compound curve must be multiplied by that value of Q’/3 which applies to
the note and material under investigation.
Since the largest maximum ordinate in all the compound curves occurs at the origin,
it will be found simplest when the greatest possible change of pitch alone is wanted to
apply at once the result obtained in § 24, replacing ¢H; by
oV pp) B8m—n 144 toni
V l+p 3(m+n) 3P int nf”
§ 45. The three curves A, fig. 2, A, and 4.,;, fig. 4, become extremely similar when
The equation for the abscissae supplying the maxima ordinates im these curves is
ele Wr 4
ee — cosa + S22) = le sina —3 feed — cos »)} = Ugoerercecocee (ei)
z l-o 7
q
aL
For ¢="5 the roots of (83) are identical with those of tanzw=w, and for all other
values of o the higher roots of (83) though less than the roots of tanw=a are very
nearly equal to them.
Thus the more remote positions of the layer answering to the maxima values of
ck./k in the case of the higher notes lie close inside the successive node surfaces, except
for the limiting value 5 of « when they exactly coincide with the node surfaces.
The first root of (83) other than zero varies from 1:2327 when c=0 to 14307
when o="5. Thus the position of that maximum ordinate which lies between the first
and second zero ordinates varies to an appreciable extent with the value of c.
There is also an appreciable difference in the lengths of this ordinate in the three
curves, these lengths unlike those of the ordinates at the origin increasing as o diminishes.
Beyond the second zero ordinates the curves would lie very close together, so in fig. 4,
curve A.., stops at this point.
For values of @ exceeding z, 6k,/ék, is very small except for such positions of the
layer as make ok, insignificant. Thus for practical purposes the dependence of dk/k on
the position of the layer, when close to or outside of the second node surface, is ap-
proximately given for the values 0, ‘25 and ‘5 of o by the curves A), Aw, of fig. 4 or
A, fig. 2, alone.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 187
Except in the case of the first one or two maxima no serious error will be introduced
by supposing the positions of the layer which supply the maxima changes of pitch to
coincide exactly with the node surfaces.
These maxima are also approximately given by the formula which in strictness applies
only when o ='5, viz.
a 1 +p V 3 (1 +2?) Bee enter eens eee ee rests ee eeseressses (84).
Here 7 is the number of the note and « is that root of (36) answering to the particular
node surface, at or close to which the layer is found.
§ 46. We shall next suppose that the layer is of given thickness. We may regard
ok as consisting of two terms given by (79) and (80). Of these the variation of 0h,
with kab is shown by curve A, fig. 1, while the variation of dk, is shown for the values
0, -25 and ‘5 of o by B,, B.., fig. 4, and B, fig. 2.
It is obvious from these curves that for values of «2 exceeding 7, 0k, is small com-
pared to 0k,, except very near the vanishing positions of the latter quantity, and the
value of dk, depends but little on the value of oc.
The exact positions of the layer supplying the maxima changes of pitch in the limiting
case represented by curve B, fig. 2, are the positions given in Table XI. for o=°5. In
this case all the maxima for any given note are equal, and their numerical values are
obtained at once from the formula
Chine Spa,
k al+p
In the third segments there is a difference only of something like 1 per cent. between
the lengths of the maxima ordinates of the curves By, B.;, fig. 4, and B, fig. 2. Also
these maxima are near the zero ordinates of curve A, fig. 1, representing the variation
in 0k,. Thus by altermg the heading of Table III. from Q to = (; 1 a
the numbers given for notes (1) and (2), we obtain what are extremely good approxi-
mations for the third and subsequent maxima, for the values 0 and ‘25 of o as well as
for o=°'5. Even in the case of the second maxima the magnitudes derived from this use
of Table III. would not be seriously in error,
) and rejecting
When kab is small the dependence of the law of variation of 0k,/k on the value of
o is so conspicuous im the figures that further comment is hardly necessary.
§ 47. For even a rough approximation to the change of pitch when the layer is
inside or but little outside of the first loop surface we must construct compound curves
for the values 0 and ‘25 of co. ‘These are formed by combining curve A, fig. 1, with
the curves B, and B., of fig. 4, m precisely the same way as the compound curves in
the case of a layer of constant volume were formed by combining curve (C, fig. 1, with
the curves A, and A.,, of fig. 4.
Vou. XV. Part Il. 25
—
wo
io 6)
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
If we suppose p very small the greatest ordinate that either of the compound curves
supplies for values of « less than 7 is very considerably less than 1, which is the approxi-
mate value of the subsequent maxima ordinates. Thus for a small alteration of elasticity
there is in the case of the higher notes no position of the layer inside of or close to
the first loop surface which can produce as great a change of pitch as the positions near
the second and subsequent node surfaces. For note (1) however none of the maxima
answering to positions near the node surfaces apply.
For ¢=0, z/7 must be less than 6626 to apply to note (1). Now it is easily
found that when p is neglected in the equation, the compound curve for ¢=0 runs very
nearly parallel to the axis of « between the values ‘67 and “66267 of « The corre-
sponding ordinate is approximately “381, and is greater than any ordinate answering to
a smaller value of a.
Also for o=0 the value of Q im note (1) is 2:253. Thus the maximum change of
pitch due to a very small alteration of elasticity, in a layer of given thickness, in the
ease of note (1) for « =0 is approximately given by
The corresponding position of the layer is at or close to the surface of the sphere. This
result is in accordance with Table V.
For «='25 the compound curve when p is neglected in its equation has a true
maximum ordinate for a value of # answering to a position of the layer at some distance
inside the first loop surface. The length of the ordinate is “58 roughly. Thus as Q when
o=°'25 has the value 1369 for note (1), it follows that the maximum change of pitch
in this case for a very small alteration in elasticity throughout a thin layer is approximately
iven b
ilies okt
The greatest possible percentage change of pitch in note (1) for given values of p
and ¢ is thus less when o equals ‘25 than when it equals 0 or °5.
When p is large the form of the compound curve near the origin will vary widely
from the form it takes when p is small. When p is positive the compound curve is
the more influenced by the form of curve A, fig. 1, the larger p is, whereas when p is
negative the influence of this curve continually diminishes as p increases numerically.
§ 48. In the case of the higher notes a pretty close approximation to the change
of pitch due to any alteration solely in elasticity, occurring in a layer outside the third
or fourth node surface and not in the immediate neighbourhood of a no-stress surface, is
easily obtained by the following considerations.
Comparing (26) and (27) we see that when kab is large u, and U;, except when
negligible may be replaced respectively by
uy =—b> coskab, U,=b? (m+n) kab sin kab.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 189
Thus, noticing (25), we see that when the elasticity alone is altered the terms in (™)*
and u,U, im (28) may in general be neglected when kab is large, and that an approximate
expression for the change of pitch is then
ok _b—c ( 1
k a \mt+n m+n
) (a1 4 1) Q Sin? Ba ..-oceceseosecsscsaees (85).
Near the no-stress surfaces the terms in (wm)? and ,U, cease to be small compared
to the term in (U;)*, but their greatest values being small compared to those of the
latter term, this limitation to the applicability of (85) is not of much practical importance.
We thus see that in the case of the higher notes when the alteration of elasticity
oceurs outside of the third or fourth node surface the change of pitch, when of practical
importance, may be regarded as depending mainly on the alteration of only one elastic
quantity, viz. m+n.
It will be remembered that when a small alteration of elasticity occurs near the centre
the change of pitch may be regarded as arising from the alteration in the single elastic
quantity m—n/3; and in the case of note (1), for a surface alteration of material, there
is for ordinary values of o a not inconsiderable change of pitch depending on the alteration
of the single elastic quantity n (3m — n)/(m +n).
It thus appears that in any purely verbal explanation of the phenomena such terms
as stiffness or elasticity would require to be used in a very elastic sense.
Note. August 7, 1891.
[When the rigidity is altered while the bulk modulus m—n/3, and so the com-
pressibility, is unaltered, the change of pitch is given, writing x for kab, by
av m—n Q
i Yt fale);
= te” Of f(a) = 9
where, as in Table VIL, fi, (7) = 2°f,; (x) = {sin x — 32 (a sin & — cos 2)}*.
ak |e = =
am+ny
V min
So in this case the change of pitch is always of the same sign as the alteration
of rigidity.
The variation of 0k/k with the position of the altered layer is shown by A or by
B, fig. 5, according as the layer is of given volume or given thickness. For comparison
with the effects of other alterations of material the ordinates of these curves should be
increased in the ratio 4: 3. When so increased the first maximum ordinate of B is the
largest ordinate in any of the curves. It answers to an abscissa of 1:247 approximately,
and so never applies to note (1). The extremely flat character of these curves near the
origin calls for special notice.]
25—2
190 Mr C, CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
SECTION III.
TRANSVERSE VIBRATIONS IN SOLID SPHERE.
§ 49. By transverse vibrations are here meant vibrations in which there is no radial
displacement,
Let p be the density, x the rigidity, of an isotropic material, and
SVSNp NMR a TEL sete etindes alana. cathe (1).
Also let Ji:;(@), J_-«4(”) represent the two solutions of the Bessel’s equation
dy 1 dy f 2 (4 2) —
aie ie es (UE 7} SU re wccccoesemmeigsesteatrsossesee (2),
where 7 is a positive integer.
Then the types of the displacements v and w, respectively in and perpendicular to
the meridian plane—or plane containing the line @=0—in a transverse vibration of
frequency //27 in a simple shell are*
v = cos kt r (sin 6) {Xi ing(hBr) + XJ (45) (HBr) } 0c eeceveerseeeenee (3),
w = cos kt r+ {wJis4(bBr) + wider (EBT) ....ceecneceececscncvessececccees (4).
Here X;, X’; are surface spherical harmonics of degree 7, while w;, w’; are quantities
connected with them by the relations
du; _ aX; duis aX i“
ip a eden ema doy ee cee }
The spherical harmonics X;, X’; must be of such a type that v is nowhere infinite,
and so—at least for a complete shell—must contain sin @ raised to some positive power.
Under (3) and (4) we may suppose included the type of vibration
hs (6):
w = cos ktr- {w, sin 0Jy (kBr) + w’, sin OJ_g (k@r),J
in which w, and w’, are constants, and so w, sin@ and w’,sin@ may be regarded as equi-
valent to the quantities w, and w’, satisfying (5). This special form of vibration will
here be spoken of as the rotatory, this term being applied to it by Professor Lamb+.
At the spherical surface separating two isotropic media there are in this case nominally
four surface conditions, viz. the equality in the two media of the two displacement com-
: ; dv ov dw w
ponents v and w, and likewise of the two stress components n (5 -%) and » ae ~ ~) ;
In consequence however of the relation (5) these constitute in reality only two in-
dependent equations.
* See Camb, Phil, Transactions, Vol. x1v. p. 319, equations (34') and (35’).
+ Proceedings of the London Mathematical Society, Vol. xm. p. 196.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. Ii
A moment’s consideration will also show that the X; and the X’; in the v displacement
in any layer of a compound solid sphere must be the same function of 6 and ¢, and
that this function must be the same for all the other layers and for the core. We may
thus represent the w displacements in the typical vibration of frequency k/27 in the
compound sphere (0.8.¢.8,.b.8.a) as follows:
In the core aurt lw: COB Kt = Alexa (KOM) mecteccenceesececcecetaececte+s+ces (7).
In the layer wri/w; cos kt = AJ: (GBir) + Bid tery (GBir) ..ecncecnecseceeceees (8).
Outside the layer
wr?/w; cos kt = (A;+0A;) Jizg (hBr) + OBi Sig) (MBI) -.ceecceecceereceeee (9).
Here 4A;, ,4;, etc. are constants whose relationships are determined by the surface
conditions, and w; is a certain function of 6 and ¢. If we suppose b—c small then
0A,/A; and 0B;/A; are of the order b—c of small quantities, and their squares are
negligible when that of b—c is neglected.
It is unnecessary to write down the expressions for the v components in the several
media as they lead to precisely the same conditions at the surfaces as the w components.
- § 50. Let us for shortness put
Sieg Br) = By © Sins (80), Sx (0B) = py © I-ceny (080)
43 (4 > iB dr i43 (PT), JS (44) 7 ~ kB dr —(i+4) (APY),
FRG Berd cnx GBT) via 3), \ Sp ennonhenueanededast (10).
F,(r.B) =n {kBrJ'_¢sy (kBr) — 35 Bry}
Then we find from the surface conditions
A: Si+4 (kBc) = A iz4 (kB,c) ate Bid 43) (kB,c),
A;F (ec : B) a iA: F (ce : B,) + BF, (c. B,),
(A; + 0A;) Fix; (kBb) + OB T_(:43) (ABD) = Aig (KBD) + Bid 1:44) (KB1b),p -..(11).
(A;+0A,)F (6.8) +0B;F,(b.8) =,4:F(b.B,) +,B:F, (6. B),
(A;+04;)F(a.8) +0B;F,(a.8) =0
Treating the first four of these equations in the usual manner, and putting
A (b = B . b’) = Sixx (kBb) J’ ey (kBb) = J i434 (kb) J _ci+y) (kBb) occcccccccccs (12),
we find
nkBbA (b. 8. b') + —{—
= — {nk? Bb? — a + (nm —n) (¢— 1) (6 + 2)} Jizy (KBD) Tex) (kB)
- (; - Je n {kBbI";.4 (kBb) — 844 (k8b)} n {kBOI (243) (kK 8b) — 8J_ 1244) (K8)} ...(13),
a nkBbA (b. 8.0!) = —— = {nk Bb? — nk? Bb? + (mn, — n) (i — 1) (1 + 2)} {Tiss (hBb)}P
x Ke --) [pe (Gb 24 (EG BV= fog (RGD) PF ee cctscestoss ck). --s- (14).
192 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
For the frequency equation of a simple sphere we find from the last of equations
(11), putting 0B; = 0,
F(0.B. a) = kBad 44 (Ba) — 8S ing (HBG) = 0 oo. ce ccensee eee eees (15)*.
From the aol of the Bessel’s function
- kB = FO. B.a) = {keBea® — (i + $)*} Jiny (ha) + 8hBad"s 4; (hBa)........(16).
ks
Supposing (15) to hold we may reduce (16) to
— kBa adi foletay 2 asew Le 1) Pape aay ee. (17)4.
Supposing (15) to hold we also obtain
FP, (a. 8) = nkBad (a. 8.0’) + Six, (kBa).
Thus, following the same train of reasoning as in Sect. IL, we conclude that if 0k
be the increase in & due to the existence of the layer, the two following equations must
be identical—
F(0.B.a) ae - {ke "Bra — (i — 1) (+ 2)} Jizy (hBa) = 0,
fO.8 we oP Bad (a. B.a@) + Ji; (kBa) = 0.
Thence we find for the change of frequency
ok 0B; kBaA (a. B.a@’) {Ji44 (kBa)}
k 7 Als k2B'a? — (i — 1) (i + 2) Cece ccc ccc ccc ccecsecnvesece
Let Es Kg.q denote the frequency of the free transverse vibration of the type (3)
and (4) in an infinitely thin spherical shell of material (p.n) and radius a; then
Rg 9 @ = (6 — 1) Gab?) BS Nt Desi |p th. dc ieee (19)8.
Also let pina i eee ), !
W, = nr (kBrJ'i44 (kBr) — 8 cx4 (kBr)}
so that w,w; coskt represents a w displacement in a simple sphere performing a transverse
vibration of frequency k/27, and W,.w; coskt the corresponding stress, both quantities re-
ferring to points at a distance 7 from the centre, and w; being a function derivable from
a surface spherical harmonic of degree 7 through the relation (5).
Employing this notation, introducing in (18) the value of @B;/A; from (14), and
noticing that
kBDA\(b.5 8. 0) = KROL G8 Ge) — 1 reenter teers eee (21),
* Cf. Transactions, Vol. x1v. p, 316, equation (47 a). + See Sect. 1. § 10.
§ Cf. Transactions, Vol. x1v. p. 320, equation (59), and (17) above.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 193
where C is an absolute constant, we finally obtain
ok (b—c) & 1 haga 35 : 2 . Wy \? 1 1\ /W;?
kh a a a p (ke — K@. 33) [toc —K .d)) | Pi (k —-K B,.0))} =) + (=- = Ge) |
This may be applied with the same limitation as in Sect. II. to the case of a central
layer.
§ 51. Inside the layer there is no change of type other than a shifting of all the
node, loop and no-stress surfaces according to the law
= OFF i Oh] RM eee Ne ane sce (23).
Outside the layer we find on substituting in (9) the values of @4,/A; and 0B;,/A;
from (13) and (14) and reducing,
w/ Aw; cos kt = ae a (kBr)
b-c/b)* ene
+90 (5) | fp G— Kran) — ps — K*y,0)} boas @. 8.7)
{1 1 ms
+ (ae -) WaiOaer "| dials Selgdelc ovaijeccieeinns (24);
where FO a B 4 r) = Sixt (kBr) J_ (+3 (kb) = J_(+y (kBr) Ji+3 et (25)
Ff (0.8.7) =Jisg (Br) F, (b. B) —I_iizy (Br) F (0. B)
The functions f have their usual meaning.
In the layer itself the displacement is given by
w/ Agu; cos kt = b*F;..3 (hBb) — kB (b — r) {bP I". (le8b) — (2h 808) J (eB) }
Dey BN aes ,
Gee (; ud = i pideree eee (26).
The change of type outside the layer, ie. the coefficient of b—c in (24), consists
like the expression (22) for the change of pitch of two terms only. The first terms in
each alone exist when the layer differs from the remainder only in density, and they vanish
when the layer coincides with a node surface. The second terms vanish when the layer
coincides with a no-stress surface. In the special case of the rotatory vibrations the second
terms alone exist when the layer differs from the remainder only in mgidity.
In the layer itself the change of type is given by the last term of (26). Thus
if there be an alteration only in density, or an alteration of rigidity occurring at a no-stress
surface, then no progressive change of type appears as we cross the layer; in other words
the layer vibrates as if it were of the same structure as the core.
§ 52. Before discussing (22) it is desirable to trace the ‘characteristic features of the
transverse vibrations of a simple sphere. The type of such vibrations is given by (3)
and (4) with X’;=0=w’;, and the corresponding frequency equation by (15).
194 Mr C, CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
If ¢ be a large integer X; may be any one of a large number of spherical
harmonies, but (15) depends solely on 7% on the radius of the sphere, and on the
material. There may thus be a large number of different forms of vibration which have
all the same frequency equation.
The displacements vary, unless 7=1, with @ and @ as well as with 7. Thus there
is a conical surface, or a series of surfaces, given by
Re TB, «cds ren al (27),
over which the component of the displacement in the meridian plane vanishes. Similarly
there is a conical surface, or series of surfaces, given by
over which the component at right angles to the meridian plane vanishes. A line of
intersection of (27) and (28) is a locus where the resultant displacement is always zero.
While the title node surface might legitimately be applied to the lines or conical
surfaces which are the intersection of (27) and (28), it will here be understood to apply
solely to the spherical surfaces over which the displacement vanishes. Such surfaces we ~
see from (3) and (4), putting X’;=0, are obtained by equating /8r to the successive roots of
Jix3 (2) ENS e cis calhet'comerscte se aie nenene eee ceeeneeee een (29).
Thus for a given sphere the positions of these surfaces depend solely on the number 7
of the spherical harmonic X;, and in no respect on its form.
In like manner there are spherical loop surfaces, obtained by equating k8r to the
successive roots of
J'44(@) — = i43(2) AM wks Sctauisaese'c nate SoG ascent ne eeere (30),
where the displacement regarded as a function solely of 7 is numerically a maximum.
There are also spherical no-stress surfaces, obtained by equating k8r to the successive
roots of
4 3
Jf i43(Z) — 5, Jits(@) = 0 See cece es ee sree essen s seen esescsescers (31),
at every point of which the transverse stress is zero.
In a given sphere the radii of the several loop and no-stress surfaces depend, like
those of the node surfaces, entirely on the number 7, and in no respect on the form
of the spherical harmonic X;.
The above equation (31) is of course identical with (15), but for certain purposes
its present form is more useful.
§ 53. Since the equations (29), (30) and (31) do not contain p or n it follows that
the nature of the material, supposed of course isotropic, has no effect on the ratios of
the frequencies of the several notes answering to a given value of 7, or on the mutual
ratios of the radii of the node, loop, or no-stress surfaces of given number, or on the
ratios of these radii to the radius of the sphere.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 195
As regards the form of the Bessel’s function Jj,,(@) we know that
edly
J, (x) = “i, = (— * — cos 2) ae ee i Kn a A, ht (32)*,
J, (a) ae \(2- 1) sin @ —? cos ol REGEN OD OLN COB ee oon CnC OCEe (33)* ;
and between any three consecutive functions there subsists the well-known relation
(2¢+ 1)Jisy (7) =2 {Ji (a) + Jizg (a)} petetalelelatelaielteletstevetetetsleisleiarelsinveretate (34).
If the value of z be large a close approximation to the value of these functions
is supplied by
Sy .
—Jisy (z)= es sin (= = z) SSO OO SE SCCOOCCECCCOOOUOO COMO SOCnG (35) +.
From (35) we see that the higher roots of (29) are given approximately, 7 denoting
a positive integer, by
Di) (2p) on Decale seiaseeets nesses acer staasee (36),
or EDC ge mis acribo aopSAnOAOS dosed DANES iba shou MOS Go GOoHCSOae (37),
according as 7 is odd or even.
Again, the roots of both (30) and (31) obviously approach more and more nearly
the higher they are to the corresponding roots of
J’ i43 (z) Saal) ors Siatepricieinrarovetaistetersvarcie ste eislearsreeys Risaisone scsarehe (88),
and from (35) it is easily seen that the higher roots of (38) le approximately midway
between consecutive roots of (29). Thus the higher roots of both (30) and (81) are
more nearly given the higher they are by
2— Yi
or 2 =(2) + 1) 7/2,
according as 7 is odd or even.
Again, from (35) it follows that those maxima values of « {J;,,(@)|? which answer to
large values of # are all approximately equal 2/7, and that the corresponding values of
g are approximately given by (37) or by (36) according as 7 is odd or even. In like
manner we conclude that the maxima values of {J;,;(7)}* which answer to large values
of x vary approximately inversely as the corresponding values of z, and these values of x
are likewise given approximately by (37) or by (36) according as 7 is odd or even.
* Lommel, Studien iiber die Bessel’’schen Functionen, p. 118.
+ See Todhunter’s Functions of Laplace, Lamé, and Bessel, Arts. 406—7, especially equation (9), p. 313.
Vou. XV. Parr II. 26
196 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
§ 54. From the data obtained for the approximate positions of the roots of the
equations (29), (30) and (31) we may draw the following conclusions :—
The pitch of the higher notes in a given sphere answering to any given value of
i increases approximately in an arithmetical progression with the number of the note. In
any one of these higher notes the corresponding no-stress and loop surfaces of higher
number lie very close to one another, and are very nearly midway between successive
node surfaces. The radii of successive higher surfaces of the same kind, whether node,
loop or no-stress surfaces, increase very approximately in arithmetical progression.
§ 55. Before discussing the general application of (22) it will be convenient to
consider the special cases when the change of material occurs at or close to the centre,
and when it occurs at the surface,
Supposing first the change of material to take place throughout a central layer, we
require to find the dimensions of the lowest powers of b occurring in (22).
Employing the ordinary formula for the Bessel in ascending powers of the variable,
we see that when Db is very small the most important terms in the coefficients of p,—p
and n,—n respectively in (22) are of orders (b—c)b***a-** and (b—c)b*™a-*), Also
(¢—1) occurs as a factor of n—xn. Thus even when 7=1, (6k/k)+(0V/V) is of the order
(b/ay of small quantities. Thus to the present degree of approximation no alteration of
material whatever, occurring throughout a central layer whether of given thickness or
given volume, has any effect on the pitch of any note of any transverse type.
Working out independently the case when the material (p,, m) forms a true core, I
come to exactly the same conclusion.
Next, making b=a in (22) we obtain the change of pitch due to an alteration of
material throughout a surface layer. Putting b—c=t, and remembering that W,=0 for
a simple sphere, we find
m—7n
ee @—P _ G1) +2)-
OR 58 |e De de I ate (39).
k a k*B°a? — (¢-— 1) (4+ 2)
When 7=1 the change in frequency depends solely on the alteration of density. For other
values of 7 it may be regarded as composed of two terms, the first giving the effect of
a surface alteration of density, the second of a surface alteration of rigidity. The denomi-
nator in (39) is essentially positive; thus the pitch is lowered when the density at the
surface is increased, and raised, except in the rotatory vibrations, when the rigidity is
increased.
Since the values of k8a supplied by the frequency equation are the same for all
isotropic materials, it follows that the percentage change of pitch due to a given surface
alteration of density is quite independent of the rigidity; and similarly the percentage
change of pitch due to a given surface alteration of rigidity is independent of the density.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 197
Putting Pith Pia lice Oona: «tates sheets, Mapeshaass waatieshemens (40),
1/1) me nnete cate acceee testa ietnerevsctecesetes es (41),
we find from (39) when g=p
ok t
Th =— me Cece cern ener ere scceecsesarerercseseseveceseee (42)
In the fundamental note answering to any given value of i greater than 1, the
effect on the pitch of equal percentage alterations in the density and in the rigidity are
fairly comparable. The higher however the number of the note the smaller is the relative
importance of the alteration of rigidity, and the more nearly is the change of pitch given by
Dp B=!
k @ p
In the case 7=1 this result is exact for all the notes.
§ 56. We shall next suppose the position of the layer to be any whatever, but the
alterations in density and rigidity to occur separately. As in either case the change of
pitch vanishes for an altered core, we may without restriction replace
(b—c) B/a® by 10V/V
pPi—p(b—c) Bb
p a
and by toM/M.
When the density alone is altered in the layer we have, according as the volume or
the thickness of the layer is given,—
ok oM
ar 7a aa Mu - {(kBb) * ie y Jig (kBb)}? Seen eee e renee seeeeeeee (44 ),
ok
7 = “i Q {(keb)? (5 ie PEAERb) ates. secthon tied (4b);
where y= ERa +P 9) ((kBa)* ei Jes (EB) |= te actcoewenee (45),
Q'= ONUBRY IE, Bade eons, le ee (46).
When the layer differs from the remainder only in rigidity we have, according as it
is of given volume or given thickness,
e ee: my — sa « ~1)(6 +2) (kab) (3) Jin (k8b)}2
+ poy (3 7)" Fisa(hBb)— § (H86)-4 (F) Feuthedy*| (47 a)
=, mm
mem" | E+) (04 (Z) Jia o8d))
+” (G86) (2) J'eax (Bd) — 9 680)-# (™) Say (180) sressse( AT B),
198 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
It will be noticed that the several expressions depend on 7% but are wholly inde-
pendent of the form of X;. Thus the change of pitch accompanying any such alteration
of material as is here considered is the same for all possible forms of vibration which
have the same frequency.
In (44a), (440), (47a) and (47b) the expression for the change of pitch consists, like the
expressions in the case of the radial vibrations, of three factors. The first measures the
magnitude of the alteration of material, the second is Q or Q’/3 according as the layer is
of given thickness or given volume, and the third gives the law of variation of the change
of pitch with the position of the layer.
The variation of the third factors with z,=k8b, may be shown by curves which apply
to all the notes answering to a given value of 7. These curves are as follows:
For a layer of altered density of given volume
4
y = (a G) Tee @ ai@ nee ee ee (48).
For a layer of altered density of given thickness
YEH) = 7s (@) ocoscnvecsoocsananoncossanan9nscccqgecoc- (49).
For a layer of altered rigidity of given volume
‘ : n m\* 7\*
y= = WO+2) ef, (0) + 2 ford (5) Jia (@) — 30-4 (5) Ig @OPaho eas (50).
For a layer of altered rigidity of given thickness
Mtr 2) =a (roccsnooassosonsenbac doocodeDscooDaINS0006 (51).
When the value of 7 is given, and in the case of (50) and (51) the magnitude of
the alteration of material, the lengths of the maxima ordinates of these four curves are
numerical quantities which are independent of the number of the note. Thus the maxima
percentage changes of pitch of any given number—i.e. the changes answering to a certain
definite maximum ordinate—in the different notes which answer to a given value of %,
vary as Q’ or as Q according as the volume or thickness of the layer is given.
Now the values of k8a for the notes of higher number are very near the roots of
(38), and so are very close to those values of # which make {J;,,(«)}* a maximum. Thus
by the same reasoning as in § 53 we conclude that in the notes of higher number
J;.,(k8a) varies more and more nearly as (k8a)~ the higher the number of the note. For
the definition of a Bessel to which (32) and the approximate form (35) relate we get
for the higher notes J;,;(k8a) =V2/rkBa approximately.
Again the factor k*8?a* + {k*B’a* — (it —1) (i + 2)} approaches more and more nearly to 1,
the larger kfa, i.e. the higher the number of the note.
We thus conclude that in the higher notes answering to a given value of 7, Q’ varies
more and more nearly as (k8a)* the higher the number of the note, whereas Q continually
approaches a finite constant value. With our definition of a Bessel we have for these
approximate values Q! =/#A?a?, Q=1.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 199
We have also seen that according as 7 is odd or even the higher values of k8a
approach to jr or to (27+ 1) 7/2, where j is a positive integer.
Thus for a given alteration of material throughout a layer of given volume the maxima
percentage changes of pitch of any given number in the case of the higher notes answering
to a given value of 7, vary approximately as j’? or (2) +1)? 7°/4 according as 7 is odd
or even. In other words the maxima percentage changes of pitch of any given number
in the case of the higher notes are such that their square roots increase approximately
in an arithmetical progression with the number of the note.
On the other hand for a given alteration of material throughout a layer of given thick-
ness the maxima percentage changes of pitch of any given number in the case of the
higher notes answering to a given value of 7 are all nearly equal.
§ 57. When the layer differs from the remainder only in density we see from (44a)
or (44b) that the law of variation of the change of pitch with the position of the layer
is always independent of the magnitude of the alteration of material.
The change of pitch vanishes when the layer coincides with the node surfaces, and
for all other positions the pitch is raised or lowered according as the density is diminished
or increased,
When the layer of altered density is of given volume the curve showing the variation
of ok/k with kab is (48). The abscissae supplymg the maxima ordinates are easily seen
to be the roots of (30). Thus the positions of the layer supplying the maxima changes
of pitch coincide with the loop surfaces.
Since the larger values of # answering to the maxima ordinates approach more and
more nearly the larger they are to the roots of (38), our previous reasoning shows that
the lengths of the successive maxima ordinates of higher number vary more and more
approximately the higher the number as the inverse squares of the corresponding abscissae.
Thus the maxima changes of pitch of higher number in any given note diminish very
rapidly as the radius of the corresponding position of the layer increases.
From a consideration of (440) and (49) we similarly conclude that when the layer
of altered density is of given thickness the positions im which it is most effective lie
outside of but close to the successive higher loop surfaces. Also the successive maxima
changes of pitch of higher number in the case of any given note are all approximately equal.
From the preceding results we may take as approximations to the maxima of higher
number in the higher notes answering to any value of i—
_ —0k 1 0M say? ; : :
for a layer of given volume a =a Gi): where b is the radius of the corresponding
position of the layer,
for a layer of given thickness — ¢k/k = Puma
200 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
o
§ 58. When the layer differs from the remainder only in elasticity the change of
pitch depends solely on the alteration of rigidity.
In this case we see from (47a) or (476) that, unless i=1, the expression for the
change of pitch is the swm of two squares which cannot simultaneously vanish except
when 2=0. Thus unless in the rotatory vibrations an alteration of rigidity occurring any-
where but at the centre necessarily affects the pitch, and the pitch is always raised or
lowered according as the rigidity is increased or diminished.
When the layer of altered rigidity is of given thickness the curve giving the variation of
0k/k with kab is (51). The form of the curve, unless 7=1, is dependent on the nature of the
material and varies with the magnitude of the alteration of rigidity. Thus in an exhaustive
investigation it would be advisable to construct two simple curves answering to the two
terms in (476). The first curve would be the same as (48), the second would be
y= atk eT J say (a) — 3a eines (Cay npn co (52).
Adding the ordinate of (48) multiplied by (¢—1)(¢+2) to the ordinate of (52)
multiplied by n/n, we should get a compound curve as on previous occasions.
For small values of 2, and so for all positions of the layer in note (1), or for
positions near the centre in the case of the higher notes answering to a given value
of 7, the contributions of (48) and (52) to the compound curve will be of like order
of magnitude,
Outside however of the third or fourth node surface in the case of the higher notes
answering to a given value of 7, the contribution of (48) to the compound curve is
always small.
On the other hand when @ is large (52) becomes almost identical with the curve
3
y= {ae (5) So CS ee te AI lee. (53),
and the successive maxima ordinates of higher number of this curve continually approach
a finite constant value, viz. 1. The corresponding values of « are close to the higher
roots of (29), which answer to the node surfaces. For the maxima changes of pitch of
higher number in the higher notes we may practically leave (48) out of account and
take as an approximate formula, for all values of 7, oa me a
When the layer of altered rigidity is of given volume we come to precisely the same
conclusion as to the relative importance of the first and second terms of (47a); and it
is easily seen that when the layer is outside of the third or fourth node surface in
one of the higher notes answering to a given value of 7, there are a series of maxima
changes of pitch answering to positions of the layer near the higher node surfaces which
depend almost entirely on the second term.
These maxima are however usually insignificant compared to the maxima which depend
essentially on the first as well as on the second term of (47a). Thus in the case of
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS, 201
a layer of given volume the comparative insignificance of the first term for large values
of « is not of much practical importance.
Unless the altered layer comecide with a no-stress surface—when the second terms in
(47a) and (476) vanish—a given increase of rigidity has less effect on the pitch than
an equal diminution, and this difference becomes more and more important in all but the
rotatory vibrations as the radius of the layer increases,
§ 59. For the rotatory vibrations we get from (20) and (32)
Ni de Wy, = : eos — cos ker) :
i ee W, = 5, {Br sin kBr —3 (= EOE Toe ier) }
ker
Also the frequency equation, obtained by equating W, to zero, is
kBa sin kBa — 8 (ae — cos ia) = (Ussoenbdunsoaseenacrenonone (54).
It will be seen that but for the multiplier / wkB/2, w, and W, are exactly the same
functions of Br and n as u, and 3U, of Sect. I. for the radial vibrations are of kar
and n, if we put m=n/3. Also (54) when @ is written for 8 is identical with the
frequency equation for the radial vibrations when m is put =n/3-
Since the condition for the node surfaces is that w, vanishes, and the condition for
the loop surfaces that w,? is a maximum, it follows that the corresponding values of kb
are identical with the values of kab answering to the node and loop surfaces respectively
in the case of the radial vibrations.
The relation n/m=8 is however physically impossible, so that the values of k8a for
the several rotatory notes cannot be identical with the values of kaa for the radial notes
in any isotropic material, and the values of k@b for the several no-stress surfaces in
the rotatory vibrations are also different from the values of kab for the no-stress surfaces in
the radial vibrations.
It follows that the positions of the several node, loop and no-stress surfaces in the
case of a rotatory note im a given sphere cannot be ddentical with the positions of these
surfaces in the case of any radial note.
The first four roots of (54) according to Professor Lamb* are given by
kBa/m =1:8346, 2°8950, 3:9225, 49385.
Comparing these with the results of Table I. Sect. II. it will be seen that the value of
kBa for the rotatory note of number (¢—1) is very near the value of kaa for the radial
note of number (7), though always slightly less than the least value of kaa, which answers
to o=0. Thus in any isotropic sphere, when 7 is large, the frequencies of the 2 radial
* Proceedings of the London Mathematical Society, Vol. x11. p. 197.
202 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
note and of the (i—1)™ rotatory note are very approximately in the ratio Vm+n : Vn.
In reality in the case of the rotatory vibrations there is a sort of suppressed note of zero
frequency as the following investigation shows.
The frequency equations for the radial vibrations, for all values of o, and for the
rotatory vibrations may be included under
Ff (a) = 27 sin @ — Ga (a sin — COS 7) =0.........secrseserseceres (55);
where g?=4n/(m+n) for the radial, and =3 for the rotatory vibrations. So long as q?
is less than 3, (55) has a root between 0 and zw. This root however diminishes rapidly
as g° approaches 3 and for this critical value becomes absolutely zero.
In what follows I shall speak of the note answering to k8a/m = 1°8346 as note (1).
The positions of all the node, loop and no-stress surfaces for the first four notes are
given in the following table. They are calculated from the values given above for kBa
and from the data already employed in Sect. II.
TABLE I,
Values of 7/a over node, loop and no-stress surfaces.
Note (1) Note (2) Note (3) Note (4)
—— ——————————— eS. —_——————E——————ee se ———— ———— ew
Node No-stress Loop Node No-stress Loop Node No-stress Loop Node No-stress Loop
surfaces surfaces surface surfaces surfaces surfaces surfaces surfaces surfaces surfaces surfaces surfaces
0 0 ‘3612 0 0) 2289 0 0 1689 0 0 1342
“1796 10 4941 -6337 65382 3646 4677 4821 -2896 3715 °3829
8494 1:0 6269 “7380 ‘7470 “4979 5862 +5934
8849 1:0 7028 =6"7943 "7995
9066 1:0
The centre is at once a node and a no-stress surface, and the number whether of
node or of no-stress surfaces is one greater than the number of loop surfaces, which
equals the number of the note, The loop surfaces lie outside of the corresponding no-
stress surfaces, and not inside them as in the case of the radial vibrations.
A comparison of the above table with Table II. Sect. II. leads to many interesting
results as to the relative positions of the node, loop and no-stress surfaces in the radial
and rotatory vibrations,
§ 60. We have already seen that an alteration of material at the centre has no effect
on the pitch of a*rotatory vibration, and that when a surface layer is altered the change
of pitch depends only on the alteration of density and is given by (43).
Supposing the layer to differ from the remainder only in density, the general formula
for the change of pitch is identical with (48), Sect. IL, writing @ for a, viz.
ok _—-t pi—p,,(sinkBb ai\_ 9M Qi 1 /sinkpb _ ‘ :
=-- ae = — cos kb) =-F + {ees (ter cos hb)} ...(56).
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 203
When the layer differs from the remainder only in rigidity we have
okt fe To (a 3 /sinkBb os ie a5
ee as Q yee kb — kBb i Bb oa cos 8b) conse ASORA A Rede RESEeN (57 a),
_ dV m—nQ (sinkBb 3 ssinkBb —__..\)? 57
= 8 | a =" onl Bb 7 COSMBD)f eeceeereeeeeeeee (57 b).
In these formulae t, M, V, ete. have the same signitications as previously. The formulae
may be applied without any restriction since @/ vanishes when the alteration of material
occurs at the centre.
Convenient expressions for @ and Q may be obtained from (38) and (40), Sect. IL.,
by writing @ for a and supposing m = 7/3.
This substitution gives
A) rs DEES) (Stone 8) (GI coaanocousenoonscaoaoonoG5ObAboE (58),
(ES (Het) PAE GIEEO) (HEIDI ooasosscococusonsunonsbapnocddeonUE (59).
From these formulae and the values given above for k8a the values of Q and Q’ for
the first four notes may be easily calculated. The results are given in the following
table :-—
TABLE II.
Values of Q and QJ.
Note (1) Note (2) Note (3) Note (4)
Q = 1:098 1-055 1020 1-015
= 36°49 85°83 154-91 243°74
A comparison of this table with Table III. Sect. II. will be found instructive.
§ 61. When the layer differs from the remainder only in density the curves showing
the variation of dk/k with k8b are exactly the same as those which under corresponding
conditions show the variation of 0k/k with kab im the case of the radial vibrations. They
are thus curve A or curve B of fig. 1 according as the layer is of given volume or
given thickness.
When the layer is of given volume the positions in which it has most effect on the
pitch of a given note coincide with the loop surfaces. The ratios of the first to the
subsequent maxima changes of pitch in the case of a given note are the same as in
the case of the radial vibrations, viz.
I: 1485 =: 0620) =; 0342... -...
The values of the first maxima are given for the first four notes in the following
table :—
Wor S°Vi, Parr Ti: :
.o)
7
t
204 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
TABLE III.
: : .—0k oM
First maximum of Eo Sar
Note (1) Note(2) Note (3) Note (4)
2°314 5-443 9824 15°457
The number of maxima is equal to the number of the note, and so all the maxinia
in the first four notes may be calculated from the ratios given above.
For notes above the fourth we obtain a close approximation to the first maximum
by means of the following formula, in which 7 is the number of the note,
—ok 0M
—— + = (1 2 6259 iaceweneassbasisaacrenosnatusas sae 60).
k V (i+ 1)? x *6259 (60),
This formula is adapted from (68), Sect. II.
When the layer of altered density is of given thickness the positions in which it
has most effect on the pitch of the note of frequency k/2a7 are obtained by equating
k8b to the values supplied for kaa for the value °3 of o in Table I. Sect. IL.
These positions are given for the first four notes in the following table :—
TABLE IV.
Values of b/a when we (¢ Ae) is a maximum.
k ap
Note (1) Note (2) Note(3) Note (4)
‘4760 3017 2226 ‘1768
6725 "4.964 3942
7560 6005
“S048
The ratios of the first to the subsequent maxima changes of pitch are the same as
in the corresponding case in the radial vibrations, viz.
Me S908 se SOpee oases.
The first maxima for the first four notes are as follows :—
TABLE V.
First maximum of = Oe (- ae) ,
k ap
Note (1) Note (2) Note (3) Note (4)
1:242 1173 1153 1144
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 205
From these results and the ratios already given all the maxima may be found for
these notes.
As the number of the note increases the formula
Ok. (e piseP\ee
; =+(C pf) = 1180 paar waite at 3 els | oo (61)
applies with continually increasing exactness to the first maximum.
For any maximum of high number in the case of one of the higher notes a close
approximation is supplied by
= (- = e) STi ayia. eines htenetien , ato (62).
§ 62. When the layer differs from the remainder only in elasticity, the change of
pitch depends only on the alteration of rigidity. In this case we see from (57a) or (57d)
that the change of pitch of a note vanishes when the layer coincides with a no-stress
surface, and that for all other positions of the layer the pitch is raised or lowered ac-
cording as the rigidity is increased or diminished. For a given numerical alteration of
rigidity the effect on the pitch is greater when the rigidity is diminished than when it
is increased.
When the layer of altered rigidity is of given volume the curve showing the variation
of dk/k with k8b, =2, is
Yi) Oe Sine — dec = (G— Sia a= COND) — fin (Eanes eee eee (63).
The first segment of this curve appears as curve A in fig. 5, and the corresponding
function of w is tabulated in Table VII. Sect. IL.
The second and subsequent segments of this curve would lie extremely close to the
third and subsequent segments of the curve A of fig. 2. The first segment answers
apparently to the first two segments of the curves of fig. 2.
The abscissae supplymg the maxima ordinates of curve A, fig. 5, are the roots of
the equation
rr ire (Ch ra) iim 7 (0) osSopnoonduososbuoaconsHseceee (64),
and the lengths of the maxima ordinates are found by substituting the roots of this
equation for # in the expression
Of ieee tO area t 1) osas element ecienetanectendececee (65).
For the first root and the corresponding maximum ordinate I find approximately
x=10638r, y='09412.
From these results with the assistance of Table II]. and the values of kBa I have eal-
culated the corresponding positions of the layer and the values of the corresponding
maximum change of pitch in the first four notes. They are as follows:
27—2
206 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
TABLE VI.
ok (AV n— : sys 3
First maximum of = = (— "ae and corresponding position of layer,
\ 1
Note (1) Note (2) Note (3) Note (4)
ok OV sm s) Ee 4
- > { ———_} = 1 14 2-695 +5 6
F ae 1145 693 60 T7647
for bla= 5799 ‘BOTS “2712 2154
In passing it may be noticed that the positions of the layer in this table coincide
with the first maximum-stress surface, i.e. the surface of least radius where the transverse
stress W,. is a maximum.
From the consideration that when 7 is greater than 3 or 4 the value of Q’ for
note (7) 1s approximately (7+ 1)?7*, we obtain as a pretty close approximation to the first
. maximum in the case of one of the higher notes of number (7)
ok OV u—n ‘
> + St (WEA XSL Bin kee Seelatedeaeameeceeen 66).
ne BE BO ( ) ee
The first maxima given in the table are considerably the largest for the respective
notes.
§ 63. When the layer of altered rigidity is of given thickness the equation to the
curve showing the variation of dk/k with k8b, =2, is
Bf A Fal(@) a) ecco a thod atngaoneetcamonseseaeeeneree (67).
The first segment of this curve appears as curve B in fig. 5 and the corresponding
function of z is tabulated in Table VII. Sect. II.
The second and subsequent segments would lie very close to the third and subsequent
segments of curves B in fig. 4, and like them continually approach, as «# increases, to
coincidence with curve B, fig. 2.
The abscissae supplying the maxima ordinates of curve B, fig. 5, are the roots of
the equation
1 = 6277 — 36> (b Sem) Tan = 0). ansoncne dectsmeneamseneee (68),
and the lengths of the maxima ordinates are found by substituting the roots of this
equation for # in the expression
Ap = (LD Saee SOR a) ee os incatenanweubleven se ocaitessasoricenes (69).
For the first root and the corresponding maximum ordinate I find approximately
2=1:23197, y=1:2339.
From these results with the assistance of Table II. and the values of kBa I have
calculated the corresponding positions of the layer and the values of the corresponding
maximum change of pitch in the first four notes, and give them in the following table :—
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 207
TABLE VII.
; p . OK tm—n
First maximum of — =/-
k a ny
) and corresponding position of layer.
ae
for b/a= “6715 4.255 3141 "2494
Note (1) Note(2) Note (3) Note (4)
Ok . (2 m2
zs )=1:355 1-280 1-259 1-249
an
As the value of Q continually approaches unity as the number of the note increases,
the first maximum in one of the higher notes is given more and more correctly the
higher the number of the note by
ok (: n,— Nn
Zh sys | a tt 70).
a7 (- = 1-234 (70)
It is obvious from (69) that the first maximum ordinate is decidedly the largest, the
length of the others approaching more and more nearly to 1 the larger the corresponding
value of w In the case of the higher notes all but the first two or three maxima changes
of pitch are given very approximately by
ay: o
et (- Pe eae a aro (71),
th if
and the corresponding positions of the layer are in the immediate neighbourhood of the
node surfaces.
SECTION IV.
RADIAL VIBRATIONS IN SOLID CYLINDER.
§ 64. If J, (kx), Y, (kx) represent the two solutions of the Bessel’s equation
CA Lies (#4) == () a eecssisiasieis cere ncet en eas (1),
dx adr Lv
then the type of vibration in a cylindrical shell of material (p, m, n) performing radial
vibrations of frequency k/27 is
B= Co ya Vald heap) 412) (Uxetp ececocedocbiod W290 sbaseOSOEane (2)*.
Here, as usual, A and B represent arbitrary constants, and
C= 1p) (ALAMO) Wa maaselseecreciPe eee eionieeoe sess ce cei ve ajepees (3).
The displacements in a compound solid cylinder (0.a.c.%.b.a.a), where b—c is
small, are as follows:
* Transactions, Vol. x1v. p. 356.
208 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
In the core i COS KES Ald; ears) G eaciette saeco cece seca een ee a ec tee (4).
In the thin layer ed 2Y ewe BOA (eH Oa ow sei (ie Sum Ja seocnca ced odosmcomonegnndcoseaones! (5).
Outside the layer u/cos kt = (A -F OA) Ja (har) == OB Wal(hair) ences cnccec su vcckhesweucsene (6).
We shall suppose terms in (b—c)*, and so in (@A/A)? and (@B/A), negligible.
Let us for shortness put
Jy (kar) = 7 2 J, (kar),
WUE acne eS @
Y{’ (kar) = ae aii dakar )
F(r.a)=(m+n) karJy (kar) +(m —n) J, (kar), :
F, (r : a) = (m 4 n) kar Yy (kar) 1 (m on n) Y, me wee ec cce cece sccccccssces ( ys
We then find for the relations connecting the arbitrary constants and supplying the
frequency equation:
Asa (hae) HA Js (ka, Bas (leet, C) Seaeece-teece sorte (O)s
SANE (Geet) "AE (Gi) = (Clay) nceceeansenenncette LO):
(A +04) J, (kab) + 0BY, (kab) = Ay, (kay) + BLY, (hay). .....ceeereeeee (11),
(A +0A) F(b.a)+0BF, (b.a) = A,F(B. 0) + BuF, (BD. Oy) essceeccseecesenes (12),
(AE OAR (Gra) fool) (G12) = Ore nnnetldee oncteteaicieeee tee eeeee (13).
The process of obtaining the frequency equation having been already illustrated in
the case of the sphere, no difficulty should be encountered in carrying it out when an
eye is kept on the expression
A (b. oy. b') = J; (keyb) Vy (keoyb) — Y, (Heeb) Ty! (Hoctb)..eccssescssevecsees (14),
which cuts out in the final equations determining ¢4/A and 0B/A. The results I find
are as follows:
4mn 4m,n, )
+ ——
“m+n m+n) dakieah) Na tkeab)
= { m+n) kab? — (m, + 1) k°a,2b?
1 4 b r (1, ) =
+(— ~—,) Fe. a) F.(b. a)+2(—"— sit, ) Wl leab)F(b.a) + Vs (bab) F(b.2))...(15),
B inch , .b-e¢
Ff (m+n) kabA (b. a.) +95
ah i) pe 4mn Anu, 1 f rab)?
= acs afb? — (m, + m) kea,2b" Ek ee (J, (kab)}
2 eee —-) (F.a)} +4("
m+n m+n
oon - io ) J; heb) FW ia) ssi (16).
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 209
It is important to notice that
[rethys (i3 C181) — (Cl cobtadaoesecnenoe nue soc oRORTAECoCec cee (17),
where C is a constant independent of k, a or 6, determined entirely by the definition
given of the Bessel’s function.
If the layer did not exist the frequency equation would be obtained by putting 0
for dA/A and oB/A in (13), whence
TK OSGISC)y SIM (Ag) Ono 2000058 ansnneccsq60eng803003000I (18).
In consequence of the existence of the thin layer, f(0.a.a) is no longer zero but
is of order b—c. Thus neglecting 0A in (13), we find for the frequency equation in
the compound cylinder
F(a.a) +4 By (a.a)=0 Eyes sre eretasieeeroclesnieesiares sistelsthosie acto’ (19).
As terms in (b—c)? are negligible, we may transform the coefficient of 0B/A im (19)
by any substitution which supposes (18) exactly true. We thus are enabled to replace
(19) by
OB (m+n) kaad (a.a.a’) _
A J, (kaw)
F(a.ajy+ Oeraeetin cece ce acene=tacien (20).
If the presence of the thin layer has raised the frequency by ok/27 then k—ok
must satisfy (18), whence, neglecting terms in 0h, we find
F(a.a)— 0k F(a.) = Op swete aecerecetnaruascettee tates (21).
Now kaa = F(a.a)=—(m +n) (ke? — 1) Jy (kaa) + (m —n) kaa J, (kaa).
As this occurs in the coefficient of 0k we may substitute for J,'(kaa) as if (18)
were exactly true. Doing so, we get
4mn
(m+n)
d 2 ry 272
kaa Kadi F(a.a)=—(m+n) {k au
KF, ita) (23),
Substituting this in (21), and then noticing that (19) and (21) must be identical, we
find ue:
ok 0B kaad (a.a.a’)
= == =i =F covncnoosooosomnbouensaNC (28).
bk A Skea? — 4mn (m+n)! (J, (kaa)}?
Let oa K (aq denote the frequency of free radial vibrations in an infinitely thin
shell of material (p, m, 2) and radius a, then
4nun ' 4mun
2 2 ee ie elscis snis 0 Silaiseis vo\sieieoe)siecin cian 24)*.
Ke a.a) (m+nyae (m+n)p ei
* Transactions, l.c. p. 356, equations (43) and (43a). Cf. also (22) above.
210 Mr ©. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
Also let
TS Ail (rr ispastotasncean soacanoessaccce Jains NOT ORES REESE OCI wie sino elses TE (25),
U,.= EG . a) =" (m+n) kar Jy’ (kar) + (mm — 2) Sy (Rar) |... eee eee (26),
so that u,cosht represents a displacement in a simple cylinder performing radial vibra-
tions of frequency &/27 and U,coskt the corresponding radial stress, both quantities
referring to points at distance r from the axis.
Employing these substitutions in the value of 0B/A given by (16), and then sub-
stituting in (23) and employing (17), we find
ok _ b—c[p(ke— FKea.w) — a (P— Ka.) (b\? (ro?
kb iv (° — Baa) C ‘a
il 1 ) & U;\" ( n ny ) b um U,
a — - Pit):
i +n m+n,/ ap (k?— K* 0.0) ( 2) see m+n m4+7/ Wp(h?—Iea.a) Ud (2)
In (27), as in (28), Sect. II., we notice the existence of three distinct terms, the
first depending on the square of the displacement of the altered layer, the second on
the square of the radial stress, and the third on the product of the displacement and
radial stress. The first term alone exists if the layer differ from the remainder of the
eylinder only in density, or if it coincide with any no-stress surface. If the layer occur
at a node surface then the second term alone exists. The signs of these two terms are
independent of the radius of the layer.
The third term vanishes if m,/n,=m/n; otherwise its sign as well as its magnitude
varies with the position of the layer.
§ 65. In the core there is no change of type due to the existence of the layer
other than a displacement of any node, loop, or no-stress surface originally of radius r
according to the law
— Or/r =Oh/I..vecees Ae oad salgrahd eaten eee (28).
Outside the layer we find by substituting in (6) the values of 0A/A and ob/A
from (15) and (16), and reducing
u/A cos kt =J, (kar)
b—e if ro 2 'e) } ah = 1 = I T
Sia +n)C lea —K (a,b) — prlk —K (ay »)} buf @. 0.7) + (a aan) Ui f(b. «.7)
{/—” n
9 a 1 (h- ) 7) 6 .
+2(7 -— 2) buf (b. a?) + Uif(b-a.7) | eS ee (29):
where, with our usual notation,
f(b.a.7) =, (kar) V, (kab) —Y, (kar) S,(hab)) (30).
f(b.a.r) =J, (kar) F, (b. 0) — Y, (kar) F(b. a)J
The loci where f(b.a.7) vanishes and changes sign are what would be the node
surfaces of a simple shell of material (p, m, 1) whose inner surface r=b is fixed and
whose frequency of vibration is k/27. Similarly the loci where /().a.7) vanishes answer
Mr ©. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 211
to the node surfaces in the vibration of frequency 4/27 in a simple shell of material
(p, m, n) whose inner surface 7=b is free.
We notice the existence of three terms in the coefficient of b—c in (29) answering
to the three terms in (27). The first two terms in (27) and (29) vanish together. The
third term however in (27) vanishes when the layer coincides either with a node or a
no-stress surface, whereas unless m,/n,;=m/n the third term in (29) can vanish only for
special values of 7 wherever the layer may be situated.
Noticing that iO 2b) =O; and! F(b% 00) — == (FTO oe cc ents stooeaees eves (31),
we find from (29) for the displacement just outside the layer
1 1 n n
oS i — = — 2 = a me coool e 2 b
ee E (0 2) \Gr +n m+ =) Us ce e ari Waser =) : nh 2
From (32) we may deduce the following expression for the displacement throughout
the layer itself:
u/A cos kt =J, (kab) —ka(b—r) Jy’ (kab)
: 1 1 oe n Ny ) uA aa
pal =o) \(- ron mm, + = Oasin? = Se Un amon : oH ana ics)
Thus, precisely as in the radial vibrations of a sphere, no change of type manifests
itself as we cross the layer if it differ from the remainder only in density, or if while
differing in elasticity it coincide with a no-stress surface and the relation m,/m,=n/m
hold.
§ 66. For a discussion of (27) we require to know the characteristics of radial
vibrations in a simple cylinder.
The type of the displacement is shown in (25). Thus there are a series of node
surfaces whose radii, r, for the note of frequency k/2m are found by equating kar to the
successive roots of
Ji GY) Oleeeroeeeease ets ui cisietsainer wenensietne sendcs (34),
viz. OF 3832, “7-016, 1013) 13:323%.2.0.,
the higher roots being of course only approximate.
The radii of the loop surfaces, where the displacement is a maximum, are found by
equating kar to the roots of
Jit (@ JO Mee ereas ote sen toss seeeceeee see cetn cee (35),
whose approximate values are 1841, 5331, $536, 11°706
The radu of the no-stress surfaces are obtained by equating kar to the roots of
(= DAL (@) (= 70) dh(@)= 0 cosanecsconsconeneobacepeene (36) ;
while by equating kaa to these roots we obtain the frequencies of the several notes the
cylinder can produce.
Vou. XV. Parr II. 28
212 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
The form of (36) depends on co. Thus when ¢=0 it is identical with (35). When
o = °25 it becomes
Bie (Bl a0 cone eee a (37);
whose roots, excluding zero, are approximately 2°069, 5396, 8°576, 11°735.......
Finally when « ="5 it becomes
whose roots are approximately 2404, 5520, 8654, 11°792.......
For the roots of (34) and (38) I am indebted to Lord Rayleigh’s Theory of Sound,
Vol. 1. Table B, p. 274. The roots of (35) and (37) I have calculated from the tables
in Lommel’s Studien iiber die Bessel’schen Functionen.
Since the roots of (34) and (35) are independent of o the ratio of the radii of any
two node or loop surfaces of given numbers in a given cylinder performing a given note
is the same whatever be the number of the note or the value of o.
The values of kaa, however, being the roots of (36), vary with the value of o; thus
the ratios of the radii of the node or loop surfaces to the radius of the cylinder vary
with the material. Still in the case of the second and higher notes the value of o has
only a small effect on the absolute positions of the several node and loop surfaces in a
cylinder of given radius.
The roots of (36) exceed the corresponding roots of (35) for all values of o greater
than 0. Thus the loop surfaces, while coinciding with the no-stress surfaces when o=0,
lie inside them for all other kinds of isotropic material.
In the case of all three equations (34), (35) and (36) the successive higher roots
come to differ almost exactly by 7, and the corresponding higher roots of (35) and (36)
are for all values of o nearly equal and are approximately half-way between successive
roots of (34).
Thus between successive higher notes there is a nearly constant difference of pitch,
and between consecutive surfaces of higher number of the same kind—whether node, loop
or no-stress surfaces—a nearly constant difference of radius. Also the node surfaces of
higher number lie nearly half-way between consecutive loop surfaces.
The positions of the node, loop and no-stress surfaces for the values 0, ‘25 and ‘5 of
c in the four lowest notes are given in the following table to three places of decimals :—
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 213
TABLE I.
Values of 7/a over node, loop, and no-stress surfaces.
e]
=) or 45 —o
= ee ret = se a — =
Number Node Loopandno- Node Loop No-stress Node Loop No-stress
of note surfaces stress surfaces surfaces surfaces surfaces surfaces surfaces surfaces
(Oy © 1:0 0 890 10 0 ‘766 10
0) B45 0 341 B84 0 B34 “435
(2) | 719 10 COMER Soko 694 966 10
ie 216 0 215 241 0 23 278
(3) “449 625 447 622 629 “443 616 6358
| *822 1:0 ‘S18 995 10 ‘S11 ‘986 10
0 api 0 ALi ‘176 0 156 204
BPA 455 327 454 “460 325 452 “468
A)ro4o-599 4 729)" 9-598" 727 | BL BOB “72d 6784
869 10 ‘S67 998 1:0 863 993 10
A comparison should be made of the above results with those of Table II. Sect. II.
In the table the axis is counted as a node.and the surface of the cylinder as a
no-stress surface, and under all circumstances the number of node, loop, or no-stress surfaces
is equal to the number of the note.
I shall refer to any such surface by its number, regarding the surface of the same
kind of least radius as number (1).
§ 67. In all the expressions for the change of pitch there occurs one or other of
the two following quantities:
o= kaa {Jy (kaa)}~
~ koa? — 4mn (m + ny?’ Badass ea cenns ca aalelcoeeients (39).
£Q’ =tkaaQ
Employing the results already recorded for the roots of the frequency equation, I have
calculated from Lommel’s tables the following approximate values for Q and QQ’ :—
TABLE II.
Values of Q and Q’.
Q’
Note (1) (2) (3) (4) || Note (1) (2) (3) (4)
(0 2275 1623 1590 1581] 0 4189 8652 13574 18:507
o=125 1868 1602 1583 1577|/¢=425 3867 8644 13573 18507
(5 1542 1565 «1568 1-569 | 5 3708 8637 13572 18507
In the higher notes the influence of o on the value of Q is small and continually
diminishes as the number of the note increases. In notes (3) and (4) the variation in
28—2
214 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
the value of Q’ with the value of o is practically imsensible. The numbers entered in
the table in the two last columns are scarcely to be relied on in the last decimal place.
The third decimal place is retained im these columns mainly with the view of showing
how remarkably small the influence of the value of o is.
The following considerations enable pretty close approximations to be found for the
values of Q and Q’ in the higher notes.
From the general formula for the approximate values of Bessel’s functions for large
values of the argument, we may when 2 is large put
J, (2) =/% cos (= - «)
approximately, employing the usual definition of the Bessel.
From the above expression we conclude that for large values of « the maxima values
of «{J,(x)}? are all nearly equal, while the maxima of {J,(«)}? vary approximately as the
reciprocals of the corresponding values of # Also the larger values of 2 supplying the
maxima whether of z{J,(x)}* or {J,(7)}* imerease very approximately in an arithmetical
progression with a common difference zr.
If now we write the frequency equation (36) in the form
, EUS =
Jy (a) + m+_n J, (x) = 0,
we see that its higher roots, whatever be the value of o, must be nearly identical with
the higher roots of J,’ (x) =0, ie. of (35). This is im fact the exact form of the frequency
equation when o=0, and the difference between the second root even of (35) and
those of (37) and (38) the frequency equations for the values 0 and ‘5 of o—is, it
will be noticed, far from conspicuous.
Thus whatever be the value of o the values of kaa for the higher notes are nearly
identical with those values of « which make {J, (z)}? a maximum,
Now for notes above the fourth the value of kaa is not less than 148, and so
—
4mn(m-+n)~ is very small compared to aa’.
Thus we see from (39) that for notes above the fourth a close approximation to
the value of Q, whatever be the value of o, is obtained by equating Q to 1 = {aXJ,(a)}?,
where x is one of the higher numbers which make {J,(«)}* a maximum. It imme-
diately follows from our recent investigation that for notes above the fourth the value
of Q is approximately constant and independent of o. No serious error will arise by
ascribing to it the value 7/2.
In the same way we find as an approximation for notes above the fourth
Q =1/{J(2)}*,
where « is one of the higher numbers which make {J,(x)}* a maximum. Consequently
(Y varies approximately as these values of xz. But we saw that these values of # increase
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS,
215
approximately in an arithmetical progression with common difference 7, and so the successive
9
values of Q’ increase approximately in arithmetical progression with a common difference
a°
“a
This conclusion is strongly supported by the numbers given in Table IL.
We
ue
are
thus entitled to assume that the value of Q’ for any note of number (7) greater than
4 is very approximately given for all values of o by
Q = 1851+ (i — 4) x (4935)
§ 68. As in previous sections I shall, before discussing the general application of
the frequency equation, consider briefly two special cases,
In the first of these the material (p,, m, 7) occurs at or close to the axis.
Writing
supposing b/a very small, but (b—c)/b still smaller, we pass to the case of a very thin
layer close to the axis of the cylinder.
By
This we shall call the axial layer.
(6 —c) b/a# = 40V/V,
we obtain the value of 0k/k in this case by retaining only the lowest powers of b/a
occurring in (27). We easily find, distinguishing this case by the suffix J,
di aV & (m—m) (m+n)
k ok, —
V 2 (m+n)(m+m)
If the material (p,, 7m, ,) form a thin core we must proceed by considering the
form taken by the frequency equation f(0.¢,.b.a.a)=0 when b/a is very small.
the following data are kept in view.
From the usual formula for
The application of the method of Sect. I. to this case presents no difficulty when
the
very small the approximate values
J; (x) = 2/2,
Bessel’s functions we obtain at once when «
Jy (@)=1/2.
Now for the other solution of the Bessel’s equation we have
1s
But when z is very small approximate values are
Y, (a) =— a7 J, (x) + log (x) J (x) — Ji(w) + powers of « above the first *.
J,(x)=1, log (a) J, (a) =0,
and we have as first approximations
Wi oes=, IG@Qse
The numerical value of the constant C of (17) is also required in this case
may determine it very simply by noticing that when # is very small
. We
(a
io —4tC =} =1,
* See Neumann’s Theorie der Bessel’schen Functionen, p. 52, equations (13), (14), and (15),
—C=2{J,(«) Y/ («)—-T! («) Y,(a)} =a
216 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
Supposing the core of radius 6 and volume oV per unit of length, so that
b/a?=0V/V,
I find, distinguishing this case by the sufhix ¢,
1 dls = oV Ym —m
k °° V2 m+n
The formulae (41,) and (41,) are not in_ general identical. When however the
alteration in elasticity is small they both reduce to
ok 0VQ m—m ;
es ae rR ;
From (41,) and (41,) it follows that to the present degree of approximation an
alteration only in density does not affect the pitch of any radial note when. it occurs
at or close to the axis.
In the case of the core the change of pitch depends entirely on the alteration of the
elastic constant m, and in the case of the axial layer the sign of the change of pitch
depends entirely on the sign of m,—m and its magnitude for any ordinary alteration of
material would not be greatly modified by the alteration in n.
If the elastic constant m alone is altered, then the formula (41;) for the axial layer
becomes identical with the general formula (41,) for the core.
If both elastic constants are altered in the same proportion according to the law
i=) Tiara By OSpamapReee soueceride Boonenob dou cose 00- (42),
the changes of pitch are given by
Le. pe oViQom eo anh) 2
A oky = 1 +p V2 (m+ n) i Ble amr @ le [eo 0 0\e 00.0 ele isla viee'elefsle es (43)),
i 2.5, iid Va onan i.
joke = P V 9 (m+n) {1 +p I wialwra(elereletalatatetsialsis[olateleva/aterateters (43,).
For any alteration whatsoever of elasticity at or close to the axis the pitch is
raised or lowered according as the elastic constant m is increased or diminished. Thus
m takes the place that the bulk modulus occupies in the corresponding case in the
sphere.
§ 69. Next suppose the alteration of material to take place throughout a surface
layer of thickness t. Then, remembering that U, is zero, we easily obtain from (27)
ggg Plse ae 4mn (= +n te ,
ok a8 t any ui p (mm at ny mo t+n,, 7 1) iS
ko a Seat anime a ak oe Sane (44)
2242 —
(m+ny
The values of 0k/k, when the dénsity at the surface alone is altered, are shown in
the following table for the first four notes answering to the values 0, ‘25 and 5 of o:—
Mr C, CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. Dalia
TABLE III.
Value of see (- ae) for a surface layer.
k a p
Number of note (1) (2) (3) (4)
| 0 1-418 1:036 1-014 1-007
op S405) 1:262 1:031 1012 1:006
le 1-0 1-0 1-0 10
If in (44) we suppose
m+ +27 oes (Dts a Pe
aS Z If cocabonooocsy pecousbMeoODOOSODBDNE (45),
then it at once reduces to
ok t
Th = i q a\u{oleleseletafeletals|ejeis/cis\elsiololsalaleisvayatsyaysieiets(elstvte|e/ae/e/eis)a/ sie (46)
Thus we derive at once from Table III. the following results for the change of
pitch due to a surface alteration of elasticity alone :—
TaBLE IV.
ok | (t (m+n am :
Value of Ea iz i ~ 1)! for a surface layer.
Number of note (1) (2) (3) (4)
0 “418 036 ‘014 ‘007
Gi 4025 262 ‘031 012 006
[5 0 0 0 0
A comparison of Tables III. and IV. leads to many interesting results as to the
relative importance of surface alterations of density and elasticity in changing the pitch of
the fundamental and higher notes.
The most important of these results is that if a thin surface layer of an isotropic
cylinder be altered in any way consistent with its remaining isotropic, then the ratios
of the frequencies of all the higher notes can only be shghtly affected; but, unless the
value of o for the unaltered material be near the limiting value °5, or else both density
and elasticity be altered in such a way as approximately to satisfy (45), the ratio of
the frequency of the fundamental note to that of any of the higher notes may be
seusibly disturbed.
§ 70. It will be necessary to restrict our discussion of (27) to some special forms
of alteration of material. We may in every case modify the function of kaa that appears
in the expression for 0k by any substitution that supposes (18) to be exactly true.
(1) Suppose the layer to differ from the remainder only in density. We have already
seen that the change of pitch is then always zero when the layer is axial. We may
thus employ without restriction a formula in which the alteration of mass per unit length
of cylinder is represented by
0M = 2b (b—c)(p:— p).
218 Mr C, CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
Denoting by ¢ the thickness b—c of the layer, and by M the original mass za*p
of the cylinder per unit length, we find from (27)
ok t pr- =
= =—- aE Qkab | J, (kab)}? = oe SPA ences tana CP)
(2) Suppose the layer to differ from the remainder me in the value of m. Employing
the well-known relations between successive Bessel’s functions, we obtain from (27)
ok _t m-—m oV m—m YY
k am+n V m+n 2
where V=7a’, OV = 27 (b—c)b.
Qkab {Jy (kab)}? =
UH etsasderoseogend (48) ;
This formula it will be remembered happens to apply for an axial core as well as
an axial layer.
(8) Suppose the layer to differ only in the Se of n. We find
Ok _ t m—n ea v
2 if }2.,
Ean eee) a ak 1g J (Rady) saves seesveanee (49),
This vanishes for an axial layer.
(4) Suppose both elastic constants to be altered in the same proportion according to
(42), then by fe for any true layer
ok _t Hye m —n J, (kab))* 4mn Jy _
ie ak a ae ie Cer m+n kab a Cees +p) kab ee
An alternative formula applicable under the usual restriction may be obtained by the
substitution
oV
= Qiad = 9S.
§ 71. Comparing the expressions (47), (48), (49) and (50), we notice that each is
a product of three factors of the usual kind.
Except in the case of (50), where the third factor is a function of o and of the
magnitude of the alteration of material, we may very easily construct curves*, whose ab-
scissae are the values of 2, =kab, to represent the variation in the magnitude of ok/k
with the position of the layer.
The equations to these simple curves are
ORR (a) == 7 (7) Je coeiencepe Mooaoageraoad onc sdélore yaa (51),
ie CAN (2))\ 375 9 Veagnneosnonopapnoonepnanae cosucosondcso: (52),
M2, I CO) = 7 @)Jassponn cbogueacopponsoooncshanoonasabos (53),
fe Aa (3) “ay Al ©) pnomenpesg apap 1b 96r1000005 0009000000 (54),
Afi f04 si) \ aH (CE) el otan setclete » asta sala elestov esrae easter (55),
= {Tana A = JqAe) or denih as dens cee rep ds sauna ened cs (56).
These curves apply whatever be the value of o in the material. Full information
as to their use is recorded in the following table :—
* On account of the difference in the values of Q for Plate V. should be increased in the ratio 7:2 for com-
the sphere and cylinder, the ordinates of the curves of parison with Plate IV.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 219
TABLE VY.
Function of x. Property of Layer of given Figure where | Letter attached
material altered curve drawn | to curve
TAD) p volume 6 A
r |
Fs(2) p thickness | 6 B
HAD) m volume 7 A
SF2(2) m thickness 7 B
FT: (2) n volume 8 A
Fs(£) n thickness 8 B
After the long discussion of the corresponding curves in the case of a sphere, it
is hardly necessary to say more than that the use of the present curves is exactly the
same as that of the previous. Each of the curves of Table V. applies to all materials
and notes. The ratios of its successive maxima ordinates are the ratios of the several
maxima changes of pitch due to the given assigned alteration of material.
Since the factor by which the ordinates of all the curves B are to be multiplied
to get the numerical magnitude of the change of pitch is Q, the curves supply us
immediately, supposing them drawn on the same scale, with a comparison of the changes
of pitch, of any given note in any given cylinder, accompanying independent alterations of
material throughout a layer of given thickness such that
(pi — p)/p =(m, — m)/(m, + 1) = (My — N)/(M FN). erreeeeesesceeceeeeeeee(IT)-
Again for the higher notes the values of @ are nearly constant and independent
of o; thus in any one of the three cases when p alone is altered, when m alone is altered,
or when 7 alone is altered throughout a layer of given thickness, the maxima percentage
changes of frequency of any given number are approximately the same for all the higher
notes and for all isotropic materials.
In the case of all the A curves the factor is Q’/2, thus the curves, if drawn on the
same scale, supply at once a comparison of the changes of pitch of any given note in any
given cylinder accompanying independent alterations of material, satisfying (57), throughout
a layer of given volume.
Also since the higher values of Q’ increase approximately in arithmetical progression
and are practically independent of c, it follows that when p alone is altered, when m alone
is altered, or when x alone is altered throughout a layer of given volume, the maximum
percentage change of pitch of any number (j) in a note of number (7), which is greater
than 2, exceeds the maximum percentage change of pitch of number (j) in the note
of number (7—1) in the same cylinder by a quantity which is practically independent of
i or of o and may be regarded as depending only on ).
Wort, 2, TEA 1D 29
220 Mr C, CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
The factors, viz. the reciprocals of kaa, by which an abscissa « must be multiplied
to supply the corresponding value of b/a are given in the following table :—
TABLE VI.
Values of 1/kaa.
Number of note (1) (2) (3) . (4)
{ 0 5431 “1876 aA 0854
o=4 25 “4532 1853 ‘1166 70852 .
5 4160 1812 1156 0848
Approximate values of these multipliers in any of the higher notes may be easily
derived from the consideration that their reciprocals kaa are nearly independent of o and
increase approximately in an arithmetical progression with a common difference 7.
§ 72. The functions of Table V., and several others whose occurrence will subse-
quently be explained, are tabulated in Table VII. For the data necessary in making the
calculations I am indebted to the tables of J,(z) and J,(x) in Lommel’s work. I have
in no case gone beyond the value 15 of « The necessity of carrying the calculations
further may in general be avoided, as the following considerations show,
We have already seen in § 67 that the maxima of {J,(«)|* when wx becomes large
vary approximately as the reciprocals of the corresponding values of 2, and so tend to
become small; while the maxima of «x {J,(«)}? tend to approach a finite constant value.
Now the same results may be proved in a similar way for any Bessel’s function J; (2).
Thus a glance at equations (51)—(56) suffices to show that the successive maxima
ordinates of any one of the curves A of Table V. diminish rapidly as the radii of the
corresponding positions of the layer increase, while the successive maxima of any one of
the curves B continually approach to equality. Consequently unless very great accuracy
is required it is unnecessary to draw either set of curves for large values of a.
The other functions occurring in Table VII. present themselves in the treatment of
(50). The form of f,(x) is given by (77), of f,(x) by (78), of f,(«) by (79) with «=0,
of fio (x) by (80) with o=0, of fi, (x) by (81), and of f(x) by (82).
This last group of functions are also represented by curves, but these must be
combined in pairs so as to form compound curves, or else apply only for special values
of c. The ordinates of these curves have to be multiplied by Q or Q’/2, and their
abscissae by the factors given in Table VI. according to circumstances.
221
CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
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Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
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224 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
>
§ 73. We may now examine the four special cases in detail.
When the layer differs from the remainder only in density the change of pitch is
given by (47). The law of variation of 0k/k with the position of the layer is thus inde-
pendent of the magnitude of the alteration of density.
The positions of the layer when the pitch of a given note is unaffected coincide
with the node surfaces for that note. When the layer is in any other position the
pitch is raised or lowered according as the density is diminished or increased.
When the layer of altered density is of given volume the curve showing the de-
pendence of the change of pitch on the value of kab is A fig. 6, whose equation is (52).
The maxima ordinates answer to positions of the layer coincident with the loop surfaces.
The first maximum ordinate is much the largest. For the ratios it bears to the
succeeding maxima ordinates, and so for the ratios of the first to the succeeding maxima
changes of pitch I find
1 ; 3539 : 2206 : -1608......
Employing these ratios, all the maxima in the case of the first four notes can be
calculated from the numerical magnitudes of the first maxima which are given in the
following table :—
TABLE VIII.
: : ok . oM
First maximum of — ie a Mu .
| Number of
Value of ¢ note (1) (2) (3) (4)
0 ‘709 1465 2°298 3133
"25 655 1463 2298 3133
5) ‘628 1462 2°297 3133
For any of the higher notes approximations to the numerical magnitude of the
first maximum change of pitch can easily be obtained by the consideration that these
numbers increase approximately in an arithmetical progression with the number of the
note. Thus for any note of number (7), greater than 4, a close approximation to the
first maximum is given for any value of o by
= as = He BEL SB ING PUAN GMOBS. «y< vscseccceveweer¥eor ct (58).
In these higher notes the next three maxima changes of pitch can be obtained
from the ratios already given in this paragraph. The maxima of higher number can be
obtained to a less close degree of approximation from the consideration that the reciprocals
of the successive maxima changes of pitch in a given note are approximately in arith-
metical progression. Thus from the values for the ratios of successive maxima already
given in this paragraph we find as a fairly close approximation to the maximum change
of pitch of number (j) in the note of number (7), supposing 7 and j both greater than 4,
ok OM 3:133+(i—4) x 835
hk M 622+ —4)x167
a chin oO ERO eee (59).
Mr CO. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 225
§ 74. When the layer of altered density is of given thickness the mode of varia-
tion of dk/k with kab is shown by curve B fig. 6, whose equation is (51).
The abscissae supplying the maxima ordinates are the roots greater than zero of
DEI) AZIM ZL) AO soccer sseestneetos cet seresGia cise (60).
Their approximate values are 27166, 5-427, 8595, 11:749...
When o«=°3 the equations (60) and (36) are identical, and so the positions of the
layer supplying the maxima changes of pitch are coincident with the no-stress surfaces.
For other materials these positions lie outside or inside the no-stress surfaces according
as o is less or greater than ‘3. For all values of o they lie outside the loop surfaces,
When o='3 one of the positions supplying a maximum of dk/k coincides with the
cylindrical surface, and for this and all larger values of o the number of maxima is
equal to the number of the note. For values of o less than ‘3 the number of maxima
is less by 1 than the number of the note. Thus in note (1) there is no true maximum,
the value of 0k/k increasing continually as the layer moves out from the axis to the
surface.
The following table gives the positions of the layer corresponding to all the maxima
in the case of the first four notes for the values 0,°25 and ‘5 of o :—
TABLE IX.
Values of b/a supplying maxima of — + (7 =P),
Bema. 9p
Note (1) Note (2) Note (3) Note (4)
See 2S ee ee ae Sa nee eee
o=0 25 a3) 0 As) 5 0) "25 5 0 25 75)
== ss ‘901 | 406 401 "392 | -254 253 "250 | 185 185 184
— = ‘983 | ‘636 633, 627 | 464 462 460
“= — ‘993 | ‘734 732 ‘729
= = ‘996
The blanks are intended to draw attention to the absence of true maxima. A com-
parison with Table I. will be found instructive.
For the ratios of the first to the successive maxima ordinates of curve 8B, and so
of the first to the subsequent maxima changes of pitch, I find
1: 947.: 940 : 938...
226 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
The absolute values of the first and largest maxima are given in the. following
table for the first four notes -—
TABLE X.
First maximum of ei (; Bae).
Sige
Boe) Note (2) Note (3) Note (4)
——
C= 5
Disie aie eds io Lae 25 5 0 "25 ‘Dy wt 0) 25NE 3B
1-418* 1:262* 1:050| 1104 1:090 1065 | 1082 1:077 1067|1076 1073 1-068
Asterisks are attached to the entries for the values 0 and ‘25 of o under note (1)
to show that they are not true maxima. They do not answer to the first maximum ordi-
nate of curve B fig. 6, but to positions of the layer at the surface of the cylinder.
From the results already obtained as to the values of @ in the higher notes and
as to the maxima of z {J,(x)}* answering to large values of «, we are enabled to conclude
that, for any note whose number exceeds 4 and for any value of o, a close approxi-
: : : ok ft py = :
mation to the first maximum of = 3\2 a) is 1:07, and to any maximum whose
number exceeds 3 a pretty close approximation is 1:00.
§ 75. In the case when the layer differs from the remainder only in the value of
m the change of pitch is given by (48). From this it appears that the law of varia-
tion of dk/k with the position of the layer is independent of the magnitude of the
alteration of elasticity.
The positions of the layer when the change of pitch vanishes are found by equating
kab to the roots of (38). They thus coincide with the no-stress surfaces when o = 5,
and for all other values of o they lie outside of the no-stress surfaces though very close
to all except the first.
When the layer is of given volume the curve showing the variation of the change
of pitch with kab is A fig. 7, the equation to which is (54).
The ordinate at the origin is, much the largest in. the curve. Thus the change of
pitch which arises when the altered material forms an axial layer is far the largest
maximum.
The magnitude of the change of pitch due to any assigned alteration of elasticity
throughout an axial layer has been already determined in § 68, the necessary formula
in the present case coinciding with (41,), The numerical magnitude is obtained at once by
dividing by 2 the values supplied for Q’ in Table II. and altering the heading from Q’ to
1 0V m —™ :
~ ok + (< HOT 2 } 7 Sek i
i 1 Ea) or a note of number (7) above the fourth we obtain from (40)
as an approximate formula
ak
(OV m, — m\
k°\V m+n
J {1S 'BL eG — 4) x 4OB5) a vanesssscennsenser (61).
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 227
The abscissae supplymg the subsequent maxima ordinates are the roots of (34).
Thus the corresponding positions of the layer coincide with the node surfaces, For the
ratios of the first to the subsequent maxima ordinates, and so for the ratios of the first to
the subsequent maxima changes of pitch, I find
ll: 11622 0908-0625
From considerations as to the values of those maxima of {J,(x)|* which answer to
large values of z, of an exactly analogous nature to those discussed in § 67, it may
be proved that a fairly close approximation to the maximum change of pitch of number (7)
in the note of number (2), 7 and 7 being both greater than 4, is supplied by
ok 4 mM, =o 18°51 + (i — 4) x 4935
is Tha —— aS 9-2 4 ( a [nr Wel (62).
k V m+n 32-08 + (j —4) x 9°87
In this formula 7 may equal but cannot exceed 7, as the number of maxima, being
equal to the number of node surfaces, including the axis, is equal to the number of
the note.
§ 76. When the layer whose m differs from that of the remainder is of given
thickness the curve showing the variation of the change of pitch with kab is B fig. 7,
the equation to which is (53).
The abscissae supplying the maxima ordinates are the roots greater than zero of
(Saal) Ina) SIG Ge) SO resent ses cece cerse cise isiieseie ois (63).
For the first two roots I find approximately ‘9408 and 3:9594.
It is easily proved that the positions of the layer answering to the maxima changes
of pitch whose numbers exceed 2 lie outside of but very close to the corresponding
node surfaces. The positions of the layer answering to the first two maxima are given
in the following table for the first four notes and the usual values of o:—
TABLE XI,
0k ( tm—m
Values of b/a where — + ) is a maximum.
k am+n
Note (1) Note (2) Note (3) Note (4)
a=0 25 “5 0 "25 5 0 “255 5 0 25 “5
ay i "455 SON ali 6, ‘174 170 | 110 ‘110 109 | 0804 “0802 ‘0798
743 734 “717 | 464 “462 458 | 338 337 336
As the second maximum ordinate is very nearly equal to all the subsequent maxima,
and is decidedly greater than the first, I have included in the following table the first
two maxima changes of pitch. For note (1) of course there is only one maximum.
Vou. XV. Parr II. 30
228 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
TABLE XII.
Maxima values of oe + (- lB =) ‘
k am+n
Note (1) Note (2) Note (3) Note (4)
a ob o 0 "25 i) 0 “25 1) 0 “25 43)
38 1099 907] ‘954 942 «920 | 935 931 922/930 ‘928 ‘928
1026 1012 9989/1005 1000 -991|:999 -997 -992
—
wo
oo
The number of maxima is always equal to the number of the note so that the
table gives all the maxima only for the first two notes.
In the higher notes for all values of o pretty close approximations are
ok (tm—-M) _ oo
7+ ¢ ae ) Soot. 2 (64)
for the first maximum change of pitch, and
ok (tm—m\_,,
z7(- mm) = 100 i ee (65)
for the second and all subsequent maxima.
§ 77. In the third special case, when the layer differs from the remainder only in
rigidity, the change of pitch is given by (49). This shows that the law of variation of
ok/k with the position of the layer is independent of the magnitude of the alteration of
rigidity.
The positions of the layer when the change of pitch vanishes are found by equating
kab to the roots of
viz.* 0, 5135, 8-417, 11°620....
The higher roots are of course only approximate. The root «=0 applies whether
the layer be of given volume or of given thickness, so that the axis is always one of
the positions where an alteration in rigidity does not affect the pitch.
Whatever be the value of o, the second and higher roots of the frequency equation
(36) are slightly larger than the second and higher roots of (66). Thus counting the
axis, the number of positions of the layer for which 0k vanishes is always equal to the
number of the note. Also these positions commencing with the second are close to but
inside of the successive no-stress surfaces commencing with the second. It seems un-
necessary to determine these positions more precisely. All the data necessary in the case
of the first four notes and the usual values of o are given above.
* See Lord Rayleigh’s Theory of Sound, Vol. 1. p. 274.
Mr C0. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 229
When the layer of altered rigidity is of given volume the curve showing the varia-
tion of the change of pitch with the position of the layer is A fig. 8, the equation to
which is (56).
There is, it will be noticed, a very close resemblance both in magnitude and position
between the segments of this curve which are most remote from the origin and the
segments of curve A fig. 7. The first segment however of the present curve would
seem to answer to the whole of curve A fig. 7 between the origin and the second
zero ordinate.
The abscissae supplying the maxima ordinates of the present curve are the roots
greater than zero of
(CPOE (Gai se otal: (3) = Osaoabasoanencansnooboscadbocucuseoce (67).
For their approximate values I find 3:054, 6°706, 99695, 13:170....
When 7 is greater than 2 the (¢—1) root of (67), omitting zero, is near but always
less than the 7 root of (34), the equation which determines the position of the node
surfaces. The first root of (67) is however noticeably less than the second root of (34).
The number of true maxima being one less than the number of node surfaces is one less
than the number of the note. In particular there is no true maximum for note (1).
The following table gives the positions of the layer supplying the true maxima in the
first four notes for the values 0 and -25 of o:—
TaBLE XIII.
ok (OV mn—n\ .. BiAard
Values of b/a where As Ga aa) is a maximum.
Note (2) Note (3) Note (4)
aa SS —_— Ms TT.
0 573 B58 ‘786 261 573 852
a 1-95 566 356 “782 260 ‘572 850
For the ratios of the first to the subsequent maxima ordinates, and so for the
ratios of the first to the subsequent true maxima changes of pitch, I find approximately
L452 2A 20GC een
The numerical values of the first maxima in notes (2), (3) and (4), and of the
greatest possible change of pitch in the case of note (1) are given by the following
table for the values 0 and ‘25 of o:—
30—2
230 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
TABLE XIV.
: : ok (oOVn—n
First maximum of 5 ie (> = =)
Note (1) Note (2) Note (3) Note (4)
———— . ee ————
c= 0 25 0 25 0 25 0 25
209" | -263* 1024 1-023 1606 —- 1-606 2190 2190
The asterisks are intended to draw attention to the fact that the entries under
note (1) are not true maxima. The influence of o in the case of the higher notes is
practically nil.
As fairly approximate values for the first and for the j maximum respectively in
note (z), supposing 7 and 7 both greater than 4, we may take
ab (@V m—n
i (+ m+n,
Ok . OV ny — _ _ 2190 + (¢— 4) x 584
k (F m+n/ 485 +(j—4) x 120
) =O100 (4 = 4) Ded nee ee (68),
These equations hold for all values of oc. For values of j less than 4 the ratios
given above should be used.
§ 78. When the layer of altered rigidity is of given thickness the curve showing
the variation of dk/k with the value of kab is B fig. 8, the equation to which is (55).
In general we see that when 7 is greater than 2 the (¢— 1)" segment of curve B
fig. 8 corresponds pretty closely in position and magnitude of ordinates to the 7" segment of
curve B fig. 7.
The abscissae supplying the maxima ordinates of curve B fig. 8 are the roots greater
than zero of
(CARS) ACD) abate GCS) e eaobngontteenaoa dosecduacoase: (70).
For their approximate values I find 3°311, 6°787, 10:0215, 13:209...._ These roots are inter-
mediate between those of (34) and (67).
For note (1) there is no true maximum, as the number of maxima is one less than
the number of the note. The positions of the layer supplying all the maxima in notes
(2), (3) and (4) for the values 0 and ‘25 of o are shown in the following table :—
TABLE XV.
Values of b/a where oe (7 ae =| is a maximum.
k amt+n
Note (2) Note (3) Note (4)
— —————— TO
nes jo ‘621 ‘388 ‘795 ‘283 580 ‘856
” [25 ‘614 ‘386 ‘791 ‘282 ‘578 854
Mr C, CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 231
For the ratios of the first to the two next maxima ordinates, and so of the first to
the two next maxima changes of pitch, I find
1 : ‘880 : *860.
The fourth and subsequent maxima are only very slightly less than the third.
In the following table are given the numerical magnitudes of the first maxima for
notes (2), (3) and (4), and of the greatest possible change of pitch in the case of note (1).
TABLE XVI.
First maximum of Be + (; ia ") F
k am+n
Note (1) Note (2) Note (3) Note (4)
a DM = OO ST ot
c= 0 5 0 oD 0 Hy 1X0) 25
‘418* -524* | 1-224 1-208 [1199 1194]1192 11189
The asterisks under note (1) indicate as usual that the entries are not true maxima.
From the table, with the assistance of the ratios given above, all the maxima in
the notes (2), (3) and (4) may be calculated.
In notes above the fourth a pretty close approximation to the first maximum will be
given for all values of o by
Chat Wat eee
Ta a G i =) PTS Digecranic oem syevessasneilave ceed stieew nese (71).
From this and the ratios given above, the values of the two next maxima may be
found. For maxima of number greater than (3) in these higher notes we may take ap-
proximately
On (: this =) EST NO OME oe as docs cae (72).
kk \am+nu
§ 79. In the fourth special case the change of pitch is given by (50). For the
hmiting value °5 of o this assumes the simple form
ok _t p eee Ola) Gy .
———— tp Qa {Jo (x)} = aya l+p DY {Jo (x)} SSOOCS OOOO OOOOOOUCLOUG (73),
writing « for kab.
This becomes identical with (48) when the factor (m,—m)/(m,+n) of that equation
is replaced by p/(1+p). Thus the conclusions already come to in the case when m alone
is altered apply also to the present case for o=°5 with merely a change in phraseology.
Except in this extreme case the expression (50) for the change of pitch is the swm
of two squares, which cannot simultaneously vanish unless b/a=0. Further we see from
§ 68 that when an alteration of material of this kind takes place throughout an axial layer
of given volume the signs of 0k and p are the same. Thus an alteration of both elastic
232 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
constants in the same proportion throughout a layer of given volume necessarily affects
the pitch wherever it occurs, and the pitch is raised or lowered according as_ the
elasticity of the layer is increased or diminished.
In considering (50) it will be convenient to consider separately the two squares by
writing
Bis SH Dik. or hee OU. ene ost (74);
where, o denoting as usual Poisson’s ratio,
dle te Weep lle
k = qb’ @—o) jesip {Jy (kab)}* eRe eee eee eee eee eee Tee eee! (75),
ok, t p ; a a Jd, (kab))*
eae ep eh {v1 (la) 3 Eo eessessneeeceatete (76).
The numerical magnitude of dk, is independent of the sign of p, whereas 0k, is numeri-
cally greater for a given negative value of p than for an equal positive value.
Again 0k, depends on the square of the displacement and so vanishes when the
altered layer is at a node surface. The more remote positions of the layer supplying
the maxima of dk, in the case of the higher notes are inside of but close to the loop
surfaces of higher number whether the layer be of given volume or of given thickness. On
the other hand 0k, depends on the square of the radial stress. It thus vanishes when
the altered layer is at a no-stress surface, and when the layer is of given volume it has
its maxima when the layer coincides with those surfaces over which the radial stress is
a maximum,
Further the law of variation of 0k, with kab is independent of the value of o, but
the maxima of 0k, diminish rapidly and become insignificant as o approaches near to its
limiting value ‘5. On the other hand so long as kab is small the law of variation of
ok, with kab depends largely on the value of c.
In the case of notes (1) and (2), or for positions of the layer inside the third node surface
in the case of the higher notes, the contribution of 0k, to the change of pitch cannot
in general be neglected. For more remote positions of the layer, however, in the case of
the higher notes 0k,/ék, is always insignificant, except in the immediate neighbourhood
of the no-stress surfaces where 04, vanishes. Thus so far as the maxima changes of pitch
are concerned the error introduced by neglecting 0k, is very trifling when the layer lies
outside of the third node surface in the case of the higher notes.
It may also be proved from (76) that the value of o has very little influence on
the maxima of ok, of number higher than 2.
We thus conclude that for practical purposes the change of pitch due to the alteration
of elasticity of the kind under discussion is given to a very fair degree of approximation
by (73) for all values of ¢, provided the layer lie outside of the third node surface of
the note considered.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 233
§ 80. When the change of pitch is wanted for positions of the layer answering to
small values of kab, it will in general be best to construct separately curves showing the
variation of dk,/k and ok,/k, and then derive from them compound curves.
For the variation of 0k,/k we have the curves
OS CNH 77.5(GeNnoccqanooceagasonbecaDa9eBeGEGbDOeG (77),
or Sa Ch OPS jg @)ococccccsoceovosocoogoo0ccennepoa000 (78),
according as the layer is of given volume or given thickness. These curves are those
styled C and D respectively in fig. 6.
Between the origin and the next zero ordinate of curve D,—which answer to positions
of the layer at the first and second node surfaces respectively—the ordinates of both
curves are far from insignificant compared to the ordinates of the other curves.
Beyond the third zero ordinate—which answers to a position of the layer at the third
node surface—I have not drawn -the curve D. Its successive segments become rapidly
flatter, as may be seen at once from the consideration that in fig. 6 the ordinate of
curve A is the geometric mean of the ordinates of curves B and D.
The curve C is drawn only as far as its first zero ordinate, answering to the second
node surface. An idea of the extreme flatness of the other segments is easily derived
from the consideration that the ordmate of curve D is the geometric mean of the ordi-
nates of curves A and C.
For the variation of 0k,/k we have the curves
fy J, 2
y= \y, Oe Or FE CA ee: (79),
a“
Ul) = aia @) Hal @ aacaoboocccnon soca cascos ones osonascescooousnbc00n¢ (80),
according as the layer is of given volume or given thickness.
These curves are drawn for the special value 0 of o in fig. 9 and are styled re-
spectively A and B. Both have zero ordinates answering to positions of the layer at
all the loop surfaces. At the origin the ordinate of curve A is precisely equal to that
of curve C, fig. 6, and for all other values of # less than 2 the ordinates of the latter
curve are the larger. In fact the ordinates of curve A do not markedly predominate
over those of curve (C, fig. 6, until the layer has passed well outside of the first loop
surface.
Curve B fig. 9 has a zero ordinate at the origin, and the first segment lies completely
inside the first segment of curve D fig. 6. The great predominance, however, of the second
and subsequent maxima ordinates of curve B over the second and subsequent maxima
ordinates of curve D fig. 6 is a complete justification of what has been said of the general
insignificance of 0k,/dk, for positions of the layer outside the third or even the second node
surface.
In the case just considered when o=0, the compound curve is constructed, accord-
ing as the layer is of given volume or of given thickness, by adding the ordinate of
234 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
curve C fig. 6 multiplied by 1+p to the ordinate of curve A fig. 9, or by adding the
ordinate of curve D fig. 6 multiplied by 1+p to the ordinate of curve B fig. 9. The
quantities represented by these compound curves are respectively
ak (av p Q dk /t p
(7 bp 9) mn BoE 250)
§ 81. As a complete graphical representation of the law of variation of 0k/k with
small values of kab for some one case when the elastic constants are altered in the same
proportion seems desirable, I have considered the most important special case, viz. when
p is so small that p? is negligible and o has the value ‘25.
In this case for layers of constant volume and of constant thickness respectively,
the curves are
= i Ops da (NAM Si) ecanccnoconsacoransaccocece (81),
8) = Bs fory) (a) Vira (G2) aor fasten seis ota lo’ sins cterefe sb siotssiaenls ttebleeeeceseeseee (82).
These are styled A and B respectively in fig. 10, and the quantities they represent are
== (FP = and — (572).
a
The marked differences between the earlier portions of these curves and the corre-
sponding portions of the curves A and B of fig. 9 are well worthy of notice.
§ 82. There is still one point worthy of explicit reference. As we have already
pointed out, 27J,(z) when @ is large is in general negligible compared to J,/ (a). Now
if we neglect «2 1J,(#) compared to J,’(w) and suppose the layer to differ from the
remainder only in elasticity, we may throw (27) into the simple form
ok _¢t ( _ m+n
My +My
r=: ) QR? eed i el (83),
a formula which is exact for positions of the layer coincident with any node surface.
Thus when the layer is outside the third or even the second node surface in the
case of one of the higher notes, the change of pitch due to an alteration in elasticity alone
may be regarded, when of practical importance, as due very approximately to the altera-
tion in a single elastic quantity, viz. m+n. This result should be compared with that
found for the radial vibrations of a sphere in § 48 Sect. II.
Note to Section IV.
The ultimate practical coincidence of the corresponding curves of figs. 7 and 8, and
the fact that their maxima and zero ordinates ultimately almost coincide in position
with the zero and maxima ordinates respectively of the curves of fig. 6 are of course
entirely due to the relations between
J,(z), Jy(a) and J, (za).
Mr C, CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 235
We have already pointed out that the successive values of x, When large, which
make any given Bessel zero increase very approximately by 7, and each is very nearly
equidistant from two consecutive values of « which make the square of the Bessel in
question a maximum.
Now from the relations between consecutive Bessel’s we have
— Ti (@) =F, (@) = 5 (Jy (x) + Soa), Wy (w) = Jy (@) — Ja (2.
Thus when J,(7) vanishes {J,(x)}* is a maximum, and when {J, (z)}? has either its
maxima or its zero values we have {J, («)}? = {J,(x)}*.
Thus the higher values of 2 which make {J,(«)}* and {J,(«)}? maxima, and the
higher values which make them zero, respectively coincide with or are very close to
those higher values of 2 which make {J,(«)}? vanish, and those which make it a
maximum. Also corresponding maxima of {J,()}* and {J,(«)|*, except the first one or
two, are nearly equal.
[November 14, 1891. If while ~ is altered the bulk modulus m—~n/3 remains unaltered, the
change of pitch is given, writing x for kab, by
epee Ae oy le
Fae oem O52 | @)- 5277, OF +
kk” a@ m+n,
3m, +n,
4 min
fa-' J, Ome
It has obviously always the same sign as 7, —~7.|
SECTION YV.
TRANSVERSE VIBRATIONS IN SOLID CYLINDER.
§ 83. In this form of vibration the displacement is at any point at right angles
to the plane which contains the point and the axis of the cylinder. Employing J,(«)
and Y,(z) for the two solutions of
d'y 1 “+ y(1-4)=0,
\ a
dhe © a ales 2
we obtain for the displacement in the typical vibration in a shell
OKs ep (sip) te 1815 (UW/S¥P))snconqsonugonocedeonooropncoes Qs;
where (BY =p |ercacente eecrrer cad dews sua terseessesuneamercees (2),
and A and B are constants.
In a compound solid cylinder (0.8.c,8,.b.8.a) where b—c is small, the typical
displacements are as follows :—
In the core ajcos kt= AJ; (KBr). -....----- ones ee cinee sesinns samsaiceresvesisie ses (3).
In the layer Heol (Uy St) ab Jen (HSH) sone tocencoseebsnoncscososonbeae (4).
Outside the layer v/cos kt = (A + 0A) Ji (k8r) + OBY; (hBT)......0...0c0receesesees (5).
Terms in (b—c)?, and so those of order (@A/A)? or (0B/A)*, are as usual neglected.
* Transactions, Vol. xtv. Equation (44), p. 356.
Vou. XV. Parr II. 31
2356 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
If for shortness we put
Fir. B) =n kBrdy (kBr) — J, (kBr)},
Fi (vr. B) =n \kBr YY (kBr) — Yi(kBr)}
then the relations connecting the arbitrary constants and leading to the frequency
equation are—
AF (c.B)= A,F (c.8:)+ BF, (c. Bi),
(A +0A) Jy (KBb) + OB Y, (&Bb) = AJ, (hB,b) + B,Y, (Bib), \.... 2.02 ceeee eee nee es
(A +04) F(b.8) + 0BF,(b.8)= A,F (b.B,) + B,F, (b.B,), |
(A + 0A) F(a. 8) +0BF, (a. 8) =0
AJ, (kBc) = A.J, (kByc) + BLY, (kB,c), |
Referring now to the radial vibrations of a solid cylinder in Sect. IV., we see that
the transverse type of displacement differs from the radial only in being a function of
k8r instead of kar. Also all the surface conditions in the transverse vibrations can be
deduced from those holding for the radial vibrations by simply writing 8 for a and
supposing m to vanish, We may thus at once deduce all the results we require for the
transverse vibrations by making m zero and writing @ for a in the results already obtained
for the radial vibrations.
The frequency of transverse vibrations in an infinitely thin shell vanishes, and thus
(27) Sect. IV. transforms into
ok b—cf pi—p/b\*(m\? fl 1\ & (Vo
—_ |- ; (7) (*) +(5-—) email) | te ee (8),
where
» = J; (kb), 9
V7, 2 rib GbT: BBY FESO 2 REBT. Cae) en (9).
Obviously 1 cos kt represents a displacement during a vibration of frequency &/27 in a solid
cylinder and V;, cos kt the corresponding transverse stress.
§ 84 In the core the only change in the type due to the existence of the layer
consists as usual of a displacement of all the node, loop and no-stress surfaces according
to the law
= OFT Ol erehs hateeeraltcine waicin usin side. costal ae (10).
From (29) Sect. IV. we find for the displacement outside the layer
¢ 2 } = 1 1 mn :
a | (pi =p) etm Os Bo) (-,) rf0.8.7)} 1S: (11);
where C has the same meaning as in (17) Sect. IV., and with our usual notation
f(.B.7) = J, (kBr) Y, (kb) — Y,(kBr) J, (kb),
f(b. 8.7) =J, (kBr) F,(b. B)— Y,(kBr) F(b. B)
In the layer itself the displacement is given by
»/A cos kt = J, (kb) — kB (b— 7) J/ (kBb) — (rv —c) (- - 2) Vigtester ae Atti Food (1l3})}
v/A cos kt = J, (kBr) + :
nv
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 237
The change of type outside the layer, ie. the coefficient of b—c in (11), consists
like the expression (8) for the change of pitch of two terms only. There is an exact
correspondence between the terms in the two equations. The first terms in each depend
only on the alteration of density, and simultaneously vanish when the layer is at a node
surface. The second terms depend only on the alteration of rigidity, and simultaneously
vanish when the layer is at a no-stress surface.
The change of type in the layer itself is the last term of (13). Thus if there be
an alteration only in density, or an alteration in rigidity occurring at a no-stress surface,
then no progressive change of type manifests itself as we cross the layer, i.e. the layer
vibrates as if it were of the same structure as the core.
§ 85. For a discussion of (8) we require to know the characteristics of the transverse
vibrations in a simple cylinder.
Taking (3) as the type of vibration, we see that the node surfaces are obtained
by equating k8b to the roots of
This is the same as (34) Sect. IV., and its roots are thus already recorded.
The radu of the loop surfaces are found by equating #8b to the roots of
This is the same as (35) Sect. IV., whose roots have been already given.
The radu of the no-stress surfaces are found by equating &8b to the roots of
dh yada (Gye dis @) S Oscaopnccconeanorocaccdonccdeeence (16).
This is the same as (66) Sect. IV., whose roots have been already given. Writing ka
for « in (16) we get the frequency equation.
Since the equations (14), (15) and (16) do not contain o explicitly, it follows that,
for any note of given number, the ratios borne by the radii of the several node, loop
and no-stress surfaces to the radius of the cylinder are the same for all isotropic
materials. Also the ratio of the radii of any two surfaces of given numbers, whether
node, loop or no-stress surfaces, in a given cylinder performing a given note is the
same whatever be the value of o or the number of the note.
Since (14) and (15) are the same as (34) and (35) Sect. IV., it follows that the
ratios subsisting between the radii of the several node and loop surfaces in a cylinder
performing one of its transverse vibrations are precisely the same as those subsisting
between the radii of the several node and loop surfaces in a cylinder performing one
of its radial vibrations.
Since, however, the frequency equation (16) would agree with the frequency equation
(36) Sect. IV. only when the physically impossible relation m/n=0 was supposed to
exist, it follows that the ratios borne by the radii of the node and loop surfaces to the
radius of the cylinder cannot in any isotropic material be the same for a radial and
31—2
238 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
for a transverse vibration. The ratios also between the frequencies of the several notes
which are produced by a cylinder vibrating radially cannot possibly be identical with
the ratios subsisting between the frequencies of the several notes produced by a cylinder
vibrating transversely, These latter ratios, it will be observed, are independent of the
value of ¢, and so the same for all isotropic materials.
Comparing (16) with (36) Sect. IV, we see that when @ is large they both approach
the form
lga(@)i—10:
Thus the higher roots of the frequency equations, both transversal and radial, ap-
proach more and more nearly the larger they are to the roots of (15). Thus the higher
notes of the two modes of vibration in a given cylinder correspond to one another in
pairs, such that the two sets of node and loop surfaces become nearly coincident, and
the frequency of the transverse vibration is to that of the radial approximately in the
constant ratio
adh n: Jm +7
A similar result, it will be remembered, was found in the case of the sphere.
The positions of the several node, loop and no-stress surfaces for the first four
notes are given in the following table. It applies to all values of o.
TABLE I,
Values of r/a over node, loop and no-stress surfaces.
Note (1) Note (2) Note (3) Note (4)
a SS a a cr —= = =
Node No-stress Loop Node No-stress Loop Node No-stress Loop Node No-stress Loop
surfaces surfaces surfaces surfaces surfaces surfaces surfaces surfaces surfaces surfaces surfaces surfaces
0 0 359 | 0 0 "219 | 0 0 “158 | 0 0 124
746 «61:0 “455 610 633 330 4420 459 ‘259 347 = = 360
834 1:0 | “604 724 “735 | “474 569 | -a17
875 ~=1:0 688 LSD ae Gol
| ‘900 10
It will be observed that the number of loop surfaces always equals the number of
the note, and is one less than the number of node or of no-stress surfaces. Also the
loop surfaces, precisely as in the rotatory vibrations of a sphere, lie outside of the
corresponding no-stress surfaces, and not inside them as in the case of radial vibrations
both in spheres and cylinders.
The axis has the curious property of being at once a node and a no-stress surface.
In comparing the transverse and radial vibrations it will be found that note (¢—1)
of the former class corresponds to note (i) of the latter.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS, 239
§ 86. I shall consider first two special positions of the layer.
Supposing in (8) b/a very small, while (/—c)/b is also very small, we obtain the
change of frequency due to the presence of a thin axial layer differing from the rest of
the material. It will be found that 0% vanishes under all conditions. The same result
may independently be proved for a core of small radius. Thus, to the present degree
of approximation, no change in pitch follows any alteration of material throughout a thin
axial layer or core.
§ 87. Putting b=a and V,=0 in (8) we pass to the case of an alteration of material
throughout a surface layer of small thickness t=b—c. For the change in frequency we
get the simple result
A surface alteration in elasticity has thus no effect on the pitch of any note, and
a surface alteration in density alters the pitch of all the notes in the proportion of their
original frequeucies, and so leaves their ratios unaffected.
§ 88. Let us now consider the general case when the density alone is altered. As
the change of pitch vanishes for an altered core we may without restriction put
b(b—c)/a? = 40V/V,
b(b—c)(p, — p)/a?p =40M/ M.
From (8) we find for the change of pitch
ak _ t p:—p kBb ie em am ie eal
i p kBa J,(kBa)S} WM 2S, (kBa)
i
a
The change of pitch vanishes when the layer of altered density coincides with a
node surface.
When the layer is of given volume, the curve showing the law of variation of dk/k
with &8b is
This is the same curve that applies in the corresponding case of the radial vibra-
tions. It appears as curve A in fig. 6. The function of x appears as f,(~) in Table VIL,
Sect. IV.
This curve has been already discussed in § 73 and the ratios of its successive maxima
ordinates recorded.
The positions of the layer supplymg the maxima, are coincident with the loop sur-
faces. The first and largest maxima, answering to positions of the layer at the first loop
surfaces, are given in the following table for the first four notes:
240 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
TABLE II.
First maximum of — = Ma
Note (1) Note (2) Note (3) Note (4)
1468 2°299 31338 3968
The number of maxima is equal to the number of the note.
The first maximum for the (¢— 1) transverse note is practically identical with that
for the 7 radial note. Also the ratios of the first to the subsequent maxima are
the same in the two cases. Thus from (58) and (59) Sect. IV. we find as pretty close
approximations to the first maximum in note (7?) and to the j* maximum in the same
note respectively, 7 and j being both greater than 4,
) ; “awe
— | + yp = 3968 + @— 4) x 835 A Ria eisutajnie(a{e/a/winte'wieteleainraieveiets tele (20),
_ 0h | 0M _ 3968 + (4 — 4) x 835 (21)
E Mpa 4) eB ols baa :
Maxima of number less than (5) can be obtained by means of the ratios given in
§ 73 for any note in which the first maximum is known.
§ 89. When the layer of altered density is of given thickness the curve showing
the law of variation of 0k/k with kab is
6 gro Bila 2) | eee P BREE Reape E CONE Soran sc sdkc soe (22).
This is the same curve that applies in the corresponding case in the radial vibra-
tions. It appears as curve B in fig. 6, and the corresponding function of « appears as
Ff;:(#) in Table VIL, Sect. IV.
This curve has been already discussed in § 74, Sect. IV.
The number of maxima is always equal to the number of the note, and the positions
corresponding to the maxima in the first four notes are all shown in the following
table :
TABLE ITI.
Values of b/a supplying maxima of — s - (; ie ae Ne
Note (1) Note (2) Note (3) Note (4)
"422 250 186 146
645 467 367
‘740 581
‘794
The magnitudes of the first and largest maxima, answering to the positions nearest
the axis in the above table, are as follows:
Mr ©. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 241
TABLE IY.
First maximum of — as (= paee. ).
keeNat Wp
Note (1) Note (2) Note (3) Note (4)
1149 1:098 1:084 1078
As in the case of the radial vibrations we find that in the higher notes a close
desta é ok ~ : :
approximation to the first maximum of iia ae pP 3 f) is 1:07, and to any maximum
whose number exceeds (4) a close approximation is 1:00, Maxima of number less than (5)
can be obtained by means of the ratios given in § 74 for any note in which the first
maximum is known.
§ 90. When the elasticity alone is altered, we find from (8) for the change of pitch
ok _t m—nkBb (J, als _oVn—n 1 aaa 23
bam kBald.(kBal ~V om, 21d.(kBat cc (23).
The change of pitch thus depends solely on the alteration of rigidity. It vanishes
when the layer is at any no-stress surface, and has for all other positions of the layer
the same sign as n,—n. Its law of variation with the position of the layer is inde-
pendent of the magnitude of the alteration in rigidity.
When the layer is of given volume the curve showing the law of variation of dk/k
with kab is
Y= 1s (G) ecsmeateenecercesedautest este esasene sc cade: (24).
This is the same curve that applies in the case of the radial vibrations when an
alteration in rigidity alone takes place throughout a layer of given volume. It appears as
curve A in fig. 8, and the corresponding function of « appears as f,(#) in Table VIL,
Sect. IV.
This curve has been already discussed in § 77, Sect. IV.
All the positions of the layer supplying maxima in the first four notes are given
by the following table. They coincide with those surfaces over which the transverse stress
is a maximum.
TABLE V.
: Pearl. ——| i 3
Values of b/a where ae a —— oo
Note (1) Note (2) Note (3) Note (4)
595 363 263 ‘206
‘797 ‘BIT 453
858 674
‘890
The first and largest maxima in the case of these notes are as follows:
242 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
TABLE VI.
: 2 ok (oVru—n
First maximum of al ( i <a ) :
Note (1) Note (2) Note (3) Note (4)
1:026 1607 2°190 2774
The first maximum of Gk/k in the (¢—1)™ transverse note in the present case
is practically identical with the first maximum of 0k/k in the 7 radial note in the case
when the rigidity alone is altered throughout a given volume, and the ratios of the first
to the subsequent maxima are the same in the two cases. We thus find, as fairly close
approximations for the first and j maxima respectively in note (7), supposing ¢ and j
both greater than 4,
ok =jeVn—n :
—+(=, —)=2774 == AN SED SAN sao deicrsieeatieciasesucecumeicees 5
: (> = ) +(@— 4) x°5 (25),
ok . Ga) 2774+ (i— 4) x 584
F 7 — V = n, - 485 + (j —4) x 1:20 Be meee eee eee eee ween een eeeee
Maxima of number less than (5) can be obtained by means of the ratios given in
§ 77 for any note in which the first maximum is known,
§ 91. When the layer of altered rigidity is of given thickness the curve showing
the law of variation of ok/k with kab is
Af =i fella (At) eines ate « othe alciseis ee eaioceSoecies saeeteteeeeeere (27).
This is the same curve that applies in the case of the radial vibrations when an alteration
in rigidity alone takes place throughout a layer of given thickness. It appears as curve B
in fig. 8, and the corresponding function of # appears as f,(#) in Table VII, Sect. TV.
This curve has been already discussed in § 78, Sect. IV.
All the positions of the layer supplying maxima in the first four notes are recorded
in the following table:
TABLE VII.
tm—n\. ;
: -) 1S a Maximum,
a %
Values of b/a where 2 ={
Note (1) Note (2) Note (3) Note (4)
645 B93 285 224
806 584 459
*862 677
893
The first and largest maxima in the case of these notes are as follows:
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 243
TABLE VIII.
: ok tn—n
First maximum of — + (2 ae
iE aie
Note (1) Note (2) Note (3) Note (4)
1-273 L207 1:201 1195
For all notes of higher number a fairly close approximation to the first maximum change of
pitch is given by
ok tm—n
k = (
g |e et Mey LO LD Ds (28).
\a ny
For all maxima of number greater than (4) we may take as a close approximation
wa (: Gis ") AIRMEN tite tins a Bae (29).
a ni
Maxima of number less than (4) can be obtained by means of the ratios given in § 78
for any note in which the first maximum is known.
pe
SECTION VI.
RaDIAL VIBRATIONS IN SPHERICAL SHELL,
§ 92. I now proceed to apply the method of Sect. I. to determine the frequency of
vibration in compound shells,
I shall first consider the radial vibrations of spherical shells.
The type of vibration and of the radial stress in a simple shell are shown in (1) and
(3) of Sect. II. From these expressions we may select the following as the values to be
assigned to the F, F,, G, G, of (1) and (2), Sect. L:
: sin /:
F(a.a)=(m+n) kaasin kaa — 4n = a | cos kaw)
ae Cos Lie) | ces (1),
cos kaa :
F,(a@.a)=(m +n) kaa cos kaw — 4n ( aa. ee haw) Monee Fates Ge eae saaae (2),
‘ sin kaa } ;
G (a,.a)= rn — Cos kaa ainfufefafaln)elefololafacaictetatstatatas afatetatalestatais\e)e}stelss/oialcis/selsiatslelcicie elo (3),
7 coskaa.
Gala) — es =P ISI CA ane naa CEE eS erE GME assis ahcteracten a aieeuckngdaess (4).
Wot, XOV, IPA IE
244 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
The form of the frequency equations in a simple shell (b.a.a@) of the types free-free,
fixed-free, free-fixed and fixed-fixed are given in equations (3), (4), (5) and (6), Sect. I.
For the present case these lead to:
J (b.a.a) =sin ka (a — b) jim +nphaab — 4n (m +n) cee + 16n? {1+ (ovat) |
—ka(a—b) coska(a—b). 4n jm + + 4n (Kea2ab)7} =O... (5),
F(b.a.a) =(m+n) kaa cos ka (a — b) + ab sin ka (a — b)}
— 4n(kevab)> {(1 + h*a2ab) sin ka (a — b) — ka (a —b) cos ka (a — b)} =0 wee (6),
J (b.a.@) = (m+n) \ba~ sin ka (a — b) — kab cos ka (a — b)}
— 4n (keaab) {(1 + katab) sin ka (a — b) — ka (a — b) cos ka (a — b)} = 0.00.00... (7),
f(b.a. @) =sin ka (a —b) {1 + (h2a2ab)} —hka (a — b) (Ka2ab)— cos ka (a — b) = 0... eeeeee ee (8).
The above expressions are the exact forms of f(b.a.a) ete. and are not reduced by
division or multiplication by any factor.
If the shell be so thin that terms in (a—b)? may be neglected the expressions
become:
f(b. a.a)=ka(a—b) (m+n) {(m +n) e2a? — 4n (Bm — n) (M+ NYA ee ceeeeeneees (9a),
= haa — 0). (ra 90) par (le BG) a sass sate e orate aaieet sl eisls nota eee ee (%),
Fas a) = (4-1) eae Fa (GB) re eos scsi svscoweassaenbnedencsanteeaseeee (10),
FO ..a@.0) == (n--n) had +- Qiea(a —b) (00) teen cessisciee eens eeieeeneeeeseeeeaaeer (ily,
AD it es ea (ID) ctw chchs nn vavh pa ody tee MRR eee Se CRE S eae en (12).
The meaning of K,.,), etc. is the same as in Sect. II]. In the coefficient of a—b
we may of course replace a by b.
Equating the coefficient of a—b in (94) to zero we get the frequency equation for a
free-free vibration. None of the three other types has in a thin shell a vibration of
finite period.
By supposing in (6) and (7) b absolutely equal to a, we find
f(G@.4.a)=F (a.a) G, (a .a)— FP, (@. a) G(@.4) =(MAN) Had oor. scecercoecnsevees (13),
f(a.a.a)=G(a.a) F,(a.a)—G, (a. 4) F(a.a)=— (MN) had ..ccrceceveceeess (14).
These quantities cannot vanish unless k, a, or a vanishes, and thus the occurrence of
either as a factor in a frequency equation does not supply a note of possible frequency.
This proves for the present case the truth of a statement made in § 9, Sect. I.
Employing the relation (13) in equation (30) Sect. I., we find
lf Gol tol oek: sel DE fet) JH Cea 2) pocket [eV (Bish (Na scncriasionnndogecona neck (15).
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 245
; Lis bys ;
We also require the value of & aii (@.a.a) under the condition that h/27 is the
frequency of a free-free vibration in a simple shell (e.a.«),
Looking on kae and kaa as independent variables, we may put
bo F(e. ah a= | hee 4 a + kaa = | fe. (2 510) odbc oe ROCCE ESC HOgOe (16).
A form of f(@.a.a) may be got by writing e for 6 in (5). It is simpler however
in obtaining the above differentials to deal with the unreduced form obtained by the
immediate substitution in (3) Sect. I. of the expressions (1) and (2) for F and F,.
It will suffice to give the work in one case. Thus
cos kze
f(e.4.a)=F(a.a) {om +n) kae cos kae — 4n ( eee + sin kae)}
— F, (a.a) |i +n) kae sin kae — 4n es — cos hae}
d
Be ae A “tae A. «)
=F (a.a) [ee a + sin hae) | —(m + n) k*o2e? + 4n} + 2 (m — n) kae cos kee]
— F,(a.a) (ae cos kae) {— (in +n) kate? + 4n} + 2 (m — n) kae sin kae| :
— (m+n) kee? + 4n} \F(. a) foe = we + sin ke) — F,(a.a) Vee — cos hze))
+2(m—n) kae {F (a.«) cos kae — F, (a. a) sin h2e}......ceeccecceeseeees (alin
Remembering that /(@.a.a) is supposed equal zero, and employing the expressions
supplied by (1) and (2) for #(e.a) and F,(e.@), we find
F(a.a) cos kae — F, (a. a) sin kae
tn 1 ‘cos kae = sin hae
~ ia, Ee / F(a.a) ( ae Gs sin kae) — F(a. a) (= Fae 7 293 lae)| :
Substituting thence in the coefficient of m—n in (17) and putting the terms together,
we find
a ona,0 2 (3m = —_
hae =~ us eee: a.a@)=—(m+n) {ire . Gas
fF (a. a) eons ae + sin hae) — F, (a.2) Ee teRe cos kae)| sdondaoocdogeobanAded (18).
Finally noticing the forms of @ and G, in (3) and (4), the expression for f(@.%.«)
supplied by (4) Sect. I, and the expression (25) Sect. II. for the frequency of radial
vibrations in an infinitely thin shell, we obtain
d P| = ae
hae do hxet (°%: Oe NaS ces a) fe (El wil) nine aroanni vie svesntine (19).
246 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
In an exactly similar manner it may be proved that
kaa d a= F(e.@.a) =— pa (ke — K%a.ay) f (C1 A.D) criccrseeneeeseens (20).
Thus
k 4 fle .a.a)=—p le (ke — Ka.) f(e.a.a) + @ ( - Lele YF (Cxein@)) ssnonc (21),
where after differentiation / is treated as a root of the frequency equation f(e.a.a)=0.
The results (19) and (20) are particular cases of the general theorem treated in
§ 10, Sect. I.
§ 93. We now possess all the data necessary for determining the change of pitch
in the radial vibrations of a spherical shell due to the existence of a thin layer differing
from the rest of the material. Supposing the shell to be (e.a.c.4.b.a.a), we have
from the general result (23) in Sect. I.
fle.a.c.m.b.a.a)=f(b.a.a) {f(e.a.c) f(E.a.b)—f(e.a.0)f(c.m.b)}
—f(b.a.a){f(e.a.c)f(E.m%.b) —f (6.0.6) f(C.0,.B)} eesccsevees (22).
Now supposing the layer (c.@.6) so thin that terms im (b—c)* are negligible, let
us employ the relations (9,)—(12) for a thin shell. Then, replacing ¢ by b in the coefficient
of b—c, we find for the frequency equation
f(e.a.c.%.b.a.a)
(m, + 1) kab
=f(e.a.c)f(b.a.a)—f(e.2.c)f(b.a.a)
be (1) a, 3m) flea. FO. 1.0) +2(m—m)f(e.a-D)flb.2.0)}
Xb-e b? (hk? — K%q,.»)) f(e.a.b)f(b.a pp i ln a.b) f(b.a.a) =0....(23)
a acai (a, 0 -A.0)7(0.a. bo ise ee a
Writing a, m, n for %, m,, m% respectively in (23), we get a similar expression for
f(e.a.c.a.b.a.a) + {(m +n) kab}.
Employing this last expression in (23), we easily find for the frequency equation
_f(e.a.c.a.b.a.a)_f(e.a.c.a.b.a.a)
(m+ 7%) kab (m+n) kab
= c : > 1 1
+ E ip (ke — Kn) — pi (I? Ke, nF 0-4-0) G20) + ( 1m +1 )fle.4 DF 0.4.0)
M+tn MmM+tNG
+(2= Bm; _ m— BM) oe ab) f(b.a.a)+2 (== oes "\re .a.b)f(b.a. | Ce anea
. ml, +1, m+n / Mm+zAmh m+n
Remembering (15) we may in (24) put
PCB C08 AO) (mn) kuch (e.0.0) Asioeseeancue (25).
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 247
Suppose now that 0k/27 is the increase in the frequency of a note due to the
presence of the layer. Then & being supposed a root of (24), k—@k must be a root of
J(e.a.a)=0. Thus assuming ok of the order b—c, the above equation (24) must be
identical with
Fe.a.a)—Ek4 f(e.a.a) =0,
Le. with
S(e.a.a)+ us p (e(? — K*a.o) f (E.a.a) +07 (hk? — Ki.) f(e.a.a)} =0...... (26).
Making the substitution (25) in (24)—replacing ¢ by b since f(e.a.a) is of order
b—c—and then comparing the identical equations (24) and (26), we find
= (m+n) kabp [e (k? — K% 0.0) f(@.a.a) +0 (k®— Kaa) f(e.a.@)} + = G
=D? {p (k? — K? 0.4) — pi (kh? — K%Qa,.»)| f (e.a.b) f(b. 0.0)
+(e PO DS O.c.0) + (MBM _ MAB") Be. a.) 6.0.0)
2 (je = as = “ i ") #(e-a.B) f(b si A600} BAER eee (27).
§ 94. Now, as explained previously, the expression for 0h/h as containing b —c¢ may
be modified by any substitution consistent with f(e.a.a)=0 being exactly true. This
enables us to put (27) into a form which brings out more clearly its physical signi-
ficance.
From (1) to (4) combined with (1) and (3) of Sect. IL, we may suppose the dis-
placement w and radial stress U at a distance r from the centre of a simple shell (e.¢.a),
performing a free-free vibration of frequency k/27,.to be given by
ON COM ir — dn — ALE (oe) tb JENGA (G56) anecodauceonrcnsbonaseccaec (28),
Oho ns SOS AUH (63) +e IBIIE (PG) pbascenocconpoueponccconce (29),
where A and B are constants independent of r or t.
In virtue of the surface conditions we have
PAH (Clicl) tay Hey (4).(Ct)) == = ACHE (apetcd) I= -u SiH (rei CL) lnceeeeeeeiaa ee eseee (30).
Thence we get
fl 9-163 82 JI (@oG)) § SIINOs) 92 = JI (C@iee)) 8 LE(@s@)) coscoocoanpaccones (31).
Employing these ratios in (28) and (29), it is easy to prove
S(e.a.b)+fe.a.@)=by +ara,
1 OC a ACIS) — 8 a (32).
f(e.a.b)+f(e.a.@) =BPU,+ ar,
f 0 .@. 0) =i (Gnas a) — 7 Uy = eu, |
248 Mr C, CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
We also easily prove
{F(@,aP+ {Ff (@.a)} 3
{F(e.a)}* + {F, (e. @)}?
(m+ n) kaa — 4n (kaa) }? + 16n? | 3
\(m + n) kae — 4n (hae) |? + 16°
T(é.a.a) = {F(e.a) Gi (e.4)— Fe. a) @ (e.a)}|
= (m+n) kae [ Risa Masterton mes (33):
and similarly
\(m + n) kae — 4n (hae)? + Een
Jil eG) 0m tty) eae Ee +n) kaa — 4n (kaa)}? + 16n2
saaetaeesen (AOS
Thus VA Ga Re) Sajal ni) = Ge BIO Wet Oaks phooneoanasa sop anonaanoncOaer (35).
Employing the results (82) and (35) in (27), we easily deduce
Ok _b-—c_ ye —(m+n) kaa
ko a pttetta [6 (? —K% ae) f(6. a. 0) + a? (k? —K% a, a)) f (€.a.0)]
x [oa (p (k? —K%a.0)) — pr (? —K%a,,0))} + PU? ( ae ee )
Mtn m+ %
+ 8bisUs ( e a )| ...(36)*,
mM+nr Mm+tnH
§ 95. The deduction from (36) of the formula for the special case of a solid sphere
requires careful treatment. Thus the term in the denominator containing
Cue (kh? — Ka.) f(E. a.)
is easily seen to vanish with e, but u,f(e.a.@) assumes the form 0x.
To avoid this difficulty we may by means of (35) replace the second of equations
(32) by
Ue f (.a.4) =— (m+ ny ha@abuy + f(D. A.A) crrecercreccseceeeneeees (37).
Thence proceeding to the limit when e vanishes we easily find
Ug if (Eis Os) = — (MAI) IAD Ug wa sisaietedin's n/stfein vs ihelw ane Soledy (38).
This leads to the same result as was obtained in Sect. II.
§ 96. The right-hand side of (36) is the product of two factors of which the second
alone is a function of b. It contains uw and U, in the same way as does the right-
hand side of (28) Sect. IL, and the physical significations of wu, and U, are precisely
the same as in the case of the solid sphere. The mathematical expressions for w, and
U,, are however, it must be remembered, different in the two cases, those for the shell
being much the more complicated.
As the first factor on the right-hand side of (36) does not contain b, it is for a
given note the same in sign and in magnitude wherever the layer may be, or whatever
be the nature of its difference from the rest of the material, The law of variation of
ok/ke with the position of the layer in no way depends on it, but only the absolute
magnitude and the sign of the change of pitch.
* See the note on p, 266.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 249
For a solid sphere we found the first factor essentially positive. A purely mathematical
demonstration that it is always positive in the case of a shell presents considerable
difficulties, but is I believe rendered unnecessary by the following physical consideration.
Suppose the layer to differ from the remainder only in density, then we have
Thus, unless an increase of density occurring anywhere except at the nodes is to raise
the pitch, the first factor must be positive. This consideration affords I think convincing
proof that the first factor is essentially positive, and that such is the case will now be
taken for granted.
§ 97. As (86) is in form so exactly analogous to (28) Sect. II. for the solid sphere,
a brief discussion will suffice.
When an alteration of density occurs at a node surface of a particular note it does
not affect its pitch, but in any other position it lowers the pitch when an increase and
raises it when a decrease.
The percentage lowering of frequency due to a given increase of density throughout
a given layer is always equal. to the percentage rise of frequency due to an equal
diminution of density throughout the same layer. The law of variation of the change
of pitch, due to a given alteration of density, with the position of the layer is independent
of the magnitude of the alteration of density. When the layer of altered density is of
given volume the positions in which it has most effect on the pitch of a given note
coincide with the loop surfaces for that particular note; when the layer is of given
thickness its most effective positions lie slightly outside the loop surfaces.
If the layer differ from the remainder only in elasticity the change of pitch consists of
three terms. Of these the first has the same sign as, and is proportional in magnitude to
n, (38m, —%) (MM +m) 1 — 2 (Bm —n) (M+ Nn).
It vanishes when the layer coincides with a node surface of the note in question.
The second term has the same sign as, and is proportional in magnitude to
(m+n) — (7m, +7).
It vanishes when the layer coincides with a no-stress surface.
The third term varies as
n(m + nyt =n (7% +%)4;
but its sign depends also on the value of &. It vanishes when the layer coincides either
with a node or a no-stress surface. It likewise vanishes for all positions of the layer
provided
TVG | tie YO sees cet le sna os oeecie cians soe esisiasieniee (39).
250 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
Thus if the uniconstant theory be true, or more generally if the relation (39) subsist, the
sign of the change of pitch accompanying a given alteration in elasticity is independent
of the position of the altered layer, and is the same as that of p. If however the
relation (39) do not hold, the sign of the change of pitch may for certain alterations of
elasticity vary with the position of the layer.
The positions of the layer whether of given thickness or given volume, when a given
alteration of elasticity has most effect on the pitch of a given note would require to be
separately determined for each possible alteration of elasticity. The first term—that depend-
ing on the alteration of »(3m—n)(m+n)"—is largest when the layer, supposed of given
thickness, coincides with a loop surface. The second term—that depending on the alteration of
(m+n)~“—is largest when the layer, supposed of given volume, coincides with a surface where
the radial stress is a maximum. As a function of } the first term varies as (w)*, the second
as (bU,)? and the third as w.bU, when the layer is of given thickness. Now from equations
(1)—(4) we see that when kab is large F(b.a) and F,(b.) are of the orders kabsin kab
and kabcoskab, while G(b.a) and G,(b.@) are only of the orders cos kab and sin kab. Thus
it follows from (28) and (29) that when kab is large w/(bU,) is of the order 1/kab of
small quantities and so is small. Consequently when kab is large the second term—that
depending on the alteration of (i +n)7%—is much the most important, and the third term
is next in importance.
Thus when the effect on the pitch of one of the higher notes due to an alteration
of elasticity is being considered, we obtain in general—unless the alteration occur close to
the inner surface and the radius of this surface be small—a close approximation to the
value of dk by neglecting altogether the first and third terms; and when the change of
pitch of one of these higher notes is of practical importance it may be regarded as due
approximately to the alteration of the single elastic quantity (m+n). The change of
pitch is in such a case greatest when the alteration of elasticity occurs at or in the
immediate neighbourhood of the surfaces of greatest radial stress.
In the case of the two or three lowest notes serious error might however arise
from neglecting the first and third terms, especially when the alteration of elasticity occurs
near a no-stress surface, more particularly the inner surface of the shell.
§ 98. I do not purpose an exhaustive investigation of (36), but one or two of the
more interesting special cases may be considered without much analysis.
Thus let us suppose the layer to be at the outer surface, so that b=a. Then by (10)
Sf (b.a.a)=(m +n) kaa,
and so the second of equations (32) becomes
Fi(E iin ©) = (it 7) Hagieu, |g) scence rcsess-sasieneee seers (40).
Hence by (35)
F (Gs 1) ==) MCA (Cig) soc sean cnonsrerneieyenesccioanel (41).
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 251
Again, owing to the surface conditions, U,= U,=0. Thus from (36), if the thick-
ness of the layer be 4 and the change in pitch dh,
Oley. Uno wiih PPR ae i 5
bE = A a (Ua)? {p (k? = Kee a») = (ih (ke = KGa) = pD SORODUUDCAIOOONUCIOY (42),
where Di ost Ues IK.) i= Ct 2 (G2 — BE (yey astas a Puhale odhiet «od ne, da Savine hat (43).
Similarly if the layer, supposed of thickness ¢, and material (p,,@,), occur at the inner
surface of the shell the change in pitch, 0k,, is given by
Ok, te : 2 : :
race {p (k? — K*(a,6)) — po (2 — Ka, «))} + pD eigistereletsfaiaiatelsinists\s(ere (44),
If the layer differ from the remainder only in density, and the mass of the shell be
increased by 0M, when the layer is at the outer surface, and by 0M, when the layer
is at the inner surface, then putting
M, = 4ra'p/3, OM, = 4ara*t, (p, — p),
M, = 4cre'p/3, OM, = 4re*t, (p2 — p),
we get 2 =— OM, (ug)? + D,)
Ral feanadtovdan ti cl) 9 (45),
oF = — AM, (1,)°k* + D! J
where D'=3 {My (ug) (2 — K*(a.0)) — My (Ue)? (le? — B2 (a ,c))}eececenceceeseees (46).
The mass of the shell when of uniform density p is of course M,—M,. From (45) we
have the elegant relation
Ye @ Bley CBWE Oia? & GLE (AF sence consoGonoceocosnoneneenededae (47).
Thus the changes in the pitch of a given note in a given shell when alterations of
density occur at its surfaces are in the ratio of the consequent alterations of the mean
values of the kinetic energies resident in the corresponding layers.
Supposing the altered surface layers to differ from the remainder only in elasticity,
we find
ef = ty (Ua)? {4m (3m, — m) (Mm, + M4) — 4n (Bm — nr) (m + 2) = pD,|
Ok. Tr gsecieer tees (48),
Te = ty (Up)® {4 my (Bm — Ng) (Mz + 2)? — 4n (Bm — n) (m + n)} + pD |
where D is given by (43).
Thus the change in pitch is proportional to the alteration in the elastic quantity
n(3m—n)/(m+n). We also notice that for equal alterations in the material at the two
surfaces
ie 8 OS 8 i(GAr & (CPF conosapnosannenconsccponnaconosadeor (49).
Vou. XY. Parr IL 33
252 Mr ©. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
Comparing (47) and (49) we see that the effect on the pitch of a given alteration im
elasticity relative to that of a given alteration of density is always more important when
the alterations occur at the inner surface of a shell than when they occur at the outer.
§ 99. Supposing the squares of 0k,/k, ok,/k and (0k, +0k.)/k all negligible, we may
take (@k, + 0k,)/k for the change in pitch due to alterations in the material existing
simultaneously at both surfaces of the shell.
We can also obtain the effect on the pitch of a note of completely removing thin
layers of the material from either or both of its surfaces by simply substituting 0 for
o, and p, in (42) and (44) respectively. .When layers of thicknesses f and ft, are
simultaneously removed we have
i] ee ees are ty air, Saraki
iB = é Pa! fag (Ua)? (ke — Oey) + Z WE (Ue)? (ke? — K* (4.6) | = D tin leiv|etefe: wieluater (50).
By supposing #, or f, negative we can obtain the change of pitch due to adding an
additional layer of thickness 4 or ¢ to the outer or imner surface respectively. This
may be regarded as obvious, supposing it be admitted that the effects of adding and
removing equal very thin layers at a surface must be equal and opposite.
As the immediately preceding deductions travel somewhat outside of strict elastic
solid principles, the following substantiating evidence may give increased confidence in
their validity.
In (50) let us suppose
and we get (6) Sie (7 i crea EI ES Si (52).
Thus our latest conclusions tell us that the effect of paring off a thickness 4 at
the outer surface and adding a thickness f,e/a at the inner surface raises the pitch in
the ratio ¢, : a: whereas an addition of thickness ¢, at the outer surface and a paring off
of thickness t,e/a at the inner surface lowers the pitch in the same ratio. Now this is
obviously a correct conclusion, because in the frequency equation of the simple shell (e.a.a),
k presents itself solely in the combinations kaa and kae. Thus the frequency equation
remains unchanged if
0 (kaa) = 0 = 0 (kae) ;
or, a being constant, if
Ole [ke = —Ory == Oe] Oh cistos cat vis seeder coer soem seem mesa (53).
Now a negative value of da means a paring off of material at the outer surface, while
a negative value of de means an addition of material at the inner surface. Thus equations
(52) and (53) are identical.
§ 100. The case when the compound shell itself is very thin may be most. easily
treated independently. For instance let us consider the compound shell
(Qy> Gh. Ags Oy Ue Oy. y),
where a,—a, is so smal] that its square is negligible.
to
or
Ww
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
By (23) of Sect. I. the frequency equation is
F(Gi-%. de) f (Ge. & . 3) f (Gs. As. Oy) +f'(dy- %~ Ae) f (Ae. % . Hs) f (As. Gs . As)
—f (dy. % . Ge) f (da. sy. ds) f (Gy. Ms . As) —f (Gh. OG . Ma) f (Ga. & .Oz) f (ds. a3. a;) = 0...... (54).
AS @,— ,, Ms — ds, ,—a; are all small, we may apply results answering to equations
(9)—(12) for all these functions. Thus neglecting products such as (a@.—) (@;— 2), we get
Keen, (2 — th) (mM, +m) {(M, + 1) k°aZa? — 4m, (Bm, — n,) (7, + HJ} K (My + Ny) kay x (Ms + Nz) kaya
4+three other terms'=O!icrrh. ech. Nteeeceeesss (55).
Here a may be regarded as the mean radius of the shell. The last term in (55), viz. that
answering to the term in (54) which contains f(@,.%.@;), is of order
(dz — @,) (4; — Az) (5 — A),
and so completely negligible. The remaining terms are of the same type as the first,
which alone is shown in (55).
Thus dividing out by the essentially positive quantity
(m, + %) (mz + Ng) (M3; + Ns) Kay aa,a4,
we obtain from (55) for the frequency equation
(a, +z a) Pi (ke? = Ke a) aba (as — My) Ps (hk? a Kee a) at (ds — Gs) Ps ( = K*q,.a)) =D ee sonance (56q),
where K(.,q)/2m represents as usual the frequency of the radial vibrations in a thin
shell of radius a and material (p, m, n).
Supposing the layers of thicknesses f,, t, t, and of masses M,, M,, M, respectively,
we may write (56,) in either of the alternative forms
Ke? = {pK (a,.a) + top2K? (a,.a) + tapsK(a,.a)} + (Lipit tops + tes). -.---eeceneeee (56;),
= {MR*,,.« + UK? .« + UK ,.a} = M+ M+ M;) ............... (56,).
This result may be extended to a thin compound shell of any number of very thin
layers, and thus in the limit to a thin shell whose material varies continuously or dis-
continuously with the distance from the centre. If M denote the entire mass of the
shell, a, and a, the radii of its. bounding surfaces, terms of order (1 —a,/a,)* being sup-
posed negligible, and the elastic constants m, n be known functions of the distance r from
the centre, we have for the frequency equation
= {[ "4 .4n (38m — n) (m + nar} eaP UT sits tpisestisss owls cations: (57).
This result for a thin compound shell could doubtless be easily—and probably in the
opinion of most authorities satisfactorily—obtained without reference to the surface conditions
by applying dynamical principles to some assumed type of vibration. Whether this has been
already done or not I do not know.
33—2
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
bo
or
C=
SECTION VII.
TRANSVERSE VIBRATIONS IN SPHERICAL SHELL.
§ 101. I pass next to a consideration of the transverse vibrations in a spherical
shell.
Employing the notation of Sect. I, and the forms given in Sect. III. for the types
of displacement and stress in this case, we have
F(r. B)=n {kr J'ix3 (Br) — 3 Jixs R87},
F, (rv. 8) =n {kBrJ iy (bBr)— 3 I—a+y Br),
G(r. 8) =Jis4 (KBr),
G(r. B) = J—i4y (KBr)
Putting for shortness
A (a. 8.6) =Jisy (kBa) J_tesy) (kB) — J~ 143) (KB) Jixy (80),
Aa’. B.b)=J'ixyhBa) Icey (KB0)— Fey Ba) Sir (h8b)| (2),
A (a. 8.0) =JSixy (kB8a) J’ (+9 (B80) — Jy (Ba) S44 (KB),
A (a. 8.0) =J':,(kBa) J ’— 245 (KBb)—F"_ iy BBO) I i4(KBb)
we find for the frequency equations of the four fundamental types in the simple
shell (b.8.a):
f(b. B.a) =72 {Re BabA (a’. 8.0’) + 3A (a. 8 .b)—3kBad(a' . B.b)— 3kBbA(a. 8 .b')}=0...(3),
=_
FG. Bia)= n {Bad (a! |B .b)— 3A (WB. b)} =O recede sees oss sa2 os0ess vsauesesangaane nee (4),
FO. Bu) =n kbA (G. B .0) = 4A. (@.,8 By} HO eee a sk ts ap pctesnaenaee saan ae (5),
FB Bs) =A BB) SO oo oss ccs hoe sate ete ceeganen ee eee aa a eee (6).
These forms of the frequency equations are easily obtained from the general formulae in
Sect. L
For a shell in which {(a—6)/a}? is negligible the functions reduce to the following
forms :—
ie a oe ee rik Bad (a. 8. a’) {eG'a*— (6—1) (64 2)}.ccsseeeese: (7),
f@:8.0= — nlead(a.8.a')(1 =o) rere (8),
f(b.8.@) =nkgad (a. 8.0/)(1 a) aek, athens. Cane eam (9),
f(b.B.a)= a BGM a, Bist) .cwiaoarences atheateeeh, Meee (10).
It has been already pointed out that
kBaA (a8 0) == \Ocoas. toss dene ee (11),
where C is a constant quantity independent of k, 8 or a.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS, 255
Equating the several functions to zero we get the frequency equations for the four
fundamental types in a thin shell. The free-free vibration is, it will be observed, the
only case in which the frequency equation has a finite root.
Supposing b absolutely equal to a we get
f@.8.a)=—f(a.B8.a)=n0,
f@.&.a)=f@.8.a) =0
Thus f/(@.8.a) and f(a@.8.@) are quantities which cannot vanish, each being the product
of m into an absolute constant.
Employing the result (12) in the general equation (30) of Sect. I, we find
i (CeisiCrS!. 0., (Bs h) — 40 aE GO) na ncecaertetee cere ator eet (13).
Another result we require is the value when f(e.8.a)=0 of
bk f(e.8.0)= | kee Lim + hBe meee ava).
where k8a and kBe are to be regarded as independent variables. By work exactly similar
in its general outlines to that already indicated in the case of the radial vibrations it is
not very difficult to prove
d 57
kbe dates © .8.a)=— pé {? — K* 2,4} f(é. 8. a),
kBa Tiga fe 8-0) == pe’ {k? — Kg} f(e.8.@)
where ar K.,, is the frequency of free-free transverse vibrations in an infinitely thin shell
of radius r and material ~.
i = =
Thus k = S(e.8.a)=—p {2 (— K°g,4) f(E.8.a)+ a (ke — Kg.) f(e.8.a)}...(15).
§ 102. We have now all the necessary data for determining the frequency equation for
the compound shell (e.8.c.8,.b.8.«a), in which b—c is small.
From the general equation (23) in Sect. I. we have
S(e.8.c.B,.b.B.a)=f(b.B.a{f(e.B.cdfE.B.b)—fle.B.Of(c. BR. db}
—f(b.B.a){f(e.B.cofE.fi-b)—fe.B-Of(c. Ar. bf =0 we (16).
Now supposing terms in {(b—c)/b}’ negligible and employing the results corresponding
to (7)—(10), we easily put (16) into the form
PCBo0-h-0-B-) _ 56.8.0) f(0.8.0)—fe-8-0f(b.8.a)
fs OFS pub? (P — Ke »)f(e.B. b)f 0.8. a)
[FCB DS0-8.0)+ if 6.8. D/O.8.0)+ Ife. 8.F0.8-0)] = 0...(17).
In the coefficient of b—c in accordance with the hypothesis that (b—c)? is negligible, ¢ has
always been replaced by 0.
256 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
Writing 8, x for 8, m respectively in (17) we obtain an expression for
GAREY DAASISD Ui -o) 3.0)
employing which we find for the frequency equation
F(e.8.c.B8,.b.8.a)
nC
eae ag nde Sa Fob ip (k* — Kp.) — pr (R— K%p,.»)} (0. 8-5) f 0.8.)
52071 a
#259 (E-Z)FE-B-DLO-B.0) =O vreeerrrresern pees 3)
But if 0k be the imerease in & due to the existence of the layer, this must be
identical with
; 4) ea aa
S(e.B.a)— 7k f(e.8.a)=0 soennnaSanrotiagauoonee6dEosc (19).
Thus remembering (13) and (15), we find on comparing (18) and (19),
ok - : : es -
h nCp {e? (k°— Kg.) f(é.8.a)+ 0° (k®?— K%6.4) f(e.8.a)}+ ale
b
= b* [p(k — K%p.») — ps (kt — K%p,.»)} f(@-B-) f 6-8. a)
+(7-7)se.8.)f0.8.a) eres (20).
This formula can be transformed into another of . greater physical significance. By
methods precisely similar to those employed in the case of the radial vibrations I find
when f(e.8.a)=0:
f(e.8.b)+f(e.B.a@) =(b/a)' x (w/w),
fb .B.a)+f@.B. a) = bie) x ove sans NE Ui (21),
f(e.B.b)+ fle. B.@) = (b/a)* x (Wr/wa), |
.B.a)+f(@.B.a) = (bey) x (W,/w)
F (QBs) FG JB) SRO oh So: oc Rie a eee ice (22),
where C is the quantity defined in (11), and w and W are the displacement and stress
in a simple shell. The form of b4w, may be got by writing b for r and 8 for B,
on the right-hand side of (8), Sect. III., and W, is the corresponding stress.
Employing these relations we transform (20) into
Ok _b-c_ te 2 oh Es Aad 1% piles
ko fae pwawe{e (k?- K%g.9) f(é.8.a) +a? (— Kg a) f(e.B-@)}
akties
x E (ws)? {p(t — K*4g.») — py (K— K%p,.»)} +B (Wo? (5 - 2) (23).
§ 103. Passing to the limit when e vanishes it may be shown without much difficulty
that
a ee eee. aesobAGuC (24).
e'w,f (2.B.a)=0
* See the note on p. 266.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 257
When these values are substituted (23) becomes identical with the result obtained
for the solid sphere, viz. (22) of Sect. ITI.
§ 104. From the same consideration as was employed in the case of the radial
vibrations we conclude that the first factor on the right-hand side of (28) is essentially
a positive quantity.
The second factor on the right of (23), which alone varies with 6, is identical in
form with the corresponding factor in (22) Sect. III, giving the change of frequency
in a solid sphere, so a brief discussion of its general features will suffice.
When an alteration of density occurs at a node surface of a particular note it does
not affect its pitch, but when it occurs elsewhere the pitch is invariably raised or
lowered according as the density is diminished or increased. The numerical magnitude
of the percentage change of pitch depends solely on the magnitude of the alteration of
density and not at all on its sign.
The law of variation with the position of the layer of the change of pitch due to
a given alteration of density is independent of the magnitude of the alteration of density.
When the layer of altered density is of given volume the positions in which it has
most effect on the pitch of a given note coincide with the loop surfaces for that note ;
when the layer is of given thickness its most effective positions lie slightly outside the
loop surfaces.
When the layer differs from the remainder only in elasticity the second factor on
the right of (23) reduces to
5 : : ap fly yal
[we (n, —n)(@—1) (+2) 4+ PW (e =| ’
ny ny
The change of pitch thus depends solely on the alteration of rigidity. Unless in the
case of the rotatory vibrations, for which 7=1, the above factor is the sum of two
squares which cannot simultaneously vanish except for b=0. Thus excluding the case
of a solid sphere, an alteration of rigidity throughout a thin layer situated anywhere neces-
sarily affects the pitch of any transverse vibration other than one of the rotatory type,
and the pitch is raised or lowered according as the rigidity is imcreased or diminished.
In the case of a rotatory vibration the change of pitch when existent has always the
same sign as the alteration of rigidity, but it vanishes when the altered layer coincides
with a no-stress surface.
In the case of a rotatory vibration the positions in which the layer, when of given
volume, has most effect on the pitch coimcide with those surfaces over which the trans-
verse stress is a maximum, but this is not exactly true of any other vibration of the
transverse type.
§ 105. Some of the more interesting special cases call for a more detailed examina-
tion,
258 Mr ©, CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
Thus suppose the altered layer to be found at the outer surface so that
b=a, W,= W,=0.
Remembering (12) we find from the second of equations (21)
fé.8.a)= nC (e/a) (we/Wa);
whence by (22
F(€.8.a) = — n€ (a/e)} (wa/r.).
Thus from (23) if the thickness of the layer be 4 and the change of pitch 0,
Oh, t os ° 9 ") ~
ae = Py as (Wa)? {p (ke = K*Q8.0)) — Pi (ke = K*8,.«)} = pD occ ceeccctererccce (25),
where D=a? (wa)? (Eh — K2@.a) — & (We) ( — Krig.e)) voseeeereescenereceneess (26).
Similarly if @k, be the change of pitch due to the existence of a layer of thickness ¢,
and material (p,, 7:) at the inner surface of the shell, we find
Oks tp - att ° vo 2 2
Bape (we)? {p (2 — Ke ig.e)) — po (2 — K%Q,.0)} PD vceereeeeereeeereee (27).
If the layer differ from the remainder only in density, and the mass of the shell
be increased by @M, when the layer is at the inner surface and by 0d, when it is at
the outer, then putting
M, = 47ra*p/3, 0M, = 47ra*t, (p: — p);
M, = 4rre°p/3, 0M, = 4re*t, (p2 — p),
we find
2 = — aM, (wt = D’,
EE (28),
= = — 2M, (wpe + D' |
where D => 3 {M, (Wa)? (k? —— K'8.a) aa M, (we) (ke? a K*@..)} Sener e ween wee (29).
From (28) we get
AEG! S SUE Gia & GlUE{ (IA) hencacdnenasconceso50n5agnNc (30).
If on the other hand the surface layers differ from the remainder only in elasticity,
we find for the corresponding changes of pitch
Hs — (m, — n) (6-1) (+2) ba (Wa)? + pD,
ap. fl WS ee] © ol reer aeenars (381),
aT = (ny — n)(t— 1) (+2) & (We)? + pD
where D is given by (26).
Thus for equal alterations of rigidity at the two surfaces
6) Se) Spee Tr IA) t(D) Gococopnpoosocnne cadsndconddacce (32).
The results (30) and (32) are identical in import with the corresponding results for
the radial vibrations, viz. (47) and (49) Sect. VL, and similar conclusions may be drawn.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 259
An exception must however be made of the rotatory vibrations as their pitch is
unaffected by an alteration of rigidity occurring at either surface.
On account of this peculiarity in the rotatory vibrations it seems worth while re-
cording the special forms taken in their case by the expressions for the changes of pitch
due to surface alterations of material, viz.
oh, He See <
i = —t, (p,— p) (we)? = {pa (We)? — pe? (we) |
a cons 1 elt Vehianmiisatin Scie
= — tz (p2— p) & (We) + {pa* (wa)? — pe (we)*}
§ 106. In the general case we may, provided (@k,+0k,)/k be small, suppose the
alterations in the material at the surfaces to exist simultaneously. Also by supposing p,
and p, to vanish we can obtain the effect on the pitch of removing thin layers from
the surfaces. Thus when layers of thicknesses ¢, and ¢, are simultaneously removed the
change of pitch is given by
= = a (Wa)? (k? — K7\¢.a)) + 2 2 (w,)? (kh? — Krp.0)| Se DDS Feetiaicisctes (34),
where D is given by (26).
Further by writing —¢, for t, and —t, for t, we find the effect of adding layers of
thicknesses ¢, and ¢, and of the same material as the remainder to the outer and inner
surfaces. A verification of these conclusions is supplied by putting in (34)
t,/e =— t/a,
when it reduces to 0k/k = t,/a.
§ 107. For a compound shell of three thin layers we have a frequency equation
deducible from (54) Sect. VI. by writing 8 for a. This leads to a result deducible from
(56,) or (56,) of that section by writing 8 for a It may also be put in the specially
neat form
ke = (t — 1) (0 + 2) (mt + Mate + Mgts) + {a? (pitr + Pate + Pats)}--ccreceeeneere (35).
Here ¢, etc. denote the thicknesses of the thin layers, (p,, 7%) etc. their materials,
and a the mean radius of the shell.
We may extend (35) to a thin compound shell of any number of layers, or to one
in which the density and rigidity vary in any manner with the distance from the centre.
The general formula applicable to all such cases is
=G21)@ 4 2) | 7 cays Paste) (36).
Here M is the mass of the shell, a, a, the radii of its bounding surfaces, (a,—«a)/a
being so small its square is negligible, and n is supposed a known function of 7, con-
tinuous or discontinuous,
Vou. XV. Parr II. 34
260 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
SECTION VIII.
RADIAL VIBRATIONS IN CYLINDRICAL SHELL.
§ 108. Employing the notation of Sects. I. and IV. we may take in the case of
the radial vibrations of a cylindrical shell:
F(r.a)=(m+n) hardy (kar) + (m —n) J, (kar),
FP, (7.0) =(m +n) kar Vy (kar) + (m —n) Y, (kar), |
OR ee G ocodonoboansaba50ands (1).
G,(7.@) = Y, (kar) |
Putting for shortness
A (a.a.b)=J, (kaa) Y, (kab) — Y, (kaa) J, (kab),
Ae ea ee) ts De eee (2),
A (a.a.b')=J, (kaa) Vy’ (kab) — Y, (kaa) Jy’ (kab),
A (a’.a.b’) = Jy (kaa) Vy (kab) — VY,’ (kaa) J,’ (kab)
we find for the frequency equations of the four fundamental types in the simple
shell (b.a.a):
T(b.a.a)=(m+ny ketabA (a’.a.b’) + (m— ny? A (a.a.b)
+(m? — n*) {kaa (a’.a.b)+kabA(a.a.b’)}=0 ......(3),
f(b.a.a)=(m+n) kaad (a .a.b) + (m—n) A (a.4.b)=0 oe (4),
f(b.a.a) =(m+n) kabA (a.a.0') + (m — n) A (G.a.b) =O... ee ec eee en ee (5),
FB teal) =D (an Or P= 0 cores dak cesta eetise Sdn seamen ge eee (6).
For a thin shell in which {(a—})/a}? is negligible the above functions assume the
forms:
f(b.a.aj= = b —— C {Kata® (m + nf — Amn}... eccseeeeseesceseeeee (a):
f(b. a.a)= 1m + ae — ae mh fas 8 elatatelofotatalalele otelotatcteiele steteievaraln (S),
f(b.a.a)=-C “a +n — —— Le —n I 5. acaitisiciteriaisteiee« aseibaict Bheiee (9),
See —b,
=— )),
HB.) =O aes crerresssonsssnsensssentenssnsnerssnseesranses (10)
where C= — Ka (G00) i. vnapsitescsescs-ereate uecereaee eases (11)
is an absolute constant, depending only on the definition of the Bessel.
The result
Fi(a.a) G,(a.a) —F, (@.a) G(@.a) =(m +N) Coro ennceonsrees seven (12),
will be found useful in verifying the conclusions arrived at.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 261
The method of obtaining the change of pitch due to the existence of the thin
layer (c.m.b) in the shell (e.a.a) is precisely the same as that already illustrated in
the case of the sphere. The relation
Ee Gf e-n-a) =—p{é(P— Keo) f(.a.a) +a? ( — K%0.q) f(€.4.G)} oes. (13)
also applies as in the case of the sphere, though of course the actual forms of the
functions are different, and the values of Kyq.,, and K,..) are to be derived from (24)
Sect. IV.
Thus it will suftice to record the result of the operations indicated, viz.,
sane,
oF im +n) Cp {e (k? — K*, 4) f(E.a.a) +a? (2 — Kaa) f (e.4.@)} + =
: \ f(e.a.b) f(b.a.a)
mM+nh m+n,
= b {p (2 — Ka») — p: (2 — K%q,.»)| f(e.4.b) f(b.4.a) + (
+2 ues Mm) (fe.a-b)fb.a.a)+f(e.a.D)f(b.a.a)) ee aes
m+n MmM+NH
Denoting by u, cos kt the displacement, and by U,cos kt the corresponding radial
stress at an axial distance 7, the following relations may be established in precisely the
same way as the results (32) and (35) of Sect. VI, the relation f(e.a.a)=0 being
supposed to hold,
F(e.a.b) = f(e.a0.@) = up/ta,
F(b.a.a) +f (E.a.a) =u/ue,
orb) = flecacc ae | J doer cage ee (15)
f(b.a.a)+f(é.a.a) =bU,/u.
i GaCin@)ERif{ Gs @ia@y = = (Wd se)? CP ocoadascsonedconodeosneua ee (16).
Employing these results, remembering that in the coefficient of 6—c we may suppose
f(e.a.a) to vanish, we transform (14) into
ok b—c _ —(m+n) aC
cis Tal Pella {2 (kK — Kao) f(E.a.a) +a? (— Kaa) f (e.a.a)}
= 1 1
x |B (wy ip (8 = Ha) —p (U8 — Kn) +0 (<E- — F)
ee tlt =e
+ 4u,U, eee 7 al © ccc civervace (17) S
§ 109. In the limiting case when e vanishes it may be shown that
Uf (€.a.d)=—(m+Nn) Cug,
Cu, (k? — K7 0.5) f(E.a.a)=0 j
and we thence obtain for the value of ok/k in a solid cylinder a result identical with
(27) of Sect. IV.
* See the note on p. 266,
34—2
262 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
§ 110. From the same consideration as before we conclude that the first factor on
the right-hand side of (17), which is independent of 6, is essentially a positive quantity.
The form of the second factor on the right of (17) leads to the following general
conclusions :—
When an alteration of density alone occurs at a node surface of a particular note it
does not affect the pitch of that note, but when it occurs elsewhere the pitch is raised
or lowered according as the density is diminished or increased. The numerical magnitude
of the percentage change of pitch is independent of the sign of a given numerical
alteration in density. The law of variation with the position of the layer of the change
of pitch due to a given alteration of density is independent of the magnitude of the
alteration. When the layer of altered density is of given volume, Le. when (b—c)b is con-
stant, the positions in which it has most effect on the pitch of a given note coincide
with the loop surfaces; when the layer is of given thickness the most effective positions
lie slightly outside the loop surfaces.
When the layer differs from the remainder only in elasticity the expression for the
change of pitch consists of three terms. Of these the first has the same sign as, and
is proportional in magnitude to myn, (m+ m4) — mn (m+n). It vanishes when the layer
coincides with a node surface of the note in question.
The second term has the same sign as, and is proportional in magnitude to
(m+n)7—(m,+7)7. It vanishes when the layer coincides with a no-stress surface.
The third term varies as n(m+n)?—n,(m,+)7, but its sign depends also on the
value of b. It vanishes when the layer coincides either with a node or a no-stress
surface. It vanishes for all positions of the layer provided
OV MOR LIL Ibs E72) con sehgoreseeee Sec ron eee cece oconeer! (19).
Thus on the uniconstant theory, or more generally when (19) is true, the sign of
the change of pitch following a given alteration of elasticity is the same as that of p
and does not vary with the position of the layer, If however (19) do not hold, the sign
of the change of pitch may vary for certain alterations of elasticity with the position
of the layer.
From the form of the expressions for u, and U, it is easily proved that when kab
is large the second term in the expression for the change of pitch due to an alteration in
elasticity alone is much the most important, and that the third term is more important
than the first. Thus in the case of the higher notes the effect of an alteration of elasticity,
when of importance, especially when the alteration occurs near the maximum-stress surfaces
of greatest radius, depends almost entirely on the term containing U;,’; and the consequent
change of pitch is a maximum when the alteration of elasticity occurs very close to the
maximum-stress surfaces.
§ 111. Confining our further remarks to special cases, let us suppose the layer to
Mr ©. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 263
be at one or other of the bounding surfaces. Remembering that U vanishes at a free
surface, we easily find for the two positions of the layer with our usual notation
Cats F & : 3 aa F
Si = : a (Ua)? {p (kh? — EG ata) = Pi (? — R0,.0)} = pD,
5 are | ee SO Pb eS (20);
a) ba s a ro
ie = 7 2 (2 (ue)? {p (k? —_— Kern = Pr» (ke? _ K*a,.0)} > pD
where DSc OG) =Le cay E (Cane = LG e 9) occonccovescoooonconsauacccod (21).
When the layer differs from the remainder only in density, let us denote the masses per
unit length of cylinders of radii a and e and of density p by M, and M, respectively, and
let OM, and @M, denote the increases in the mass of the shell per unit length due to the
existence of altered layers at its surfaces, so that
M, = 77a’p, 0M, = 2rrat, (p, —p),
M, = re’p, OM, = 2rret. (p2 — p).
In this case (20) reduces to
oie =- 0M, (my kh + D’,
‘ Soohedouorbna Hostodbandccnaogano ose (22),
ee =—0M, (u,)k? + D' |
where D’ =2 {M, (Ua)? (2 — Ka.) — Me (te)? (2 — K2Q0.0))} cveecvecececseeees (23).
From (22) we get
Ohh Glogs = OMG (tg con lg Ebel) eter Sosa sensei cicnp fidence one (24).
If on the other hand the surface layers differ from the remainder only in elasticity
we find
Oi 1 » § 4m, 4inn ) J
Th Ore ea P.|
Gia ita, | 4anins 4mn )
ae | - + pD |
Mye+tN M+N
where D is given by (21).
Thus for equal alterations in elasticity at the two surfaces we have
Oi SOE min Qe 8 C-YA(G Piecaconbacenscoocceboawesnooos (26).
Comparing (24) and (26) we find
(0k,/Ok,), p altered, : (0k,/0k.), elasticity altered, :: a? : € ..........00005 (27),
supposing the alterations in density and in elasticity to be the same at the two surfaces and
to occur there throughout given layers, Thus relatively considered, an alteration of elasticity
at the inner surface is more important than a like alteration at the outer surface.
§ 112. Supposing (0h, + 0k,)/k small we may suppose the alterations at the surfaces to
occur simultaneously, Also by supposing p, and p, to vanish we may find the effect of
264 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
removing thin layers from the surfaces. Thus when layers of thicknesses ¢, and f, are
simultaneously removed the change of pitch is given by
ok h, 2/18 ro leg pa asiris 72
ET \: @? (Ua)? (2 — K2,a.a)) + aioe (2g) BY (FBO) een Rios ee eaeenncasce aes (28),
where D is given by (21).
By changing the signs of 4 and # im (28) we get the effect of adding layers of
thicknesses # and f¢, to the bounding surfaces, the added layers being of the same
material as the rest of the shell. As usual a verification is supplied by putting in (28)
t./e = —t,/a,
when it reduces to ok/k = t/a.
§ 113. For a compound shell of three thin layers the equation (54) Sect. VI. applies
without any change in form. From it we easily obtain results identical in form with
(56,) and (56,) of that section. We may also write the expression for the frequency in
the form
4m,n, 4mm, 4mgns
1m, +N. 7 Ma+tN, >= Ms+N;
) sah ( pit pi eae (29).
This result may be extended to a thin compound shell of any number of layers, or
to one in which the density and elasticity vary in any manner with the distance from
the axis. The general formula applicable to all such cases is
Sa [% mn
~aM m+n is
al
Here M is the mass of the shell per unit length, a,, a, the radu of its bounding
surfaces, {(a,—,)/a,}* being negligible, @ the mean radius of the shell, and m, n are
supposed known functions of the axial distance r.
SECTION IX.
TRANSVERSE VIBRATIONS IN CYLINDRICAL SHELL.
§ 114. Employing the notation of Sections I. and V., we may take in the case of the
transverse vibrations of a cylindrical shell:
F (r.B)=nkBrJy (kBr) — J, (kBr)},
F,(r.B)=n|kBrYy' (kBr) — r.er,|
G (r.8)= Si (k8r), |
G,(r. B)= Y, (kBr)
Now these expressions and likewise the expressions for the displacements and stresses
can be at once derived from the corresponding expressions in the case of the radial
vibrations by simply supposing m to vanish and writing B for a. Thus it is unnecessary
to go through the mathematical work by which the expression for 0k/k is arrived at,
because with 0 substituted for m and 8 for a each step of the analysis in the case
of the radial vibrations applies to the present case.
Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS. 265
The very same constant quantity C that occurred in the case of the radial vibrations
occurs here also, though it presents itself under the form
C=— kBa {J, (kBa) Y1' (kBa) — Jy’ (kBa) Yi (hBa)) .... cece eee eee ee (2).
In transforming the expression (17) Sect. VIII. for the change of pitch it must be
remembered that, as shown in Sect. V., Aig.) is zero.
We thus find for the change of pitch in the transverse note of frequency k/27 in
the shell (e.8.a) due to the presence of the thin altered layer (¢.@,.b) the equation—
06 ea = na
ie G3 prvak? ef (é€.B.a)+af(e.8.a)}
—~b (wy) k (pr — p) + (Vs)? (- -*), TAS meth (3)*.
ny
The forms of v and V are given by
v, = AG (r.8)+ BG, (r. 8),
rV,=AF(r.8)+ BF, (r. 8),
the value of B/A bemg determined by one of the surface conditions.
§ 115. For the limiting case when e vanishes we have
Uf (€.8.a) = —nCrg, )
ev. f(é.8.a)=0
and we thence obtain for 0k/k a result identical with (8) of Sect. V.
§ 116. The first factor on the right-hand side of (3) is independent of 6 and may
by the same consideration as in the previous types of vibration be seen to be essentially
positive. The second factor, which shows the variation of the change of pitch with the
position of the layer, consists of only two terms, of which the first depends only on the
alteration of density, the second only on the alteration of rigidity.
When an alteration of density alone occurs, the pitch of a given note is unaffected
when the layer coimcides with one of its node surfaces, but for all other positions of the
layer the pitch is raised or lowered according as the density is diminished or increased.
The numerical magnitude of the percentage change of pitch is mdependent of the sign
of the alteration of density, and the law of variation with the position of the layer of
the change of pitch due to a given alteration of density is independent of the magnitude
of the alteration. When the layer of altered density is of given volume per unit length
of cylinder, the positions in which it has most effect on the pitch of a given note
coincide with its loop surfaces.
When an alteration of elasticity alone occurs, the change of pitch depends solely on the
alteration of rigidity. The pitch of a given note is unaffected when the layer coincides
with one of its no-stress surfaces, but for all other positions of the layer it is raised
or lowered according as the rigidity is increased or diminished. The law of variation
with the position of the layer of the change of pitch due to a given alteration of rigidity
* See the note on p. 266.
266 Mr C. CHREE, ON SOME COMPOUND VIBRATING SYSTEMS.
is independent of the magnitude of the alteration; but a diminution of rigidity is more
effective in lowering the pitch than an equal increase is in raising it. For a given
alteration of rigidity throughout a given volume the change of pitch has its maxima when
the layer is at the maximum-stress surfaces, .
§ 117. For the cases when the layer coincides with the surfaces of the shell we
have with the usual notation
ok, __ hppa (my
k a@ p Dy [ (5)
Ae es. (recy Gitioag oars ;
Ki 2255 D
where 10 OO Cp) stessaosedsobosn Gabe oscar mdcadosnudoone (6).
A surface alteration of elasticity has thus no effect on the pitch, and if 0M, and 0M,
be the alterations in the mass of the shell per unit length due to alterations in the density
at the outer and inner surfaces respectively, the corresponding changes of pitch have
their ratio given by
A) GS OB GUE GAS 8 GUE (GHP deecsadocnnacenooa se-odchocesosoo0es (7).
When alterations exist simultaneously at both surfaces we have with the usual
limitation
Ok = Ok, + Oke.
When layers of thicknesses f, and ¢, are simultaneously removed the change of pitch
is given by
ok ty :
== {2 .@ (Ug)? + A 2 (oh See HD aie vaieate ste sees elec sTa opel eR (8),
where D is given by (6).
By changing the signs of t and t, we get the effect of adding surface layers of
thicknesses ¢, and ¢, of the same material as the remainder.
The frequency of the transverse vibrations of a composite shell when very thin is
always zero. In other words no such vibration has a physical existence.
[December 1, 1891. The factors independent of 6 in the general expressions for @k/k in
shells can be put into simpler forms. Replace (36) p. 248 by ak/k = (b—c)p 1D" x [last factor]
...(a), (23) p. 256 by ck/k=(b—c)p D* x [last factor]...(b), (17) p. 261 by @k/k = (b—c) p* D™ x
[last factor]...(c), (3) p. 265 by 6k/k=(b—c)k-*p- D~ x [last factor]...(¢), where D is given:
in (a) by (43) p. 251, in (6) by (26) p. 258, in (c) by (21) p. 263, in (d) by (6) p. 266.
The modes of reduction are all similar to the following for case (a). Using the notation
of pp. 247—8, we have
f(@.a.a) F(a.a)G,(e.a)—F,(a.a)G(e.a) BG,(e.a)+AG(e.a) eu,
(m+n) koa F(a.a)G,(a.a)—F,(a.a)G(a.a) BG,(a.a)+AG(a.a) au,’
and therefore by (35) p. 248, f(e.a.a)+(m+n) kae =—au,/eu,.
In case (b) use nC =F (a. B) G,(a. B)—F, (a. B) G (a. B), and similarly for (c) and (d).]
VIL On Pascal's Hexagram. By H. W. Ricumoyp, M.A., Fellow of King's
College.
In the volume of the Atti della Reale Accademia dei Lincei, published in 1877,
there are two important memoirs on the subject of the Pascal Hexagram: the first, by
Professor Veronese, contains geometrical proofs of all previously known properties of the
figure together with a large number of new properties discovered by him. The second
memoir, by Cremona, obtains proofs of many of the theorems given by Veronese from
a new standpoint, viz. by deriving the hexagram from the projection of the lines which
lie on a cubic surface with a nodal point, the nodal point being the origin of pro-
jection.
It is my purpose in these pages to attack the subject by the methods of Analysis,
adopting Cremona’s point of view. I have recently been led to notice a new form of
the equation of a nodal cubic surface which has the advantage of giving the equations
of the lines on the surface in perfectly symmetrical forms,—that is to say im forms
where each line is represented by exactly similar equations: using this form of equation
to the surface, I propose to develop briefly a few properties of these lines, and others
connected with them, and then by projecting these lines upon an arbitrary plane to
obtain analytical proofs of theorems relating to the Pascal Hexagram.
There are three other references which I wish to make to papers on this subject.
The second volume of the American Journal of Mathematics contains an interesting
paper by Miss Christine Ladd, in which the chief properties of Veronese are explained
in a concise form and his notation improved and simplified; some new results are given
connecting the Pascal Hexagram formed by six points on a conic with the Brianchon
Hexagram formed by drawing tangents at those points: in the second place, Professor
Cayley has published two papers in the Quarterly Journal of Mathematics, Vol. IX.,
pp. 268 and 348, of which the latter contains some results whose form is strikingly
suggestive of the forms obtained here, though the connexion is not apparent: lastly, im
the volume of the same periodical for 1888 will be found a short paper written before
I had obtained the simpler form to which the equation to the cubic surface can be
reduced, which forms the foundation of the present discussion.
Won, io 1A I 35
268 Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM.
The nodal cubic surface.
Let the nodal or conical point O be taken as one vertex of the tetrahedron of
reference for a system of four plane coordinates, so that the equation to the surface is
of the form
(*KUa, y, zP+w(*KYa, y, zP=0.
It is clear that there are six straight lines on the surface which pass through O the
nodal point and that these lie on a quadric cone; they are in fact the lines of inter-
section of the two cones
(KG, y, 2 =0,
and (* Ga, y, 2° =0.
Denote these lines by A, B, C, D, £, F; then any plane which contains two of them,
as for example C and 2, must cut the surface also in a third line which does not pass
through the nodal point; this line may be called CZ.
We have thus found on the surface six lmes which pass through O the nodal point,
and fifteen other lines which do not pass through O, and these form the complete system
of lines on the surface. For the plane through any line on the surface and the nodal
point O must cut the surface also in a curve of the second order having a double point
at O, ie. in two straight lines which pass through 0: hence, since only six lines on the
surface pass through O, there can only be fifteen other lines on the surface. Two lines
such as CD and CE cannot intersect since they both meet the line C; but it may be
shewn that any two of the fifteen lines which are not met by the same line through
OQ must intersect. For if we take a series of planes through one of the lines, AB, these
eut the surface also in conics which are found to break up into two straight lines for
three planes of the system besides the plane OAB; further it is seen that the pairs of
points of intersection of these conics with AB are in involution. It is therefore necessary
that these three planes which pass through the line AB should contain respectively the
pairs of lines CD, EF; CE, DF; CF, DE.
There are therefore fifteen planes, known as tritangent (or triple tangent) planes,
which cut the surface in three straight lines and which do not pass through O; three
such planes pass through each of the fifteen lines, and moreover the eight points on any
line AB where it is met by the lines CD, EF; CE, DF; CF, DE; and by the lines A
and B are in involution.
Equation to the surface.
Taking nine lines such as AB, AU, AF, DB, DC, DF, EB, EC, EF, we see that they
lie by threes in six tritangent planes;
AF, BD, CE, lie in a tritangent plane «=0,
AC. BE Di se ee y=0,
AB CD Hite eee eee z=0,
2B. OE. DE. eee w=0,
AD, BD, ER AA ee ee v=0,
AB OBE, CD, is 5o.veduataentereensncaheteee w=0.
Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM. 269
Hence the equation to the surface must be
vyz=k. ww.
Further, since none of these six planes pass through O the nodal point, we are at
liberty to assume that at O
L=y=Z=U=Vv=w.
Therefore k=1 and the equation to the surface is
LYZ = UW.
yp Jo 3
The equation to the tangent plane at (a’y/2’u'v'w’) is
y 2.0 vow
oa SSP ST
y Z u Vv WwW
TAT ae Son ae
zZwvw') be the coordinates of O, this will give an identical relation in #yzuvw,
av
SeaF
a“
if now (ay
Viz. atyt+ z=utvt+w.
But a second identical linear relation must connect these quantities, such as
Letmy+nzt+put+quv+rw=d,
where L+m+m%4+24+%04+7=09,
since at O f= Y=2Z=uU=v=w.
Hence (1, +A) a+ (m +A) y+ (mM +A)Z+(p.—A)U+(GH—A)V+ (1 —- AV W=D
for all values of 2.
We can now find one finite value of X such that
(2, +X) (m, + A) (7%, +A) + (i — A) (H—A) (7, — A) = 0.
Give 2 this value and replace
L+A, m+A, M+EA M—-A GU-A, m—-A,
by Lm, 1p; anit
and the second linear relation takes the form
le+my+nz+ pu+qv+rw=0,
where l+m+n+p+q+r=0,
and lmn + pgr = 0.
Equations of the fifteen lines.
It has now been shewn that the equation to the surface can be brought to the form
DY ZULU eraa\escneeie eects esse (i).
where BAY ZS WUEVEW vorcccsceccceescceeee (2),
le+my+nz+ put qutrw=0......... (3) Witereeeensnenauacue hedentee IN
L+m+n+pt+qtr=0 ween (4),
ED EY UG I gneeopondce opooeeneodalbosqaberne (5)
35—2
270 Mr H. W. RICHMOND, ON PASCAL'S HEXAGRAM.
and at the nodal point 0,
L=YH=sZ=U=v=w.
It is now possible to obtain the equations of all the fifteen lines AB, AC, etc. Nine
of them have already been found, viz.
AB, z=0, wu=0; 1DM5}, HS}, 7 SWs EB, y=0, w=0;
AG, y=0; » =0; DCS 2 —05 ew — 0: EC, «=0, w=0;
AF, «x=0, w=0; IDK, WSQ, 2 ]OK Ee: —0)oi—10)
The equations of the remaining six lines are derived from (3): the three planes
lx + pu= 0, my + qv =0, nz+ rw = 0,
intersect in a straight line which lies on the surface, and which meets the lines
z=0, w=0; or EC; y=0, v=0; or AC; 2=0, w=0; or DC.
Hence it is the line BF, and the remaining six lines are identified as follows :—
BF, lz+pu = 0, my + qu =0, nz+rwu=0;
FC, le+qu = 0, my + rw = 0, nz+pu=0;
CB, le+rw=0, my + pu = 0, nz+qu=0;
AD, le + pu = 0, my +rw= 0, nz+qvu=0;
DE, le +rw= 0, my +qu = 0, nz+pu=0;
EA, lxa+qu = 0, my + pu = 0, nz+rw= 0,
Also the fifteen tritangent planes are made up of :—
Six such as z=0;
nine such as lx + pu=0.
These equations are obvious modifications of Schlafli’s equations for the lines on an
ordinary non-singular cubic surface; by means however of a simple transformation it is
possible to bring the equations to all the fifteen lines and all the fifteen tritangent planes
to absolutely symmetrical forms.
First let 21 =b+¢, 2nm=c+a, 2Qn=a+b,
2m=et+f, 2q =f+d, Q2r=d+e,
Then DE DE CO Ate — lO bewiareinaciscaasacisoanl marnemene eects (i),
and (a +b)(b+c)(c+a)+(d+e)(e+f)(f+d) =0 oo ceereeeecrrer eens (ii).
But by (1) (at+b+cyf+(d+e+fy=0;
that is C+ 4+04+3(a+b)(b+c)(cta)+h+e+f°+3(d+e)(e+f)(f+d)=0.
Therefore OO OOO Ff? = Oise os ionencas cnsavnnssrssossseden soe (ili).
Mr H. W. RICHMOND, ON PASCAL’'S HEXAGRAM. 271
Again let 2le =B+y, 2my = y +4, Qnz =a+B.
2pu=e4+6, 2qv =€+6, 2rw=d+e.
That is eal eu arky ete.
b+e c+a
Thus Gat G eb iey +: Otel GO saree neers sesencseeineseseseessiae ces (iv),
and, as in (11), the equation to the surface
(B+) (y +4) (a+ 8) + (+e) (e+ 2) (E+ 8)=0
is equivalent to CPaL /ePoey eee aa a(G ==) GoonacdoosnsanesspnonosdsonoanOseone (vy).
Seas. +8 ote met goatee
b+e cta'at+b d+e e+f' f+d’
(a+B+y)(a+b+cPp—a@a-VB—-cy (6+e+f) (dt+tet+fy—db—ee—f°F
(b+ c)(c+a)(a +b) = (d+e)(e+f)(f+d)
Also
The two denominators are equal and opposite, and
(a+b+cPp=(d+et+f);
hence by (iv) this is equivalent to
7at+bhB+ cy+d%+ee+f°F= 0.
Lastly at O the nodal point,
L=y=Z=uU=v=W;
B+y_yta_at+B_e+§ 648 S+e
that is, fale —C-itne Geo Cag) aed) DCEsee
ROM MEG ee “IS
or Gb Ca em
The six planes a=0, 8=0, etc. appear to have hitherto escaped notice: I shall
speak of them as coordinate planes or fundamental planes.
The complete system of equations is now as follows :—
Equation to the surface
Caen] Cie Sor eC o)oc a au Bt S| Seep se eso Ooane (1),
where a+B+yt+o+e4+6 = Oren tba detec: (2),
Wat+bB+cey+a@4+ e+ fF=0......... (B)o Sobisteassatane sends scinas..ti9% B,
a+b+c+d+et+f =())5penasc (4),
@4+B4E4+P+e4+f? =(dosesane (5)
and at the nodal point
GEN ery) 210) 216) 5) Ge: CONC sO em Cut/s
Each of the fifteen tritangent planes is now represented by an equation of the form
a+B8=0, a+6=0, and each line of the surface by three equations such as
a+B=y+d=e+C=0,
272 Mr H. W. RICHMOND, ON PASCAL’'S HEXAGRAM.
The equations of the fifteen lines and fifteen tritangent planes are given below :—
AB,a+8=y+8=c+ ¢=0, a+@8=0 contains AB, CD, EF;
AC, aty=RB+e=d4+f=0, | aty=0 w.. AC, BE, DF;
AD, a+8=8+f€=y+c«=0, @+O=0 ......0.. AD, BF, CE;
AF,ate=B+8=y+6=0, | ate=0 ou... AE, BD, OF;
AF a+6=B+y=8+e=0, | @+C=0 2. AF, BC, DE;
BC, a+ €=8+8=y+e=0, /3) 75710) Goawesode AF, BD, CE;
BDia+ <= Bp Py =o6=0, 4 BOS 00 ann: AL, BC, DF;
BE, @+y= peste 00 9 |b Behr ee AC, BF, DE:
BF, 2+&=B+e=y7+6=0, (325 {8 caacdaooe AD, BE, CF:
CD, a+B=7+6=64+ c=0, 97 tO 0 qanccaae AB: CF. DE
CE, 2a+6=B+y=€+6=0, 5745 G10) secocoase AD, BC, EE.
CF, at+e=B4+E=y74+ 6=0, GA GEA aceraanice AE, BF, CD;
DE, a+£=BR+e=y+8=0, EES AF, BE, CD;
DF, a+y=B+8=€+6=0, eG = Ome neces AC, BD, EF;
EF, a+B=y+¢=84+6=0., CFC 10 ok await AB, CE, DF.
These equations have been arranged in such a way as to shew a certain correspondence
between the English and Greek letters; but this correspondence is soon lost sight of in
the subsequent work.
This system of equations having been obtained, the properties of the fifteen lines and
fifteen planes may be discussed. It should be explained that the names of the various
points and lines which present themselves will be borrowed from the projections of those
points and lines in the Pascal hexagram.
(1) In each tritangent plane, as a+ @=0, lie three lines AB, CD, EF, which form a
triangle denoted by Ajg, or sometimes merely by A; the vertices of this triangle are called
P points; thus CD, EF intersect in the P point
a+B=yt+e=64+F6=y74+6=8+6e=0,
or a+B=0, y=b=-—c=—6.
There are forty-five of these P points, each lying in five tritangent planes, and on
each line lie six of these points, which were seen to fall into three pairs of points in
involution.
The fifteen tritangent planes pass by threes through the fifteen lines of the surface,
and any plane is met by six others in lines which lie on the surface.
(2) Although the six fundamental planes a=0, B=0, etc. appear to have hitherto
escaped notice, yet the fifteen planes given by equations such as a=, were known to
Pliicker, and are usually spoken of as Pliicker planes; two Pliicker planes pass through
Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM, 273
each P poimt; for example through the P pomt a+8=0, y=d=—e=- € pass the two
Pliicker planes y=6, and e=€.
Each of the fifteen Pliicker planes corresponds to one of the fifteen tritangent
planes, thus the Pliicker plane @=y corresponds to the tritangent plane B+y=0:
two such planes pass through the line of intersection of two of the fundamental planes
8=0, y=0, and are harmonically conjugate with respect to those planes.
(3) Two triple tangent planes «+ 8=0, a+ y=0, which do not pass through a
common line on the surface, intersect in a lme —a=8=y, which must meet the surface
in three poimts. But the complete intersection of a+ 8=0 with the surface is the three
lines AB, CD, EF, and the complete intersection of a+y=0 with the surface is the three
lines AC, BE, DF; hence this tne —-a=8=y must meet AB, CD, EF, the sides of
A.p, in the same three poimts it meets AC, BE, DF, the sides of A,,: hence the line
—a=8=y must pass through the three P points which are the intersections of AB and
DF, CD and BE, EF and AC.
Such a line is called a Pascal line or an / line and there are sixty such lnes in
all, each given by an equation similar to —a=8=y, and each the common line of
intersection of two tritangent planes and one Pliicker plane. Eight h limes le in each
tritangent plane, and four in each Pliicker plane.
It has been seen that each h or Pascal line passes through three P points; thus
the h lime —6=e=€ passes through the three P points
—d0=ce=C=—4, B+y=0, ie. AF, BD,
—d=e=f=—8, yta=0, ie. BE, AC,
—d=ce=6=-y, a+8=0, ie. CD, EF.
Conversely, through each P point pass four h lines; thus through the intersection of
AB, CD, ic. the P pot a+B=0, y=—d=e=—€ pass the four / lines
~y=B=f; —3=yae; —= b=; b= 76
(4) It is clear that besides intersecting by fours in the P points, the h lines inter-
sect by threes in various other points: thus the thre -a=B=y; -a=y=68; —a=B=6
are seen to meet in the point
-a=B=y7=6.
Such points are known as Kirkman or H points, and are sixty in number: each les
on three tritangent and three Pliicker planes, and through each H point pass three h
lines and on each / line he three H points.
The notation employed bemg absolutely symmetrical shews that a correspondence exists
between the h line —a=8=y and the H pomt —a=d=e=€; it is easily verified
that if three h limes meet in an H point, the corresponding H points lie on the cor-
responding h line; but a more convenient method of defining the correspondence is the
followmg :—
274 Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM.
The five tritangent planes
a+P=0, a+y=0, 2+8=0, at+e=0, a+ f=0,
contain all fifteen lines of the surface and form a pentahedron which may be called
the ‘a’ pentahedron: there are then six such pentahedra the faces of each being tri-
tangent planes, and any two pentahedra have one face common: any two faces of a
pentahedron intersect im an h line, and the three remaining faces are found to intersect
in the corresponding H point; thus each of the six pentahedra has ten edges which
are h lines, and ten vertices which are the corresponding H points; in other words the
sixty h lines and sixty H poimts may be subdivided into six groups of ten points and
ten lines, the lines and points of each group being the edges and vertices of a penta-
hedron.
(5) There are twenty other points in which three / lines intersect, which complete
the system of the imtersections of the tritangent planes, viz. points such as
a=B=y=0.
These are known as Steiner or G points, and are twenty in number; two such as
a=8=y=0, and 6=e=f€=0 are said to be conjugate to each other, so that the twenty
G points fall into ten pairs of conjugate points. The G points are therefore the twenty
vertices of the hexahedron formed by the fundamental or coordinate planes a=0, B=0,
etc. and must therefore lie by tens in these planes, and must also lie by fours in the
edges of the hexahedron.
The Steiner or G points therefore lie by fours in fifteen lines such as a=8=0,
called Steiner-Pliicker lines or 7 lines, each 7 line being the intersection of a tritangent
plane with the corresponding Pliicker plane.
If six lines such as AB, BC, CA, DE, EF, FD, be omitted from the fifteen, the
remaining nine lines may be grouped into three plane triangles A in two distinct ways:
for if the lines be arranged in a square thus, -
Bans Cy SB ay,
a+d AD BF CE
ate CF AE BD
6+e BE CD AF
they may be grouped into triangles either by the rows or columns of the square, and the
plane of each triangle is shewn at the end of the row or column. The three planes
of either group of three triangles intersect in a G@ point, and those of the other group
intersect in the conjugate G point.
(6) It was noticed in (4) that if three lines meet in an H point, the three
corresponding H points lie in an fh line; it is also true that if three A lines meet in
a G point, the corresponding H points lie in a line. For if we take the G@ point
a=B=7=0, the three H points are
~a=8=e=; —B=8=e=; -y=b=e=5
and clearly lie on the line 6=e =€.
Mr H. W. RICHMOND, ON PASCAL'S HEXAGRAM. 275
There are twenty of these Cayley-Salmon or g lines, each corresponding to one G
point; thus the line 6=e=€ corresponds to the point a=8=y=0, and moreover the
g line which corresponds to a @ point passes through the conjugate G point.
When four G points lie in an 7 line, the corresponding g lines are found to meet
in a point: thus corresponding to the four G points which he on a=8=0, are the
four g lines 6=e=£; y=e=6; y=S=C; y=S=ec; which meet in the Salmon point
or IT point
yedSc= 6
There are then fifteen of these J points, through each of them pass six Pliicker planes.
The rest of the lines and points of intersection of these systems of planes do not
appear to be of sufficient interest to be worthy of separate mention here: their projec-
tions are of interest in the theory of the Pascal hexagram, and will be treated of in
fuller detail in connexion with that theory; moreover, since it will be found that the
development of the theory of the Pascal hexagram is so closely related to that of the
limes on a nodal cubic surface, that from each proposition relating to the former theory
an analogous proposition relating to the latter is at once deduced, it seems better to
obtain the properties of the Pascal hexagram first, and to state where necessary the
corresponding properties of the cubic surface as corollaries.
Before passing to the projections of these lines, I wish to mention certain quadrics
which pass through sets of six of these lines of the surtace.
(7) Any set of six lines such as AD, DE, FA, BC, CF, FB, must be generators
of a quadric surface, since each of the first three intersects each of the last three;
and the nine planes in which pairs of intersecting lines lie may be concisely shewn by
means of the table
BF. JAG! CB,
ADa+6, B+& yts
AEy+& ate 6+6,
DEBt+e yt+s, at+6.
The equation to the quadric is found by equating to zero any minor of the deter-
minant
) a+, B+% ye
lyt+& ate, Bt+S
| Bre, y+6, at€
Another more symmetrical form of the equation may be deduced; for if
(a+ 8)(a+e)= (B+ f)(y+ 8),
that is w@+ad+aet+ de= 67468 + Sy + By,
then (a+ d+eP+e—8&-C=(€4+8+y7%4+0C—-—R-Y.
Vor, XV. Panroil 36
bo
~J
(or)
Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM.
But (at d+e~=(S+B +4)
Hence the equation to the quadric may be written
e+ 84+yr=F4+ 64+ C6.
There are ten quadrics such as this, whose complete intersection with the cubic
swrface consists of six of the fifteen lines on the surface; any two such quadrics have
two common generators, thus the quadric
4+ E+ Pet F+ FS,
which passes through the six lines AC, CD, DA, BE, EF, FB, has the two generators
AD, BF in common with the former quadric. The complete intersection of the two
quadries is contained in the two planes a+6=0, of which the former contains the two
common generators AD, BF; hence the remainder of the curve of intersection of the
two quadrics consists of the plane conic
a=6, Bt+yHeE+l.
THE PASCAL HEXAGRAM.
As has been stated above, Cremona has shewn that by projecting the lmes and
points derived from the consideration of the lines on a nodal cubic surface, we obtain
the figure of the Pascal Hexagram.
Adaptation of equations. The equations we have made use of in discussing the cubic
surface are readily transformed into others which are applicable to the plane figure; for
since at 0, the nodal point
Clap cheaei fi 30/yseGe 2. Seca)? (Sha GS ee if,
we can always find the equation to the plane which passes through O and any line
whose equations are known, or to the line that joms O to any pomt that has been
determined.
It is now only necessary to imagine that this system of lines and planes, all of
which pass through O, is cut by an arbitrary plane YW, and the projection of the three-
dimensional figure upon this plane W will have been obtained. It is not desirable that
any particular plane should be selected as the plane of projection, but, for the sake of
the nomenclature, I shall consider that the section by a plane W has always been made:
thus, although < WftY really represents a plane which passes through O, the conical
es ' OS As ee bere : 2 NC Sen Set fam Eee
point, I shall be justified in speaking of the line Pgh ee if it is always understood
that the system of lines and planes is cut by the plane W in a system of points and
lines. In the same way, when I speak of a conic, the equation used will really re-
present a quadric cone whose vertex is at O, the conical point.
Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM. 277
The six lines A, B, C, D, E, F, which pass through the conical point O, were found
to be the lines of intersection of a cubic cone and a quadric cone: projected from
the conical point upon a plane W, they appear as six points A, B, C, D, E, F, which
lie on a conic.
The fifteen lmes AB, AC, ... each of which meets two of the six lines, are pro-
jected into the lines which join by pairs the six points A, B, C, D, EF, F, and thus
furnish the foundation of the figure of the Hexagram.
Equations of the fifteen lines AB, AC....
The equation to the plane which passes through O the nodal point and the line AB is
ag yb ® ee
a+b c+d e+f’
hence this is also the equation of the line AB in the projected figure. Expressions such
eae and a= 2
a+b a—b
at once to replace them by simpler symbols.
Let att be represented by the symbol (48),
will occur so frequently in subsequent work that it is convenient
a—B
a—b
and be represented by the symbol y (a).
Thus in three dimensions, each tritangent plane is given by an equation such as
(a48)=0, and each Pliicker plane by an equation such as y(a8)=0, and at O the nodal
point
B
b
The equations of the fifteen lines AB, AC, can be at once derived from those on
p. 272; they are
a
a
=7 ... = (a8) = (ay)... =x (a8) = x (ay)... -
AB (a8) = (75) = (eb);
AC (ay) = (Be) = (88);
AD (a8) =(BE) =(ye);
AE (ae) =(B6)=(y¥6);
AF (af) = (By) = (6e);
BC (af) =(88) =(ye);
BD (ae) =(By) = (88);
BE (ay) =(8&) = (6);
BF (a8) =(Be) = (78);
CD (a8) = (yf) = (6e);
CE (ad) = (By) = (eb);
CF (ae) =(8f) = (98);
DE (a) = (Be) =(78);
DF (ay) = (88) = (ef);
EF (a8) = (ye) = (86).
36—2
278 Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM.
The equation to the conic on which A, B, C, D, H, F lie is
da? + 062 4- oy? + do? ee + fE HO) ee (1),
and the complete system of equations is
Bb ty tid ae C= Oop - et cictateda al Ss op Sones (2),
@a KE BAs cy 08d +e Af 26 = 0) oc. cheat seeps ee es Sones es (3),
aber Pet f= OLR0..SUA. Wi. See tans oleh ot (4),
PAGE ie ides FS = Ona vowaea soos sare cqnememececncetes (5).
> : ‘ a+
Further, (a8) is defined as - +f og saialaceiite sais Stata ra maetnerniaesee ace ae (6),
a—B
x (a8) acces eoncceescoce a= 5 A BOB OSTEO On Doom Oheoe non nacorco- Osfooanod (7).
It follows that
ss a £B e :
if aa each is necessarily also = (a8) = yx (a8),
if (G@B)=H(YO) cacacsecctmsceencecenecetescis = (eC),
if (G8) =(@y)) ceccccescctcmsasecncceemoeee: = x (By),
if WA (cfs) =o A (7) Rececseoseensde0s- 06003-05066 =x (By),
and at the nodal point
2 UB te
ee: = (a8) = (ay)... =... ¥ (@8) = x (ay) -.--
Before I pass to the Pascal hexagram, it is convenient to discuss in two lemmas
some properties of the figures formed by projecting on any plane the lines of intersection
first of five planes and secondly of six planes in three-dimensional space.
I. Take five planes in three-dimensional space,
u=0, v=0, w=0, x=0, y=0,
forming a pentahedron, with ten edges and ten angles; take also a point O not situated
on any of these planes as origin of projection.
We may introduce factors into the functions u, v, w, #, y, so that at O,
u=V=W=—L— y.
Further, the five quantities 1, », w, z, y, must be connected by an identical linear
relation
Pl AGU SoBe 1Cyl Olepiaientae= cele malcie(re sis aieltesiaslerrer sei (1),
where pt+qtrt+s+t=0.
Then “=v represents a plane passing through OU and the line of intersection of
u=0, v=O0 ete.
Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM. 279
If now we consider the sections of all the planes which pass through 0 and through
one of the ten edges of the pentahedron, by an arbitrary plane W, we obtain the
required projection. The figure is shewn below.
It consists of ten lines which meet by threes in ten points, and three of these
points lie on each of the ten lines. Selecting any point w= v=w, the three lines
U=V, V=W, w=u pass through it; six of the remaining lines form two perspective
triangles, viz. w=a, v=a2, w=a; and u=y, v=y, w=y, and the tenth line z=y is
the line of perspective on which corresponding sides intersect.
There is a certain conie such that each of the ten points is the pole of the corre-
sponding line, viz.
ji AR ODP Seni seeke Se Gap =) vissoecocnddoooseobosssanpuaguosoonabed (2).
For the polar of the point (wvjworoyo) 18
Ply + QV + TWW, + Sax, + tyy, = O.
If now u%=v,=wW, the polar is
Uy (pu + qu+rw) + sxx, + tyy = 9,
or by equation (1)
Up (— sa — ty) + saa + tyy, = 0,
=f one (a FF MU) at ty (Yo = Us) = 0.
ca PUlo + Qo + PW + SL + ty = 0,
(PHqO+7) (Wi) + 82) + ty = 0,
or (—s—t)u,+ sx, + ty = 0;
S (a — Uy) +t (Yo = Uy) = ()
280 Mr H. W. RICHMOND, ON PASCAL’'S HEXAGRAM.
Hence the polar of the point u=v=w is the line e=y,.
The figure may be called a Projected Pentahedron.
Il. Taking next six planes
uw=0, v=0, w=0, z=0, y=0, 2=0;
we project their mtersections from the point O at which
U=V=W=H=Y=z.
The six quantities wu, v, w, 2, y, 2, ave connected by two linear relations
putqvtrwtsettyt+kz=0,
put+qv+rwt+se+ty+kz =0,
where ptqatrt+s4+t+k =0,
p+ dtrt+s +t +h =0.
The projection consists of fifteen lines #=y, ... which meet by threes in twenty
points «=y=z, and four of these points lie on each line.
The figure, which may be called the figure of a projected Hexahedron, is shewn
below.
Mr H. W. RICHMOND, ON PASCAL'S HEXAGRAM. 281
If we select any point, eg. «=y=z, through which pass the three lines «=y,
y=2, z=, nine of the other lines group themselves into three perspective triangles,
viz, Z=U, Y=U, Z=U; L=V, Y=V, Z=V; L=w, y=w, z=w; and the three lines of
perspective in which corresponding sides of any two triangles intersect are the remaining
three lines w=v, v=w, w=u, which meet in the pomt u=v=w.
If we start with the point w=v=w, the nine sides of the three perspective triangles
are the same nine lines as before, but differently grouped.
(a) Two points such as r=y=z, w=v=w are conjugate with respect to any of
the conics
(p + rp’) w+ (q + Aq’) V+ (7 +0") w+ (5 + As’) + (EFA) Y? + (K+ DK) 2 =O.
Denote the coefficients by P, Q, R, S, T, K, then two points (%, %, Wo, Lo Yo, 20)
and (%, %, W:, 4%, 1, %) are conjugate if
Pug, + Quy, + Rww, + Sar, + Tyy, + Kaz, = 0.
If now m=%= Ww and #,=y,=4%, the condition of conjugacy is
uy (Pu, + Qu, + Rw,) + 2, (Sx + Ty, + Kz) = 0.
But we know that at any point
Put Qvt+ Rw + Sx+ Ty + kz =0,
v. (P+Q4+ R)u, + Sx + Ty, + Kz) = 9,
and (Pu, + Qv, + Ru,) + (S+ 7+ K) a, = 0.
Also (P+Q+R)+(S4+ 7+ K)=0.
Hence the condition is satisfied and the points are conjugate with respect to any
conic of the system.
(8) The system of conics above consists of all conics which pass through four fixed
points which for the moment may be called P, Q, R, S. If the diagonals of the quad-
rangle PQRS meet in L, M, N, it follows that the lines from any one of these points
such as £ to any two conjugate points, as w=y=2 and w=v=w, form an involution,
the double rays being the lines which pass through the four points P, Q, R, 8S.
If the conic
Pu + Qu? + Rw? + Sx? + Ty + K2=0
break up into two straight lines, which intersect in the Point (UV WoL VYo%o),
then Pum + Quo, + Rww, + Sax, + Tyy, + Kzz,=0;
*, Pu =ap + BP, Sa, = as + BS,
Qu, = aq + BQ, Ty, = at + BT,
Rw, = ar + BR, Kz,=ak+ BK;
hence substitutmg in Plot To +... = (()
pp gg rr ss tt kk
we have jae a) Oe gS . eat oh
282 Mr H. W. RICHMOND, ON PASCAL'S HEXAGRAM.
. pp ; iil i tt’ kk
that is PP 5 spe Sata or ba at =0
ptrAp gtAq r+" sts t+ att k+vk {
an equation which gives three finite values of X.
Giving X these three values in succession, we may find the coordinates of the three
points LZ, M, N.
The six points where any line of the figure «=v is met by the six lines w=2, y=z;
w=y, L=2; w=z2z, ©=y; are conjugate in pairs with respect to one conic of the system,
viz. that for which
P+Q or p+Ap' +q+4+rAq' =0.
For the condition of conjugacy being as before
Puc, + Qur, + Ruww, + Sav, + Tyy, + Ka 2, = 0,
if we have
Up =U, h =, Wy = Zo, Nn = 2,
the condition becomes
(P+ Q) wan + wm (Rw, + Sa.) + x, (Ly) + Kz,) =0.
Also (P+Q)u+(R+ 8) w+ (Ty + Ka) =,
(P+ Q)u, + (Rw, + Sz,) +(7+K) y, = 0.
If then P+Q=0, the condition is satisfied, since
(R+8)+(7+4 K)=0.
I now proceed to deduce from the properties proved for the cubic surface the
analogous properties of the plane figure.
Ill. The fifteen lines AB, AC,... which jom by twos the six points 4, B, C, D, EH, F,
group themselves into fifteen triangles A, on whose sides lie all the six points A, B, C,
D, E, F: such a triangle is AB, CD, EF, to which as in section (1) I give the name
A,s: any line AB belongs to the three triangles AB, CD, EF; AB, CE, DF: AB, CF, DE;
and further since the other sides join the four points C, D, EH, F, it follows that ‘the
six vertices of triangles A which lie on AB are in involution.
The vertices of these triangles are called P points and are 45 in number.
IV. From (3) we infer that
AB meets DF
CD meets BE
in three points which lie on the h or Puscal line (a8) = (ay) = x (By).
EF meets AC |
And sixty such lines exist.
Consider now the six lines just mentioned: if we arrange them in the order
AB, BE, EF, FD, DC, CA, it is clear that they are sides of a hexagon ABEFDC
inscribed in the conic, and we have shewn,
Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM. 283
‘The opposite sides of any hexagon inscribed in a conic intersect in three collinear
points.’
There are sixty different hexagons which we can form by joining the six points
A, B, C, D, E, F, in different ways, and from each hexagon is derived one of the sixty
h lines.
On each fA line lie three P points, and through each P point pass four A lines;
thus through the intersection of AB, CD pass the four h lines derived from the hexagons
ABECDF, ABFCDE, ABEDCF, ABFDCE.
V. The sixty h lines intersect by threes in sixty H or Kirkman points
(a8) = (ary) = (28) = x (BY) = x (78) = x (88) 5;
and on each fh line lie three H points.
The three concurrent f lines are derived from the hexagons ABHFDC, ACEBFD,
ADCEFB, respectively: it was pointed out that to each / line corresponds one 4 point:
now the sides of these three hexagons are composed of nine only of the fifteen lines
AB, AC...; and the six lines omitted are the sides of the hexagon AHDBCF from
which is derived the corresponding / line (ae) =(a€).
VI. The edges and angles of each pentahedron are projected into ten hf lines and
ten H points, forming a figure of a projected pentahedron discussed in I.: it follows that
a conic exists such that each of the h lines which form the figure is the polar of the
corresponding H point.
The sixty A lines and sixty H points fall into six groups of ten lines and ten
points; and with each group is associated a conic such that each h line of the group
is the polar of the corresponding H point (which always belongs to the same group)
with respect to it.
There is no difficulty in finding the equation of this conic,
(a+ 8)+(a+y)+(a+8)+(ate)+(a+ 6) = 4a
and B(at+8)+C(aty)+...=(P4+EC4P4+E4+f2?-@)a,
(P+ +P+e+f?— a’) ((at+8)+(a+y)+(at+8)+(at+e) +(a+)}
=4[P(a+8)+e(aty)+@(atd)...... Ik
That is (a? + 3b? — c? — d?— e& — f*) (a + b) (aB) +...... =(0).
Hence the equation to the conic is
(2 +382 — @&—d— & — f?) (a +b) (a8) +......=0.
VII. The sixty / lines also intersect by threes in twenty Steiner or @ points,
4 =F = = (a8) = (ay) = (81) =x (08) = x (ay) =x (9).
Vou. XV. Parr II. Bi
284 Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM.
The three concurrent h lines are derived from the following hexagons:
(a8) = (ay) = y (By) from ABEFDC,
(a8) =(By)= x (ay) from AFECDB,
(ay) =(By)=x (a8) from ACEBDP,
in which the first, third and fifth letters are the same, and the second, fourth and sixth
are cyclically interchanged.
The twenty G@ points fall into ten pairs: with the point above is associated the
point
Do = p=) = (6b) = (BE) = (Be) = Heb) = x BE),
in which intersect the three h lines derived from the hexagons ABECDF, ACEFDB,
AFEBDC, where the first, third, and fifth letters are again the same as before, while
the second, fourth, and sixth are derived from those of the former hexagons by non-cyclical
interchanges,
VIII We may apply the results of Il. to the figure formed by the projection of
the intersections of the six tritangent planes
(a8) = 0, (By) = 0, (ya) =0, (Se) =0, (ef) =0, (£6) = 0.
The figure is simpler than that in II. inasmuch as one of the linear relations con-
necting
(48), (Bry), (ya), (Se), (ef), ($6)
is (a8) + (By) + (7%) = (de) + (ef) + ($0), see page (271),
so that the three lines such as
(a8) = (e); (By) = (eb); (ya) = (£8)
are concurrent.
The second linear relation is
(b + ¢) (By) + (¢ + a) (ya) + (a +b) (a8) + (d + €) (Se) + (0 +f) (ef) + (f+ d) (€8) = 0.
The system of conics in II. comprises all conics which pass through the four points
common to
(a8)° + (By)? + (ya)? = (6e)* + (eb)? + (66)?
and (a+b) (a8)+(b+c) (By) + (c+ a) (ay) + (d+ e) (Se) +(e +f) (ef)? + (f+ d) (S6)?= 0.
The former of these two is the fundamental conic on which the six points
A, B, C, D, E, F, lie, and can therefore be reduced to
ao? + DB? + cy? + d& + ce + fC =0.
Two G@ points such as
which have been called conjugate G points, are therefore conjugate with respect to the
fundamental conic.
Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM. 285
4
By simplification of the second equation, it may be shewn that these two @ points
are conjugate with respect to all conics which pass through the four points common to
aa + b3? + cy? + d& + ee + fo? = 0
and (be + ca + ab) (a+ 8 ++)? — bea? — caB? — aby’?
= (de + ef + fd) (S++ $) — of & — fde — deg,
and, further, nine pairs of P points such as
(@B) = (ay) =(€6); (de) = (8) = (48);
the intersections of AB, DF, and of AF, DB are conjugate with respect to all conics
of the system.
Again by II. (8), we see that any side such as (a8)=(ef) or AB, is met by the
h lines (By) =(yx) and (Se) = (8), which are derived from the hexagons ACEBDF,
ACDBEF, in two points which are conjugate with respect to the fundamental conic,
and therefore form with A and B a harmonic range.
On the side AB there must lie six such pairs of conjugate points, each pair forming
a harmonic range with the points A and B.
IX. The fifteen lines in the figure of this projected hexahedron are composed of
six h lines and nine sides of the triangles A, which join two of the six pots A, B,
C, D, E, F; consider the grouping of the eighteen points where the nine sides of
the triangles are met by the / lines.
On each fh line, as (a@8)=(ay), lie three of the points, viz. the points where this
line is met by AF, CE, BD, the sides of Agy. The points fall into two groups of
nine, according as the h line they lie on passes through one or other of the @ points.
Arrange the points thus:
(a8) = (ay), (Bry) =(€&) | (By) = (8%), (ya) = (ef) | (72) = (7B), (a8) = (€8) |
(a8) =(ay), (By) =(£8) | (By) =(Ba), (ya) =(88) | (y2)=(7B), (@B)= (66) | .....(A);
(a8) = (ay), (By) =(8e) | (By) = (Ba), (7%) = (Se) | (y2)= (98), (48) = (8)
(Se) = (88), (ef) =(By)| (ef) =(c8), (8) =(By)| (68) = (Se), (8e) = (By)
(Se) = (88), (e£) =(y2) | (eb) = (€8), (68) = (ya) ($8) = (Ee), (Be) = (74) |---- (B).
(Se) = (86), (e€) =(a8)| (ef) = (68), (68) = (a8)| (8) = (Ee), (8) = (48)
Taking either group, the nine points form three triangles, if we take them in rows,
and lie by threes on the h lines, if we take them in columns. The conjugates to
three points of either group which form a triangle are three points of the other group
which lie on an h line.
The sides of the triangles of the first group are
(a) + (ary) = (€6) + (By); (8a) + (By) = (5) + (ay); (y) + (YB) = (8) + (48);
(af) + (ay) = (£8) + (By); (Ba) + (Bry) = (88) + (ay); (v2) + (VB) = (88) + (48) 5
(a) + (ay) = (Se) + (By); (Ba) + (By) = (Be) + (ay); (72) + (7B) = (Se) + (48).
Thus the corresponding sides of any two triangles intersect on an h line which
passes through the second @ point.
37—2
286 Mr H. W. RICHMOND, ON PASCAL'S HEXAGRAM.
Again, since
(a8) + (By) + (y%) = (Se) + (ef) + (88),
the equation to each of these lines may be written in a new form: for example
(a8) + (ay) = (ef) + (By)
is equivalent to
2 (Bry) = (de) + (8E).
Hence this line passes through the P point
(By) = (Se) = (8£) = (af) =(ae), Le. the intersection of AF, BD,
and further it forms with the h line (8) = (8) and the two sides AF, BD a harmonic
pencil.
X. Corresponding to the three h lines which meet in the G@ point
Byey,
Wi A
a —
i=
are the three H points
(a8) = (ae) = (a5) = x (Se) = x (eb) = x (8);
(88) = (Be) = (BE) = x (Be) = x (eb) = x (88);
(78) = (ye) = (yb) = x (Be) = x (eb) = x (58);
which are seen to lie on the Cayley-Salmon or g line
x (Be) = x (ef) = x (68).
This g line corresponds to the G@ point above, and passes through the conjugate
G point. There are twenty such lines in the hexagram, on each of which lie three H
points and one @ point.
XI. Four @ points such as
4 =F = = (a8) = (ay) = (By) = x (a8) = x (an) = x (89);
@ =F _5 = (af) = (a8) = (88) = x (28) = x (28) = x (89);
@ = 8 © < (af) = (ac) = (Be) = x (aB) = x (ae) = x (Be);
428 8 _ (apy = (at) = (BE) =x (a8) =x (ab) = x (88);
a as an i + x x x J
lie in one of fifteen Steiner-Pliicker or 7 lines such as
«=F = (a8) =x (a8),
which pass by threes through the twenty @ points.
Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM. 287
The twenty G@ points and fifteen 7 lines form the figure of a projected hexahedron,
discussed in II., viz. the projection of the hexahedron formed by the six fundamental planes
a = 0...ete.
Any two conjugate G@ points are therefore conjugate with respect to all conics which
pass through the four points common to the fundamental conic
aa? + DB? + oy? + d& + ee + fC? =
ae [ss yf? &
and = Seo!
KG
45+ By
The conic
(«+*) w+ (b+ 5) B+ (c+ *) y+ (a +7) 8 +(e +>) é+ (f+ 5) o=0
will break into two straight lines if (as may be deduced from II.)
1 & 1 as 1 - 1 a it a 1 =
r n r x py Neues
nee — th = ae == =
Usa b ge aes d+— ie Its
a b c d e if
or ERR SR GEN GEN PEN EN
whence
5A? — 35.17 + 5A +5; = 0,
where
S. = ab+ac+ad+... ;
8, = abed+ abce+... ,
8; = abcdef.
But if = : + : tat : + Us 0, any value of 2 satisfies the equation. In this case
ip
however, the constants a, b, c, d, e, f are equal and opposite in pairs, and the funda-
mental conic degenerates into two straight lines.
XII. Corresponding to four G points which lie on the 7 line
= = (a8) =x (a8)
a
a
are four g lines which meet in one of fifteen Salmon or J points
x (78) = x (ve) = x (75) = x (Se) = x (8f) = x (€8),
and this J poimt corresponds to the 7 line above.
XIII. The projection of the figure formed by the five planes
x (48)=0, x (ay) =0, x (ad) =0, x (ae) =0, x (al) =0
gives the figure of a projected pentahedron discussed in I.
288 Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM.
The ten lines are here g lines and the ten points are J points: but each g line
is common to three of the six figures and each J point is common to four figures, and
in different figures different g lines correspond to the same J points, and different Z
points to the same g line. .
With each figure of ten g lines and ten J points is associated a conic such that
each g line is the polar of the J point which corresponds to it in that figure; the
equation to the conic is found to be
YL(@+P+e+d+e+f?— 6b’) (a—b)x(as)|?=0.
XIV. In the three-dimensional figure consider the A lines which pass through the
vertices of the triangle A.g formed by AB, CD, EF.
Through the intersection of CD, EF pass the four h lines
—y=e=6; —S8=e=€; —e=y=8; —f=y=6.
Through the intersection of HF, AB, pass
Se on we ee 8 — a
Through the intersection of AB, CD, pass
Si Oe eee oe
These twelve / lines intersect by threes in four H points
—y=b=e=6; —8=e=f=y7; -e=f=y=86; —C=y=b=e:
and in four G points
See pee ahs C=) = 0 S03 VSO=eS0)
such that the four conjugate G points are collinear.
The H point —y=8=e=€ is joined to the G point =e=f=0 by the g line
6=e=€, and is joined to each of the other three @ points by an A line which passes
through a vertex of A,g: also the four g lines intersect in the J point y=S=e=€.
Hence the tetrahedron formed by the H points and that formed by the G@ points
are perspective with respect to four distinct centres, viz. the vertices of A,g and the I
point y=d=e=€.
The corresponding property of the hexagram is, The quadrangles formed by the H
points
(78) = (ye) =(78); (By) =(8e) = (88); (ey) =(€8) =(€6); (Sy) = (88) = (Ee);
and the G points
(Se) = (ef) = (£6): (ye) = (eb) = (78); (78) =(8£) = (98); (v8) = (78) = (6),
respectively, are perspective with regard to four distinct centres of perspective, viz. the
vertices of A,g and the J point
xX (79) =x (ye) =x (7S) = x (Ge) =x (85) =x (€6).
Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM. 289
I proceed to consider the complete figure formed by the projection of the lnes and
points of intersection of the tritangent planes, the Pliicker planes, and the six coordinate
planes: from this figure are deduced nearly all the properties of the hexagram given by
Veronese, and one or two new properties. It will be convenient to treat first of the
intersections of the tritangent and Pliicker planes, and to introduce the six new coordinate
planes later.
XV. Consider the projections of the eight h limes which he in a tritangent plane
a+ 8=0:; they form two quadrilaterals
(48) = (ay); (@8)= (28); (@8)= (ae); (48) = (ab),
(Ba) =(By); (Ba) =(86); (Ba) =(Be); (Ba)= (RE).
The six vertices of each quadrilateral are H points, and corresponding sides intersect
in @ poimts which lie on the 7 line 4 _F = (a8) = x (a8); while sides which do not
correspond meet in the twelve P points which lie on the sides of Aj,g but are not
vertices of that triangle.
The lines which join corresponding vertices of the two quadrilaterals are called
v lines; for example the two vertices
(a8) = (ay) = (a8) = x (By) = x (88) = x (78),
(Ba) = (By) = (8S) = x (ay) = x (a8) = x (79),
are jomed by the v line
(a8) = x (78).
The hexagram contains ninety of these v lines, each the projection of the inter-
section of a tritangent plane a+@8=0 with a Pliicker plane y—6=0: on each v line
le two H points; through each P point pass two v lines, and through each H point
pass three such lines.
The six v lines derived from the two quadrilaterals given above pass by twos
through the vertices of the triangle A,g; their equations are
(a8) =x (78); (aB)=x (ye); (4B) =x (75);
(a8) =x (ef); (aB)= x (£8); (a8) =x (e);
and therefore they intersect by threes in four points which for the present I call H,
points, such as
(a8) = x (8) = x (ye) = x (8e),
each of which lies on one of the g lines which pass through the J pomt corresponding
to Ags. It follows that the diagonals of the quadrangle of H, points are the sides of
Axe, and hence
The two v lines which pass through any P point form a harmonic pencil with the
sides of the triangle A which intersect in that P point.
290 Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM.
There are sixty H, points in the hexagram, lyig by threes on the twenty g lines:
there is clearly a correspondence between the sixty H, points and the sixty h lines
and the sixty H points; thus the H, point
(a8) = x (¥8) = x (ye) = x (8e)
corresponds to the H point
(Sy) = (88) = (Se) = x (¥8) = x (ve) = x (Se),
and to the h line
(Sa) = (€8) = x (a8).
Each H, point is joimed to the corresponding H point by the g line which passes
through it.
XVL It will be seen that the h limes which correspond to two H poimts of a v
line meet in a P point, and are the projections of two h lines which lie in a Pliicker
plane. The four H points therefore which correspond to four h lines through a P point
such as (a8) = (ve) = (yf) = (Se) =(8f) le on two v lines, (y5)=y (a8), (eC) =x (a8); and
these intersect in a Y point
a_B
aie
= (a8) = (y8) = (ef) = x (a8),
the intersection of the 7 line which corresponds to the triangle A,g with the side of
the triangle opposite to the P point.
The Y points number forty-five and lie by threes on each side of a triangle A and
on each 7 line.
The six v lines which pass through the intersections of the diagonals of the quad-
rangle C, D, E, F, are
(48) =x (y8); (y8)=x (a8); (eb) =x (48);
(4B)=x (eS); (y8)= x (eb); (eb) =x (79),
and intersect by twos in the three Y points of the line AB. Since the v lines through
the intersection of CD, HF form a harmonic pencil with CD, HF, it follows that the
six P points of any side AB form harmonic ranges with two of the three Y points of
that side.
XVII. To the forty-five Y points, where a side of a triangle A,g is met by the
corresponding 7 line, correspond forty-five y lines which join the opposite vertex of the
triangle to the corresponding J point.
The y lines are seen to be given by equations such as
x (78) = x(€6),
this being the line which corresponds to
(48) = (78) = (eb) = x (a8).
Mr H. W. RICHMOND, ON PASCAL’'S HEXAGRAM. 291
To three Y points which lie in an 7 line correspond three y lines which meet in
an J point; and to three Y points which lie on the side of a triangle A, as
(a8) = (8) = (¢b)
correspond three ¥ lines which meet in one of fifteen R points
x (28) = x (¥8) = x (€6),
to which I shall have occasion to return later. The three y lines which meet in the
R pomt which corresponds to the side AB, pass through the intersections of the diagonals
of the quadrangle CDEF.
Each y line is the projection of the line of intersection of two Pliicker planes:
through each P point pass two Pliicker planes, which intersect in the y line, and each of
which contains two h lines, and one v line passing through the P point: the four lines
in each plane form a harmonic pencil.
For through the P pomt a+8=0, y=é6=—e=—€ passes the Pliicker plane e=£,
and this meets the four planes
yte=0, 6+6=0, (yt+e) +(8+0)=0,
in four lines which form a harmonic pencil, whose rays are the four lines spoken of.
The projections of these lines also form a harmonic pencil.
XVIII. It was shewn in XV. that the four points H,
(a8) = x (Se) = x (eb) = x (88),
(a8) = x (ve) = x (eb) = xX (78),
(a8) = x (78) = x (88) = x (78),
(a8) = x (¥8) = x (Se) = x (ve),
form a quadrangle whose diagonals intersect in the vertices of the triangle Ajg: hence
the lines joining these four points to any other poimt and any two of the lines which
join the point to the vertices of A,g, form three pairs of lines in involution.
In particular, if the point chosen be the Z point, which corresponds to Ajg, we have
the property that
The four g lines through an / point and any two of the three y lines through
the point, form three pairs of lines in involution.
XIX. The y lines intersect by threes in sixty = points, such as
x (a8) = x (8) = x (ve) = x (Be)
which lie by threes on the g lines, and correspond to the sixty h lines and H points.
Consider the quadrangle formed by this = point and the three J points
x (a8) = x (ad) = x (ae) = x (88) = x (Be) = x (Se),
xX (aB) = x (ae) = x (ay) = x (Be) = x (PY) = x (er),
x (a8) = x (ay) = x (48) = x (By) = x (BS) = x (78).
Vou. XV. Part II. 38
292 Mr H. W. RICHMOND, ON PASCAL'S HEXAGRAM.
The JZ points are joimed by the g lines
x (a8) = x (av) = x (BY),
x (a8) = x (ad) = x (88),
x (a8) = x (ae) = x (Be),
and the = point is joed to the three J points by the y lines
x (a8) =x (Se); x (48)=x (er); x (4B) =x (79).
These lines must cut any transversal in involution; take as the transversal the
g line
x (48) = x (ab) = x (86).
Hence the three J points and the three > points of any g lines are in involution.
XX. The ninety v lines also intersect by pairs on the / lines in 180 £ points
(a8) = (ay) = x (By) = x (5e);
each £ point is the intersection of two v lines, one h line, and one y line; and three
E points lie on each h line and four on each y line.
If we take as the transversal which meets the sides of the quadrangle in XIX. the
h line (af) =(8E)=y(a8) we see that the three # points of any A line and the three
H points of the line are im involution.
XXI. The only other points furnished by the intersections of the tritangent and
Pliicker planes are ninety NV points, given in the hexagram by equations such as
a “ = (a8) = x (a8) =x (78);
a
each the intersection of a v line, a y line, and an 7 line.
XXII. To complete the figure formed by the tritangent, Pliicker and coordinate
planes, it is necessary to consider only the intersection of one coordinate plane with the
planes and lines discussed above; for the line of intersection of two coordinate planes
is an @ line and has already received notice.
Each coordinate plane, as a=0, is met by ten of the tritangent planes 8+ y=0 in
a line called a o line a=B+y=0. The projections of these sixty o lines
a
are noticed by Veronese who shews that if three y lines meet in a = point, the corre-
sponding Y points lie in a o line which passes through a @ point.
The line in which a coordinate plane a=0 is met by a Pliicker plane B=y may
be called a pw line: there are sixty mw lines, each containing three N points and one
( point: the projections of these pw lines
; a, ¢ (By),
have the same property.
Mr H. W. RICHMOND, ON PASCAL’'S HEXAGRAM.
XXIII. There are further in the three-dimensional figure
60 F points, 2=0; B=y7=5;
180 J points, «=0; B=y=—6;
90 K points, «=0; B=y; S=«;
180 ZL points, a=0; B+y=0; d=;
whose projections give in the hexagram
60 F points, © = x (By) =x (78) =x (88),
Gk
180 J points, 7 = (88) = (98) =x (By).
90 K points, — =x (By) =x (de),
3
a
180 Z points, = (By) = x (Se).
oh
It will be as well to pause here and enumerate the various lines and points which
compose the hexagram as far as we have at present discovered them. There are
15 sides of hexagons............... (a8) = (78) = (€€) ;
60 h (Pascal) lines...............-.. (aB) =(ay)= x (By);
20 g (Cayley-Salmon) lines ...... x (a8) =x (a7) =x (By) ;
15 7 (Steiner-Pliicker) lines ...... (aB) =x (48) = : =? ;
0X0) a Imei osasasoasuseosdoogodenacasder (a8) =x (¥8) ;
AS WAMlINES ts scisclenieleseaesreenae secs x (a8) = x (78) ;
(XO) cp llint Gs cossiqosocrdoccceccoognbobene ; =(By);
‘ a
Gy emlin CSeeneseeriesectiscle tice tee: eX (By).
6 fundamental points...... AL, 1B CID JIG IH
15) JE qaosbaiis! Gapdeonopasseeedor (a8) = (ye) = (yf) = (Se) = (88) = x (78) = x (€8) ;
60 H (Kirkman) points ....(a) = (ay) = (a8) = x(By)= x (88) = x (78);
20 G (Steiner) points ...... (a8) = (ay) = (By) = x (48) = x (av) = x (By) =— =
a
15 J (Salmon) points ...... x (a8) = x (ay) =x (a8) = x (By) = x (85) = x (78);
38—2
Cc
a Bo,
b
294 Mr H. W. RICHMOND, ON PASCAL’'S HEXAGRAM.
GQ Asspointshs sso eee (a8) =x (75) = x (ve) = x (8e) ;
AS Yi POMS xececcencavessssnes (a3) = (8) = (ef) = x (a8) = - = :
Fe POMGS ce ceswe ee ees eg: x (a8) = x (8) = x (€8) ;
COPS pointe’ cs...... scence eee x (a8) = x (y8) =x (ye) = x (Se) ;
TOO EE POIMts sccveca.cuncsnaveiss (a8) = (ay) = x (By) = x (Se) ;
SOPN] pomts:....--<eessecrnans (a8) = x (a8) = x (78) = - = S :
CO pots: oinck-..sceschons x (By) = x (BS) = x (78) = i ;
USOnS Points... saeeseecee essere (88) = (8) = x (By) = - :
90: Ke. points: be. h.asaes onteae x (By) =x (de) =« 3
TSORE points s-5--eeececsecese (By) = x (8e) = = :
Of the lines and points derived from the figure formed by the tritangent and Pliicker
planes, all receive notice in Veronese’s Memoir except the NV points and RF points; but
of the intersections of these planes with the coordinate planes only the o lines are
mentioned. In the case of two kinds of points I have altered Veronese’s notation, the
H, points and = points being called by him Z, points and ¢ points respectively; and
for the sake of brevity I have spoken of H points, G points and J points, h lines, g
lines and 7 lines where Veronese uses the names of the mathematicians by whom they
were discovered.
In the three-dimensional figure, the limes and points of which the projections have
been given are as follows:
15 sides of hexagons...a+P=y+6=e+6=0;
GOTAUNES cane nmseec tener —a=B=y7;
20g) Littes 22: -.cnp sss sae a=B=y7;
D5.¢: Mes has, sesnatehbe a=B=0;
90. v) lines. ac-paaccneeN a+B=0; y=6;
453) MNCS 5. jestaeoes ner a=—; y=0;
GOko: lines: Beeeeee ee a=0; B+y=0;
(Uppy gl Ho A pepersnncrice a=0; B=y;
Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM, 295
25) JO Joyo) oocaopnoooc0s at+B=0; y=o=-—ce=—6;
GOMHM poms ences ieeer —-a=B=y=5;
20) G, points scesseeeens 2 —7 08
15) JE Fao ccocaoso anne C= /9 = So:
60) He pointsie esses at+tB=0; y=d=e;
AS Ve sO beaters sees a=B=y+6=e+f=0;
16) JR) [OOS eon opeoonac a=8; y=6; «=6;
GO) Se pointisy canes soca a=B: y—=o=e;
alisX0) 72s jotesbals)) ce ss5enccboae —a—p—y; O64;
90g points ese. --p reer a=/G=0e G/=Os
(0) 22" TOGMIHS coonocospdce: a=0; B=y=6;
180 J points ............. a=0; B=y=—-—5;
XO) L&C Forortauiis) -oodscaa008 aA=3s /s=578 Gee
Ilfs{0) JD, jaOUMANS eonaooooseoos a=0; B=y: d+e=0
There are two sets of lines and points which Veronese has noticed, viz. m lines
which are the projection of lines such as
a+B=0, 2aty+o=9,
and 7 points which are the projections of points such as
a+B=0, S5=6 2a+y+6=0;
but these do not appear worthy of further mention.
It was poimted out in (7) that the six lines AD, DE, EA, BC, CF, FB are gene-
rators of a quadric surface, viz.
GEE REC Oe tk OSE Ie! ooo sonugnnouodsoorusnoceadsooocoReT (1).
It follows that the planes which pass through O the conical point and these six
lines touch the enveloping cone from O to this quadric
(2+ B+ y— oe — @— 0) (e+ P+e—-2-e@—f*)= (aa + b8 + cy—d8 ee — ft),
or (GPC Nr (ils (0.2) nepmnonitoon sgceccbae see lernebochon (2),
where on the left hand side are the nine squares in which one of the three letters
a, 8, % is associated with one of the three 6, «, €; and on the right hand side are the
remaining six squares.
Hence in the hexagram, the six sides of the triangles ADH, BCF touch the conic (2).
Again, the projection of any plane section of (1) is a conic which has double
contact with (2). Hence it is inferred from (7) that the projection of the conic
a=6, B+ypH=e+l?
296 Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM.
has double contact with the two conics which touch the sides of the triangles ADE, BOF,
and of ACD, BEF.
This conic clearly passes through the eight points
SO Rowe 7st 6
8
and a=6, B=+G y=te
Hence it is inferred
The four P points a=d6=—B=—e; y+f=0; or CD, AL,
a=6=—-B=-—£; y+e=0; or BC, EF,
a=d=—y=-e; B+6=0; or BE, CF,
a=d=—y=-—¢; B+e=0; or AC, DE,
the two Y points a=—o=0; Ste—O0; Wec—0:
=o —) ee G—0r rye 0)
and the two & points f— oo ery —
Bao} (ISS OSE
lie on a conic which has double contact with the conic which touches the sides of the
triangles ADE, BCF and with that which touches the sides of ACD, BEP.
The remainder of Veronese’s memoir, of which I wish now to give the analytical
equivalent, treats of certain systems of lines and points (called by him 2,2;... limes and
Z.Z,;.... points) which correspond in many ways to the h lines and H poimts and may
be grouped into six sets of ten lines and points in a similar manner: as stated above
the Z, points of Veronese have been spoken of as H, points.
XXIV. It was shewn in XV. that the six v lines which pass through the three
vertices of a triangle A intersect by threes in four points H, such as
(a8) = x (y8) = x (ye) = x (Se),
that there are sixty such points in the hexagram which lie by threes on the g lines,
and that further the point above corresponds to the H point
(Sy) = (66) = (Se) = x (8) =x (Se) = x (ve),
and to the h line
(Sa) = (£8) = x (a8).
In the three-dimensional figure, the six v lines which pass through the vertices of
a triangle A and lie in its plane, intersect by threes in four H, points, such as
C605 roe
which corresponds to the H point -€=y=8=e, and to the h line —f=a=8.
‘If three H points lie in an h line, the corresponding H, points lie in a line called
an A, line.’
Mr H. W, RICHMOND, ON PASCAL’S HEXAGRAM. 297
Taking the three H points which lie on —€=a=A, the corresponding H, points are
d6+€=0; a=P=y¥7;
e+ty=0; a=B=8;
yto—0; a——p—e;
and these are seen to lie on the h, line
a=B=ytd+e,
which is equivalent to
€+3a=64+38=0.
Thus corresponding to the h line a+8=a+y=0, and to the H point
a+d=ate=a+C6=0,
are the h, line
a+38=a+3y=0,
and the H, point
at+36=a+3e=a+3f=0.
XXV. Thus, both in the three-dimensional figure and in the Hexagram, the sixty
H, points and h, lines correspond to the H points and hf lines; when three h lines
meet in an #H point the corresponding h, lines meet in an H, point, and the cor-
responding H points and H, points lie on an A line or h, line respectively; while if
three h lines meet in a G point, the corresponding /, lines meet in the same @ point,
and the corresponding H points and H, points le on the corresponding g line. Hence
the @ points and g lines and therefore also the J points and 7 lines are common to
the two systems (1) of h lines and H points, (2) of h. lines and H, points.
Thus from the figure formed by the five planes
a+38=0; a+3y=0; a+36=0; a+3e=0; a+3f6=0;
it is clear that in the hexagram the ten f, lines and H, points which correspond to
the ten h lines and H points of a projected pentahedron (as in VI.) themselves form
another such figure which has associated with it a conic such that each H, point of the
ten is the pole of the corresponding A, line; and the h, lines and H, points may be
grouped into six such figures.
But the relations between two or more figures of h, limes and H, poimts are not
identical with those existing between the corresponding figures of h lines and H points;
for the latter are derived from the projections of the intersections of fifteen planes,
while the h, lines and H, points cannot be derived from fewer than thirty planes; thus,
in the three-dimensional figure each pentahedron of / lines and H points is contained
by six out of fifteen planes, and each plane occurs in two pentahedra; but in the case
of h, lines and H, points, each pentahedron is contained by five out of thirty planes,
and no plane occurs more than once.
298 Mr H. W. RICHMOND, ON PASCAL'S HEXAGRAM.
XXXVI. The intersections of these thirty planes however furnish a second similar
system of limes and points, which for the present may be distinguished by the
suffix 3.
Corresponding to two H points which lie on a v lne a+8=0; y=6; viz the H
points
a+ 8=at+y=a+6=0; a+B=8+y=84+5=0:
we have the two A, lines
a+3e=2+36=0; B+3e=84+35=0;
which meet in a point called by Veronese a V point (or later a V., point)
LBS 6 Gwar o: — ae — Tl
lying on the y line a=f, e=€.
From these V points may be derived the second system of h; lines and H, points.
For through each V point pass two h,; lines of the second system, viz.
3e+a=3e+ 8=0,
3§+a=36+8=0.
Thus in the projected figure, each line of either system contains three V points;
thus the first system determines the V points, and these determine geometrically a
second similar system of lines and points which has all the properties of the first
system.
XXVII. These results may at once be generalised. Consider the system of thirty
planes such as Aa+p8=0, where 2 and pw are definite constants.
Two of these planes pass through each of the fifteen 7 lines, and are harmonically
conjugate with respect to the tritangent plane and the Pliicker plane which intersect in
that line.
Let the line in which two of these planes such as
Aat+uB=0; rAa+py=0
intersect, be defined as an hyn line, and let a point such as
Aa+ wb =a+ we =Aa+ wO=0
be defined as an H, , point; and let their projections in the hexagram bear the same
“
names.
Thus the /, lines and H, points are equivalent to h,,; lines and H,,; points, and the
h, lines and H, points to h,, lines and H, , points.
Then it is clear that we have a system of sixty h, , lines and H, | points which
» ay
has all the properties mentioned in XXV. as possessed by the /, lines and H, points
and further that a second similar system of h, , lines and H, , points may be deduced
Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM. 299
in the same way that the h, lines and H; points were deduced from the h, lines and H,
points.
The h, , limes and H, , points with the h, , lines and H, , points together form
a system which may be called the (Aw) system: each system (Aw) contains ninety
V,, points, such as
a+ me=Aa+ wO=AB 4+ we=AB4 pl6=0
two of which lie on each y line and are harmonically conjugate with respect to the P
poimt and the Z point of that line.
In the hexagram, the projections of these lmes and points have analogous properties,
and the V, _ points serve to connect the h, , lines with the h, , lines.
2 By
XXVIII. Veronese connects the systems for different values of X : ~ by a method
which leads to a curious analytical equivalent.
The Ve . points were obtained as the intersection of two h, 4 Imes which correspond
to two H points of a v line. If instead, the corresponding H, _ points are taken, the
line which joins them may be called a v, | line.
Let the v line be a+8=0; y=6; then the two H, | points are
Aa + pB=rAa + py =ra + pd= 0,
AB + wa =AB+ py =AB+t pd = 0,
and the v, , line which joims them is
Thus there are ninety v, _ lines, which intersect by pairs in the forty-five Y points,
and form harmonic pencils with the v lines and J lines.
But since a+ @B+y+6+e+C=0, the line may be written
i Bey Se ee aS
PS por we Oe sae
and hence belongs equally to a system (XV, w’) for which
that is
Thus from the system of h, | lines and H, , points ninety v, , lines are determined,
from which in turn a second system of H, ,, poimts is determined, viz. as points of
concurrence of three v, _ lines, which belong equally to the new system.
Thus in the hexagram, from the system of points and lines distinguished by sutfix
(4, #1) may be deduced by means of V points a second system given by the suffix
(4, ry).
Wo, AY, Ieee IMT 39
300 Mr H. W. RICHMOND, ON PASCAL'S HEXAGRAM.
Hence by means of v lines a new system (Aj, fy) is derived, where
He
Sar
Ay fh
and hence again a system (2, 4).
From this is obtained a system (As, us) where
and so on ad infinitum.
Again it is possible to reverse the process, hence as a rule from any system a
series of other systems extending to an infinite number may be deduced in two ways,
the whole forming one complete series of systems of sixty points and sixty lines.
The solution of the equation
Pn os Xa 4
And Kn
Hn 1A (V3 41)" 4+ BWW3-1)™"
Xn 2 A(VB+ 1)" + B(V3— 1)"
Hence whatever system be chosen to start from, the limiting value of x is always
either 2+./3 or 2—/3.
From the system of h lnes and H points for which ~,=2,, Veronese deduces a
series of systems, given by values of the above fraction when A= B.
is
XXIX. There is one special system of the h,, lines and H,, points which has
not been noticed, and appears to deserve attention. Corresponding to four A lines which
pass through a P point, as for example
yte=y7yt+F=0; et+y=ce+8=0;
b+e=64+6=0; F+y=64+6=0;
are the four H, , points
Ay + pa =Ay + UB = Dy + pd =0;
rb + pa =AS + wB=AS + py =0;
Ne + pa=Ae+pB=re+ pl=0;
AC + pa =AE+ pB=2AC 4+ pe = 0.
The first two are joined by the line
a B_yt+8_ e+
oN ea ee
and the last two are joined by
a B_e+€ vyté
el ee
If now A—p=p"—3A, that is 2\=yp, these two lines are identical, and therefore,
corresponding to four 4 lines which meet in a P point, there are four H,,. points which
lie on a line conveniently called a p line. Thus to the P point
‘ a+B=0, y=s=-e=-6
corresponds the p line
a=B8=—(y¥+6)=—-(e+6).
Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM. 301
To three P points which are vertices of a triangle A,g correspond three p_ lines
which lie in the plane a=, and form a triangle whose vertices are R points, while
to any P point on one of the fifteen lines AB,
a+B=7+6=e+6=0
there corresponds a p line which passes through the & point
G=/5, We, 6C=E
The p lines may also be obtained as the lines of intersection of the ten planes
such as
at+B+y=6+e4+6=0,
which may be called ® planes. Each p line, the intersection of two ® planes, lies also
in a Pliicker plane; also each h, line
a+2B=a+ 2y=0
is the line of intersection of one of the ten ® planes with a Pliicker plane
a+B+y=6+e+6=0; B=y.
The H,, points are points of intersection of three of the ® planes; for if
a+26=a+2e=a+ 26=0,
then Btyto=B+y+e=B+7+F=0.
The ® planes intersect by twos in the forty-five p lines, by threes in the H,,
points, and by fours in the fifteen R points: each plane contains two conjugate G points
and is met by the six coordinate planes in six o lines.
These ® planes would furnish by their intersections with one another and with the
tritangent, Pliicker, and coordinate planes many new lines and points of interest in the
theory of the Hexagram; I have however no wish to increase further the already un-
wieldy number of lines and points of the plane figure. In three dimensions, it has been
seen a comparatively small number of planes is sufficient to determine the complete figure,
and the confused intricacy of the plane Hexagram is avoided. With this brief mention then
of the ® planes, which appear to stand next in importance to the tritangent, Pliicker,
and coordinate planes, I shall leave the subject.
It is clear that in the figure of the Hexagram, the lines and points obtained may
be grouped into figures of projected pentahedra and hexahedra in a very large number
of ways; for if from the planes in the three-dimensional figure any five are selected
of which no three intersect in a common line and no four pass through a common
point, their intersections will give a figure of a projected pentahedron, and any six planes
selected under the same conditions will give a projected hexahedron; should the con-
ditions not be satisfied, the figure of the projection will be modified. It may be worth
while to examine one or two of these figures.
(a) Taking the five planes (a8)=0, (ay) =0, (ad)=0, (de) =0, (8) =0 the pro-
jections of the edges of the pentahedron will be found to consist of six h lines, and the
four lines AC, CD, BE, EF; and the ten vertices are the two H points (a8) =(ay)=(a8);
(ad)=(Se)=(8£) and eight of the nine P points in which the sides of the triangle ACD
meet those of BHF, the point of intersection of AD and BF being omitted.
302 Mr H. W. RICHMOND, ON PASCAL’S HEXAGRAM.
The figure is given below.
Since (a+ d)=(a+8)+(at+y)+ (d+ 6€)+(64 6)
or (a + d) (a8) = (a+b) (a8) + (a +e) (ay) +(d + e) (Se) + (d+ f) (88)
it follows that each of the ten points of the figure is the pole of the opposite line
with respect to the conic
(a +d) (a8) = (a +b) (@B)? + (a + ¢) (ayy? + (d+ €) (Se? + (A +f) (85).
(b) Taking the six planes
(48) = 0, (78) =0, (ary) = 0, (88) = 0, (ad) = 0, (By) = 9,
we have a figure of a projected hexahedron, whose fifteen sides are made up of the
three sides of the triangle A,, and twelve hf lines, and whose twenty vertices are made
up of four H points, four G points, and twelve P points which he on the sides of the
tangle A.- but are not vertices of that triangle. The four H points are conjugate to
the four G points, and the P points are conjugate in pairs with respect to any conic
which passes through the four points given by
(a+ b)(aBy + (ce + d) (v8 = (a +.€) (ay)? + (b +d) (Bb = (a + d) (ad)? + (b+) (By?
or (a+b) (e+d) [(aP) —(yé8)}? =(a +c) (b + d) [(ay) — (88) = (a +d) (b +.¢) [(ad) — (By).
Veronese also obtains many properties of harmonicism and involution which I pass
over, as in no case does the proof present any difficulty.
Vol. XV
lL. Soc. Trans
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VIL. The Self-Induction of Two Parallel Conductors. By H. M. Macponatp,
Clare College.
In § 685 of his Electricity and Magnetism, Vol. 1, Maxwell gives the relation
L b
— 19
7 = (H+) + 2p log — ,
as that existing between the self-induction LZ of two parallel infinite cylindrical con-
ductors, radii a@ and a’, the distance between their axes being b, mw, mw’ their magnetic
permeabilities and p, the magnetic permeability of the swrounding medium. It was
remarked by Lord Rayleigh in the Phil. Mag., May, 1886, that this expression is only
true when p=pw’ =p. The following is a solution of the cases when the p’s are not
all equal.
1. F, G, H the components of the vector potential at any point a, y, z satisfy the
equations
a nel fo + 4oruu = 0
0) oy? | 02 ey
CG CG C&G
=A ple SOUS ceeaectttadswatesneeatbea send taccedears 1
0a? aE oy? cs 02? eA @)
CH eH 0 H hedeepx 20
3a? * Gy? * oe
throughout space, wu, v, w being the components of the total current at the pomts a, y, z
and « the magnetic permeability of the medium at that pomt. At the bounding surface
of two media for which p is the same, F, G, H satisfy the equations
aF ak |
apt oe a
0G = oc’
ger Eee ee) Sone ee 2
apt Oye ee (2),
OH | OH" _
ov on
v, v being the directions of the normals drawn from the bounding surface into the two
media.
Won, xOVG 1PAaw IDI, 40
304 Mr H. M. MACDONALD, ON THE SELF-INDUCTION
Equations (1) and (2) were given by Maxwell, Phil. Trans. 1865. The equations
which hold at the bounding surface of two media, magnetic permeabilities w, uw’, may be
shown to be
1 OF {1 OF _,
m Ov im Ov’ a ’
eoG yl neGs
. ar ae == TS Ser csemene a anonencacsod sad adoobascasnsacod (8),
1a 1 OH’_,
fw Ov pm Ov
exactly as the analogous equations at the bounding surface of two media, for which the
specific inductive capacities are K and k', are proved in electrostatics, by taking
Kyu=1, K’p'=1 and remembering that
F= | ii | = dx’ dy’ dz’, ete.
2. Applying these to the case of two infinite parallel conductors with circular
sections, taking as plane of ay a section perpendicular to their lengths, as axis of y
the straight line joining the two limiting points of the circles in which they cut the plane,
and as axis of x the straight line bisecting this at right angles, we find the equations
eH CH
Oa + oe + 4rrpw = 0 wae v eee eecccscsectecuscncevensevecsecases (1),
Up el il llal
to determine H, while F and G@ are constant or zero. ‘Transform these equations by
the relation
x+y =c tan} (&+ ue),
2c being the distance between the limiting points of the circles. Let »=a be the
bounding surface of one conductor, p its magnetic permeability, 7 =— 8 the bounding
surface of the other, »’ its magnetic permeability, and «, that of the surrounding medium.
Equations (1) and (2) become
CH fH 4p” :
A . =... 5 ‘ => =
oF % on? * (cosh 7 + cos Ey (5) OMA eat Te
fH, . &H, .
=... i = =—
ae + On )...(4) from =a to 7 B,
fH’ CH’ Aarp’w'c? ows
oF t On? + (Gosh =e cos Ey =0...(5) from »=—B to n=—-&,
H =H, and 2 wes md oH, ...(6) when 7=a,
On fy ON
H, =H’ and , = 2 (a ...(7) when 7» =— 8,
by On pw’ On
OF TWO PARALLEL CONDUCTORS. 305
where H is the vector-potential inside the first conductor, H, in the surrounding medium,
and H’ in the second conductor, w and w’ the densities of the currents in the two
conductors.
8. To solve these equations assume
y sinh (a — -1) me x ;
H=A,+8B eosin teGs +3 Cait (CArmiCOS 11-1 d A SIMU ie) eeeeeereee rece elclerse(e (8),
ao
H, = Aj + Bin + (A,’ cosh ny cos nE€ + B,! sinh ny sin nF
1
+ C,/ cosh ny sin n€ + D,' sinh nn cos n€) ...(9)
WaA’4 py Soh (Si)
EGaEIAEINCOS et =, ae Ni CA COBH UE I-12 By USI 77) sper eaecteeeeseee (10).
Equation (8) satisfies (8) and is finite when 7=%, (9) satisfies (4), and (10) satisfies
(5), and is finite when »=—a. Further by differentiating (8) and (10) and_substi-
tuting in (3) and (5), we obtain
9 2
pes 2rrmwe” |
sinh a a
re, ad Qmplwe SSTIGRECeCOnEADOL Hana cebacie Roc REO Bae RECpOeE (11)
sinh 8
To determine the remaining constants (6) and (7) give
nD
Aly ue Ye" (A, cos nE + B, sin n€)
a
=A, + Bia+ (A, cosh na cos né + B,’ sinh na sin n€ + C,/ cosh na sin n&é
1
se JD). fill 706 "EOS EE) ocogoonscopoposuc .(12);
B a J
———. — > ne" (A, cos nE+ B, sin n
~ cosha+cos— 7 (4n f - 5)
a By +22 (A,/ sinh na cos né + B,’ cosh nasin n& + C,/ sinh na sin n&
Fo 1
+ D,, cosh na cos np) ...(13),
A,’ + > e"8 (A, cos né + B,” sin né)
1
=A,— B/B+(A,’ cosh nP cos né — B,’ sinh nf sin n€ + C,’ cosh nf sin n&
1
— D,’ sinh nB cos n€) «2.02.4. (14)
Be
—__—— S ne-n8 A,’ cos nE + sin 7
coahB teow Bt M07 (An” cos né + By” sin n€)
EIB +3 + =n (—A,’ sinh nf cos nE + B,’ cosh nB sin nE —C;/ sinh nf sin n&
Sais
+ D,/ cosh nB cos nf)} sor (1G).
40—2
306 Mr H. M. MACDONALD, ON THE SELF-INDUCTION
whence Ags bi Sen By cast eve bs Litton beeen. cero (16),
,__ fm 3B .
B= ene ue (17),
Ag = Ay = BeBe Oe et OR AER (18),
’ He Br’
Bie i seh Sie thee (19).
Now
sinh a
: = TO psaens By eae)
cosh a + cos & 1 — 2e cos + 2e-*« cos 2F — etc.,
therefore from (12) and (13) we obtain
A,e™ = A,’ cosh na + B,’ sinh nz,
B,e-™ = B,/ sinh na + C,' cosh na,
2B )
—_ n(A,/ sinh na + D,,’ cosh na
sinh a (An " ), [>
Ko Aen =
i" (- nA,»e Tian (-)"e na.
. (— ne~* B,,) = n(B,/ cosh na + C;,,’ sinh na)
hence
2Be™ Ho
A, =A, 6, | cosh na +o * sinh na ) +(—)" - “ sinh ia,
nsinha’
B,’ = Bne-™ =e cosh na — sinh net) ,
daaeeates (20),
GC =B,e—= ia sinh na + cosh na)
2Be-™
D, =A,e—™ (—sinh na — x cosh na )— (—)” — .,— . cosh na
nsinha” “
(2
also from (14) and (15)
A,’e-™ = A,’ coshnB — D,’ sinh np,
B,’e“* = — B,' sinh n8 + C;' cosh np,
sinh 8
Eo nB,/e? = n (B,/ cosh n8 — C,' sinh nf)
“
0 ” 2, | as a ‘J ,
Z (nA, eB + (—)” Bee Jan (— A,’ sinh n+ D,’ cosh nf), /,
whence by (20)
B,, = Bye = 13p = Cy =0,
: 2B"e"8 ;
Apri: 4 5—np Mo ant h\T — zy
ay ane (cosh np + 4 sinh np) +(—) The sinh mB, (21).
: 2B" ep
= W 5—np Ko 4 ; = n “+ 0 h
Di —P Algae (% cosh n8 + sinh nf) +(-) Wank” Fike np
OF TWO PARALLEL CONDUCTORS. 307
Solving (20) and (21) for A,” and A,, we find
2B —na by ; 2B” —np
—(-)" g \e sinh n (a+ 8) + cosh nie) =(—)" Ee 2)
een nsinha’ pw |p ) n sinh Bo p'
€ - Ho! ») sinh n(a+8)+ = + He) cosh n(a+)
iu bow, (22)
2BeEe" Deane Ms {Ms Nag?
a f= \n = el (—) 0. 6G | sh
Pena 2! nsinha’ uw Nn sinh 8° studs Cea eee Ce: a) j
n ae
(1 + Ho ) sinh n (a+ B)+ nc Es ) cosh n (a+ PB)
ao Bo KS
The current in the »=—f conductor being the return current to that in the n=a
conductor, we have
| udndy + | |w'de'dy’ =0,
: f ap lEd palf 2 AS Elm’ F
that is w | | mee —» + Ww | sae = 0;
0 (cosh n + cos €) eae » (cosh 77’ + cos —'P
now fs _ 4 eee d [" de se
Jo (cosh n + cos €P sinh 7 dy Jy cosh n + cos & ’
therefore Di RDN Se =0,|
sinh?a@ sinh? 8 | (23)
B Be |
aa ee + oe = 0|
wsinha pw’ snhB
the latter of the two equations bemg obtained from (11).
Hence from (20), (21), (22), (23), we have
2B ena—8) > sinh n (a +8) —coshn(a+)
Bes a, aha a
ie (1 Sales ) sinh n(a+8)+ ‘. ip Eo) cosh n(a+)
My Bo pe)
2Bu P\ sinh n(a+ 8) +cosh n(at B) — er'8-2) |
A,’ =(—-)" —— . aa ae aa ea
n s h >
Beene oe ae “) sinh n(a+8)+ (@ ~ i) ) cosh n (a+ 8)
bp ro matty
. eh ore ene (24).
oR me (cosh nat+/ sinh na ) —e"4 (cosh n+ Es sinh np )
A, =(—)" = oe ; $$ ___ i |
B (i+) sinh n(a+p) + (+) cosh n(a+8)
pe’) He I
/ /
OB — ¢ 8 (sinh na + Po cosh nat) —e@n (sinh nB + eu) cosh nf)
D — (—)" creo \ BK \ K
y ny sinh a”
(1+) sinh n(a+)+ (™ +4) cosh n(a + 8) |
bye pep
308 Mr H. M. MACDONALD, ON THE SELF-INDUCTION
Therefore from (11), (16), (17), (18), (19), (21) and (24), we find the expressions for
H, H, and H’ to be
oe 2rpywe Qerpwe? sinh (a— 7)
—** fae sinh a © cosh 7 + cos &
n(a-8) — 4 sinh n — cosl
Amrpwue? & (—yte“™ e Fe inh n (a+ 8) — cosh n (a+ 8)
sinh? a a Wir in wose eg ten ih CGR eT Lk n€,
- (1 +f :) sinh n (a+ 8) + Gr + | cosh n(a+ 8).
lid Bw pe
> ae 4 oo es eae Coane leah n(a—)+ Po sinh n (a— all
a. LT MWC TMyWC” S =) if PP j
= = Ui] =e ane a ae
sinh? sinh? me ee 0, Mo
ate oe Pes (1 46 ) sinh xn (a+) + S at =) coshn(a+ 8)
bee bp
fo _-
¢ a cosh n(B+n) +> sinhn (8+)
ems Gee eee ae ee ee
sinh?a 7 n of =a) 3 Tes eee
1+ —)sinh n(a+t+ + (H+) cosh n a+)
( wp alia 5, near
27p’we* sinh B sinh (8+ »)
; , Qrpowe-
HAL 3
sinh? @ cosh n + cos &
sinh? @
Ko sinh « s} _ pn (B—a)
__ druue? & (=) mene (a+8)+ cosh n(a+P)—e
sinh?a 7 n Pf
—
(1 se “) sinhn (a+ P8)+ (= 35 ) cosh n (a +P)
be BM
A, can be determined from the condition that H, vanishes at an infinite distance
from the conductors.
4. Let JZ be the whole current in either conductor, then J=-we?/sinh?a, and
further let Z be the coefficient of self-induction of the current, then
LP= | | Hvdedy
ae Hdédn es _ Hd&dy
= we I[ (cosh 7 + cos £)? we If (cosh » + cos &)
die dEdy , , Lar pwe? Qorwe? sinh (a — ») )
oat J. |, (coshn+cos€? (7 " "sinh? a *~ sinha cosh n +cos€ 4 te
—o T dédn
ha ‘A,’ —et pale
ewe iN (cosh + cos £)? ! etc. ;
Now by (23)
we [| sees +u'e? [ | d&dn
}} (cosh 7 + cos Ey !J} (cosh n + cos &) 7
Again
1 [" [" 2rpwea d&dn nits [= & Sar werB dédy
~ Ja Jo sinh?a (coshn+cosé) -p I, sinh?a (cosh y + cos &)*
= 2y,I? (a + B),
OF TWO PARALLEL CONDUCTORS. 309
also
[ 2 i sinh (a —
|. Jo (cosh n + cos
dé i ie sinh (a — n) d i 1 ad T )} ee T
De at ae sinhn = * dy (sinh y dy a= fo" = asinh? a’
"™ p71 eOS nEdEdn _ mo ew (— —)r ean (—)? eo 2na
and i ih (cosh 7 + cos &)? Tal sinh Pe : ( sinh - 7 =) dy = 2 sinh? a *
Therefore
LI? = I? \2u, (4+ 8) +4(u+p)}
x ("2 + eB) cosh n (a+ 8) + (2 pas Se ei) em) sinh n (a+ 2)
+ 4ynl?S =. a ee
y 2
(ae + od sinh n (a+ 8) + . +H) cosh x (a+ 8)
be
2 1 er (a+B)
— Sy,J? > 2
Bo : Ko
(1 4. = sinh n (2 + 8)+ i + Hy) cosh n ( (a+B)
Tn
that is
L=3(w+p’) + 2u, (a+ B)
£ el (a—B) (a a a) + e” (B—a) (1 ae Ms) +e (a+3B) (2 zs He) +e” (8a+B) @ fa) a —ev (a+B)
[ art 0 0 \ Fl
1 n i(1 af = (2 te Hs) ef (a+B) ae (1 a = (1 = e | e7 a+B) }
Vo" be a KM j
When p=p’=p,, we have
x ei {a—B) + et (B—a) __ Qe (a+B)
DL = py + 2m, (a+ 8) 4+ Spy > > ae
Mecitaees na — 2e7 2 (a+B)
= fy + 2py (a + B) + 4h, > > al
sinh? (a + 5
— 2u, log — , ;
nL °8 sinh a sinh 8?
now if b is the distance between the axes of the conductors, @ and a’ their radii,
then
a=ccosecha, a’=ccosechB, b=csinh(a+ )/sinh asinh 8;
theretore i — ai Pais SE
i ae aa’
and the force between the conductors tending to increase their distance apart is 24,l2/b
per unit length. These results agree with those given in § 685 of Maxwell’s Electricity
and Magnetism.
5. When p’=p,
it e (a—B) @ fi ts) ae e7 {a+38) € oo Hn) ae Den (27h) 4e—” (a+p)
L=3(u +p) +2 (a+) + 4u, = E =
: 2Qn (a + ne (ats)
be
310 Mr H. M. MACDONALD, ON THE SELF-INDUCTION
ol e72n8 — Na bs p—2n (a+B)
= 4 (4+ pm) + 2pm (at 8) + 4p, 5 +
1
2n
—p % ene 4. ert (2a+48) = Deen (a+B)
4y& eS —
B+ 1
sinh?(¢+8) 9) b= Mo atts sinh* (a+ 8) _
sinhasinh8 7 +, © sinha sinh (a+ 28)
2n
d(u + po) + 2, log
0
B= Fo
lo a
M+ py Sa
be
9
a tea
= 3(M + fo) + 2p log =
This gives L when one conductor is iron, the other being any substance whose
magnetic permeability is the same as that of the surrounding medium.
The repulsive force between the conductors is
5, ff eh a ; :
2 fo (= —————_ -__.——,_ | J? per unit length:
b wtp b(F=-a*)
These results shew that Maxwell’s formula makes Z too small in this case, the error
being of amount
= U2
eT ele Gy 2?
B+ py ° ba
and makes the force between the conductors too large by an amount
Dy. ane ae 2
Ss TSI (0) 2
Taking the case of conductors of equal section, the following table shews how the
variable part of the coefficient of induction varies with their distance apart.
2 es 2 ; my, L—50°5
b log ea ’ Bl eiog & _,| Imerease | Z-50°5 | som above
aa Mt+poy —b?-a?"| percent. | Maxwell. iPS
| |
2u 138629 282007 20°3 2:77258 333659
3a 219722 ‘117760 53 | 439444 462996
40 2°77258 ‘063260 22 | 5:54516 567168
5a 3°21887 039829 1:2 643774 | 651739
6a 3°58351 027583 nf 716702 | 722218
Tu 389164 ‘020211 9) 778328 782370
8a 415888 015936 3 831776 $'34963
9a 439425 012131 2 878850 | 881276
10a 460517 009851 2 921034 9°23005
|
The first column gives the distances between the axes of the conductors, the second
the values of half the variable term in Maxwell's formula, the third half the term which
has to be added to it, the fourth the increase per cent. of the variable part due to the
term neglected by Maxwell, the fifth and sixth the values of the variable part of the
induction in both cases; pw, being taken to be unity and ~=100. The table shews
OF TWO PARALLEL CONDUCTORS. 311
that the term neglected is considerable when the conductors are near one another, and
decreases rapidly as they move apart at first and afterwards more slowly.
Again taking the conductors touching one another, the following table gives the
maximum values of the correction as the radius of the iron conductor increases.
5 | u— 0 be Increase L-50°5 EeeveD
a 4 HE aa | aa log b= a?’ | nee cent. | Maal: shee
= [bsee |
a 2a’ 138629 | 282007 20°3 2°77258 333659
2a’ 3a =| «21750407 =| 576147 | 383 | 3:00814 416043
3a 4a’ 167397 | *810307 48:0 3°34794 496855
4a 5a’ 1°83257 1001419 546 366514 566797
ba | 6a’ | -1:97407 1162144 588 394814 6°27242
6a 7a | 210005 1300593 61:9 420010 6°80128
Ta’ 8a’ 2°21297 1°422097 64:2 442594 | 727013
8a’ 9a | 231447 1530317 66:1 462894 768957
9a’ 10a’ 2°40794 1:627843 67°6 481598 807166
10a’ lla’ 2°49320 1716587 688 498640 | 841951
| | | |
The first column expresses the radius of the iron conductor in terms of that of the
other conductor; the remaining columns are as in the preceding table.
The expression for the force between the conductors
py (1 bam @&
b bth P-&
can be made to change sign by choosing the radii of the conductors so that 0? is
somewhat less than 2a*, thus making the force attractive instead of repulsive.
It may be noticed that the part of the above formulas depending on the size of
the conductors and their distance apart is but slightly altered whether we suppose jp to
be 100 or 1000.
6. When p=w.
DL = w+ 2n, (a+ £)
x i] se Hs (e” (a—s) + @ G52) + (1 a 2) (e™ (a+38) 4+ e-nsa+B j= 4e-” (a+8)
+ 4y, = z :
u nN {( + ea en (a+B) _ { — Ho \” en eet
[2 ( B/ )
/
be NED
H+ fo
?
putting
T= w+ 2p, (a+ 8)
a—B) + e” (B—a) sh) (Ex* (a+3B) +e (a+) — 7) (X ae 1 je (a+8)
n (e” (a+B)) <5 Ne {@+B))
ao nr |
+ yy (X+1) 3 ©
Vou. XV. Part III. 41
312 Mr H. M. MACDONALD, ON TWO PARALLEL CONDUCTORS.
2 be sinh? (a + ee dog sinh‘ (a+ 8) _
=e sinh asinh 8 + 2p © sinh a sinh 8 sinh (@ + 28) sinh (2a + 8)
4.. x21... Sinh (a + 8) sinh 2 (a + 8)
+ ph" log inh (a+ 28) sinh (2a + 8)
: eee : sinh* 2 (a + 8)
fauaalee sinh (a + 28) sinh (2a + 8) sinh (22+ 38) sinh (3a +28)
eegnt lope sinh 2 (a+ 8) sinh 3 (a+ £) Whi
8 sinh (2a + 38) sinh (3a + 28)
From the relations
_ sinh (a+ 8)
= © Sinha sinh B ’
a sinh a=a' sinh B = ¢,
we have
bs
L = p+ 2p, log ~ “+ Zpor log & @-@) =a)
b? (b? — aw — aw”)
+ 4d? log cores
bt (b> — a — a”)
+ 2d} log — @) (8 — a”) (b—@p — ab} (B= a2 = ao} *
If we take p and q, so that
1
+-— = 2 cosh (a— ),
pt p (
i
q+ Pmietaesy GF)
then
1
- (p+ =) (1 + Ag") — 2(X +1) q”
L = wy, log q+2u, +1) 2 —L+—__________
1 ES 2nnr
nls nq")
ree a+a? @ =a7) Heese
dear P> Sad’ 2beaa’ baa”
; a ee ata? be
1~9qa' 2aa’ aa’
. JG —a—a" 22 — 4a'a?
=F 2b
The repulsive force between the conductors is
ye { Aebg as +1) (a =a) pt SL =p) (+ 9") 4"
(1 — q*) ad’ bad’ (1 — p*) 7 (1 — r2q?") prt
4p, At 1) bq? a 4(A+ 1) 7 ep + p”) ‘hare! ae BAG" + Vg" — Aq ae)
aa’ (l—q’) 1 p” (1 —24q")?
Hh
IX. Changes in the dimensions of Elastic Solids due to given systems
of forces. By C. Cures, M.A., Fellow of King’s College.
[Read March 7, 1892.]
§ 1. Ler e, f, g, a, b, ¢ denote the strams, and wz, w, =, 7%, =, % the corresponding
stresses in an elastic solid referred to a system of orthogonal Cartesian co-ordinates.
Then the most general form of the stress-strain relations is:
He = Cy + Cof + Cg + Cyd + Cisb + Cy6C,
TY = Cn + Cf + Cog + Coy + Cosd + Cog,
Cn @ + Cf + Cy + Crh + Cy5D + Cope,
= Cn + Co f + Cg + Cy@ + Cid + Cy,
et = C€ + Cx f + Cog + Cys + C5sb + Coe,
ay = Cn + Cof + Cos + Cos + Cob + Cope
c.7
ne
ll
where the coefficients c,; and cy, are equal. The notation is that employed by Professor Voigt*.
If the solid be homogeneous, in the sense that at every point it has the same _ pro-
perties along directions fixed in space, then the 21 mdependent coefficients appearing in
(1) have everywhere constant values.
Let Il denote the determinant of 6 rows and columns formed by the 21 coefficients,
and in it let C,, be the minor answering to c¢,;, the order of the suffixes being immaterial.
Let strains with suffix 1, e.g. e,, answer to 7=1 with all the other stresses zero, strains
with suffix 2 to 7% =1 with all the other stresses zero, and so on for each of the
other six stresses in order.
Thus for instance answering to 7 =1, with all the other stresses zero, we have
@=Oy/Tl, fr=Cr/T, go= n/T, (2);
EERGISITT AS = (GNITIE” ER (aNTT OP a aR 5
while answering to 7=1, with all the other stresses zero, we have
A= (Chae f= C/U, Ga— Can) Ul (3)
RTPI gta fT pe GATT J a
* Cf. Wiedemann’s Annalen, Bd. 34, p. 981, 1888.
314 Mr C. CHREE, ON CHANGES IN THE DIMENSIONS OF
We shall also employ the following notation:
a= 1/E,, fo= 1/E,, 9s =1/E;,
a@&=1/n, b, = 1/ns, Coins
(4).
RRRISRSNICOOOOOIOOIOOOOIOUIOIO OOOO OSG igSO OOO UOOO OOOO Origin i
The quantities #,, #,, EZ, are Young’s moduli for longitudinal traction in directions
parallel to the axes of 2, y and z respectively ; while m, m, m; are moduli of rigidity.
The quantities », when the suffix does not contain 4, 5 or 6, are values of Poisson's
ratio. For instance, ny; is the ratio of lateral contraction parallel to « to longitudinal
expansion parallel to z for longitudinal traction parallel to z. The order of the suffixes
is not in general immaterial in 7».
§ 2. Let KX =F (CP ASY + GF + AYZ + DZ + CHY) coccrrecreesensecneeeeesens (5),
and let the suffixes 1,...6 attached to the coefficients have the same significations as
above. Thus for instance the coefficients in
2ys = eg? + fay? + gs2* + ayy + dyza + cyey
are the strains answering to =1, with all the other stresses zero. The quadric surface
M = CONSUANE -. reneeeeacceensarncenensne syocooabncanceBosedne (6)
is what is termed the elongation quadric. In general the elongation quadric varies in
form from point to point of the solid, but when the strains have everywhere constant
values a single form of elongation quadric shows the strain at every point. This is the
case in the present applications, and we shall suppose the quadric to have its centre
at the origin of co-ordinates and may regard its dimensions to alter so as to enable any
point we choose to lie on its surface. When the strain is pure, as in the present appli-
cations, and is also small, as is required for a legitimate application of the elastic solid
equations, the displacements a, 8, y at any point may be derived as follows. Take the
elongation quadric (6), where x has the form (5), supposing its centre at the origin of
co-ordinates and its magnitude such that it passes through the point in question, then
dx
a=—*
dx’
x _&
ere Re RL
This may be at once verified, as it obviously gives
da da dp da dp
ee Ta eae iy de ete.
The physical meaning is that the direction of the resultant displacement is along the
normal to the elongation quadric.
ELASTIC SOLIDS DUE TO GIVEN SYSTEMS OF FORCES. 315
Answering to ==1, and all the other stresses zero, we have
d 1
a; = aa == E, (nat + $135Y + $0352),
d 1
Bs = ae = (KE tog® + Yao +E MegZ)) Pecerecreccesceercrecanecase (7).
d 1
= Ges BH Pe —dmay +2)
Similarly when 7 =1 and all the other stresses are zero,
dy, ies
a= — =— a (nat + 3M6Y + $7452),
dx, 1 L 8
i n, 6— BI — May + $2); no raceoandosoDsoaoDEsanN6or (8).
Ue tae a FNst + FY — 12)
The values of 4's, n’s and n’s may all be expressed as above in terms of the 21 elastic
constants occurring in (1).
§ 3. There is another case we require to consider, viz. when there is everywhere a
uniform normal tension equal to 1. In this case
w=yw=2z=1, xy=x=y=0.
Let the suffix 0 distinguish the corresponding strains and the corresponding form of y.
Then by (1)
& =(Cu+C.2+C)/Tl, fo=(C2t+ Cot Cz)/T, go= (Cis + Cs + Cy)/ TI,
= CaCO Al aby = (Oy BCs OAL, sepa (Og Oy 40) ai OP
The corresponding uniform dilatation A, is given by
Dia Fah Geil ath: eke tense Nees ME Lae RE (10),
where =| (Ca Caer Cac 205 20 REE LCL) serene eee cc oreo cetene (11).
From its physical meaning k, the bulk modulus, is necessarily an invariant whatever be the
directions of the co-ordinate axes.
§ 4 Let X, Y, Z denote the component bodily forces at any point per unit of
volume (including the reversed effective forces
aa d* dy
— Pe Pde? Pat
where there is vibratory motion), and let F, G, H be the component surface forces per unit
of surface. Then the bodily and surface equations in the elastic solid are each 3 in number,
of the respective types:
= dix dz dz\ \
Zasis Way sirgpeete) | MMM ein oF sve e008 (12)
njejelaiofotalarevintelole'elcteiereimaiare emveieiniars |
ee ee ee ee )
F=)ii + pay + viz, |
sch fake dare a BE Beara Sa paces oSaenest (LG)
316 Mr C. CHREE, ON CHANGES IN THE DIMENSIONS OF
where A, «, v are the direction cosines of the outwardly-directed normal at points on the
surface, or surfaces if there be more than one. The strain energy W per unit volume at
any point of the solid is a quadratic function of the 6 strains, and is obtained in terms of
the strains by substituting for the stresses from (1) in
W=h (ere +f + 9= + ar + b= + cm).
Suppose that a second system of bodily and surface forces acting on the same solid,
with the accompanying displacements, strains, stresses and energy, are given by dashed
letters, X’..., F’..., a..., e..., @..., W. Then Professor Betti* has established the equality
of the following four expressions for any two systems of force:
[ [Xa + V8 + Zy) dedyde + | [Fer + Gp" + Hy) as ee eee (1),
[|] ee + ap + Bg + Ral + BU + Be) dandy de eee (IL),
I | | (Ret Wf + 2g t+ Ha + Wb + Fc) dacdydz..cc..ccccccesneseseees (IID),
i [fix ‘a+ VB + Z'y) dadydz +{ [era + GB + H'y) dS ....00.00(IV).
The volume integrals are taken throughout the entire volume occupied by the solid,
and the surface integrals over its entire surface, or surfaces if there be more than one.
Professor Betti’s mode of proof is very simple. Multiply the equations (12) by a’, B’, 9
respectively. Then integrating the right-hand sides by parts, using (13) and adding, we
at once establish the identity of (I) and (II). Then remembering that
SW. es ee
sacge ae voey TH = —— 2... YX = assy
de da de da
and that W and W’ are quadratic functions of the strains possessed of the same coefficients,
we deduce the equality of (II) and (III). Then (III) bears to (IV) the same relation that
(II) bears to (I). The equality of (I) and (III), with the reversed effective forces supposed
zero, is the relation that is made use of here.
§ 5. In passing, attention may be called to the relation that exists when we suppose
the two systems of applied forces the same, so that the dashed and undashed letters are
equal. Then (I) gives the work done by the applied bodily and surface forces acting
through the displacements answering to the position of statical equilibrium, while (II).
represents double the work done by the elastic stresses as the strains increase from zero
to their equilibrium values. If then the applied forces suddenly commence to act, the work
they have done up to the instant when the body passes through that position of strain
which answers to final equilibrium—assuming all elements to reach this position simul-
taneously—is double the work done by the stresses. Thus the energy communicated to
the solid is at this instant half potential energy of strain and half kinetic energy of
motion.
* Annali di Matematica Pura ed Applicata, Ser. 1. Tomo v1. pp. 102-3.
ELASTIC SOLIDS DUE TO GIVEN SYSTEMS OF FORCES. 317
§ 6. The use that is to be made here of the equality of (I) and (III) is in deter-
mining the mean values, throughout the volume of an elastic solid, of the equilibrium
strains and dilatation answering to any assigned system of bodily and surface forces.
Suppose, for instance, we wish to find the mean value of the strain g when the forces
X, Y, Z, F, G, H are given, then we have only to put =’ in (III) equal to 1, with all the
other stresses zero, and to substitute in (I) the corresponding displacements from (7).
Thus representing this mean value by g, and denoting by v the volume of the solid,
we have
ug = [[[oaeayae =|[ I(x oe +YV e +Z “) dxdydz
where the volume integral is taken throughout the whole space occupied by the material,
and the surface integral over its entire surface or surfaces. Sometimes it is convenient to
retain the y, but in other cases it is better to insert at once the expressions for the
displacements. Thus we have
Bye =[[][X — nae dnwy — 452) +¥ (— 42 — nay — $mu2) +2 (beh may +2)] dedy de
+ [fore +4 ) +H jas
where the coefficients of F, G and H are respectively the same as those of X, Y and
Z. Similarly for the mean value @ of the shearing strain a, putting 7 =1 and all
the other stresses zero in (III), and substituting the corresponding displacements from
(8) in (1), we find
w= {i f(x ar ‘« +28) dedyde + |f(F eG ee Bee) ag ee (16),
or
nwa = {ffx (= nut —$ny — 452) + Y (—4ngt — ney +42) +Z (— $904 dy — 9y2)] dudydz
+ |] FC y+ G( )+ H( yas
adlacis dase g38 (17)
For the mean value A of the dilatation
A=e+f+g
we put a = yy =a — I
and Ye =x =n’ =0
in (III), and substitute m (I) the corresponding displacements from (9). Also we notice
vA= |[[Adwayaz OU bone POSCGHOR ERODE: LO aOR cREPAr tae (18),
318 Mr C. CHREE, ON CHANGES IN THE DIMENSIONS OF
where v is the increase in the whole volume occupied by the solid. Thus we find
ih wi rr ey dy, , dx, 7 xs) = 3
w= By =| f](x G+ VOC + ZR) dedyde
+(e re , GD) iS ae cio ween (19),
or =8v= Miles (qx + sey tbo2)+ Vdawe+foy + daz) + Z (dbo + kaoy + Goz)] dedydz
+ |] FC + ) + HY )] a8...(20),
where @...¢, are given by (9).
For the case of isotropy the expressions for the mean strains are of course much
simpler. Thus
oe | il | (Ze —n(Xw+Vy)} dadyde+ | [Hem n Bet Gy) Dees (21),
jita = | i [(ve4 Zy) dedyde + i | Ge Hiergyads felt ab doe (22),
3k30 = I | | (Xa+ Vy + Ze) dudyde + | | (fe 2 Gy + Beds (23),
The mean values of the strains in the case of isotropy for given surface forces—i.e.
results such as (21) and (22) with X = Y= Z=0—were given I believe by Professor Betti*
in his original paper. But this I have unfortunately been unable to consult. I may add that
I arrived quite independently at (23) and (20) when unacquainted with Professor Betti’s
results, having been led to their discovery by what seemed a curious coincidence in
the expressions for the changes of volume produced by rotation in certain solids (see
(32) below).
§ 7. One very general result as regards the mean strains—as we may call
é,...d,... A—is obvious from the formulae containing the functions y. Taking, for instance,
the strain g, we see from (14) that g vanishes if
YX oe rag M+ dys _ 0,
s ie re aie eee (24).
PYG ate =
This signifies that if the resultant of es applied forces at every point, both in the
interior and at the surface, lies in the tangent plane at the point to the elongation
quadric, for the stress =1 with all the other stresses zero, which passes through the
point and has its centre at the origin, then the mean strain g vanishes. A similar result
applies for each of the other mean strains. These results obviously follow from the property
of the elongation quadric mentioned above in § 2. Attention may specially be called to
the fact that (23) implies that the change of volume in an isotropic solid vanishes when
the bodily and surface forces have their resultant at every point perpendicular to the
radius from the origin.
* Nuovo Cimento, 1872.
ELASTIC SOLIDS DUE TO GIVEN SYSTEMS OF FORCES. 319
§ 8. In some cases the formulae for the mean strains can be put into neater forms.
For instance, if the applied surface forces be everywhere normal to the surface of the
solid, then denoting the normal force by N and its direction by n, we have
[]#Qe+e +n Be) ds = { [are 5 ad (25).
Again, if the bodily forces be derived from a potential V, we obtain, noticing that
ox = e;, ete.,
| I(x aX +Y Be +2 ss) copie
[[v % 1s — [[[Ve@+riroo CERO: cocaosnesnoosnoasoe (26),
=| ~~ 7 as—[ [[x¥Vdedydz ye a ee (27).
The form (26) might prove convenient when the surface of the solid is an equi-
potential surface for the bodily forces. In applying it to determine the change of volume
the relation (10) should be noticed. The form (27) seems likely to prove convenient when
V is the potential arising from gravitational forces whose origin lies outside S, for the
volume integral would then vanish since V?V = 0.
In the case of the change of volume in isotropy we may replace the volume integral
in (23) when a potential V exists by
[[evas—3 ||[Vaedy ae
where p is the perpendicular from the origin on the tangent plane to the surface of
the solid.
§ 9. Owing to their physical meaning the expressions (I)—(IV) must remain equal
however their forms may be altered by changes in the system of coordinates. We may
for instance suppose the forces, displacements, strains and stresses occurring therein to
refer to any set of orthogonal coordinates,—such for instance as 7, @, @ in polars—and may
thus, at least in some cases of isotropy, determime the mean values of the corresponding
strains throughout the solid. In an aeolotropic material, such as (1) refers to, the
constants in the stress-strain relations in coordinates other than Cartesians would vary
from point to point, owing to the variation of the directions of the coordinate axes.
There may however be some solids in which the values of the elastic constants are the
same at different poimts not for parallel systems of axes as in (1), but for some other
orthogonal system. And it is conceivable that in some such cases the mean values of
strains referred to this orthogonal system may be obtained by means of the equality
of (I) and (III).
Vou. XV. Part III. AQ
320 Mr ©. CHREE, ON CHANGES IN THE DIMENSIONS OF
Determination of the compressibility.
§ 10. In an isotropic solid we may by means of (23) determine the bulk-modulus,
and so the compressibility, by measuring the change of volume produced by any known ~
system of forces in a body of any shape. Suppose, for instance, a block of the material
to rest on a perfectly smooth plane and to be subjected to vertical pressure over its
upper surface, supposed horizontal. Taking the plane ay through the base of the block,
with the origin at any convenient point, and supposing the upper surface at a height
h above this, we find from (23), denoting the total pressure by P,
SRM oP osc cs ees ee (28).
If the block have a uniform horizontal section, and p be the mean pressure per
unit of area of the upper surface, this becomes
SOUT BH aebeissistie cenissennssen aecoaestiuntaeseecees (29).
Thus for a given total pressure, 6v increases with h, but for a given pressure per unit
of surface 6v/v is independent of h.
§ 11. Since no plane is absolutely smooth it would appear desirable im practice to
have the base of the block as small as is consistent with the stress-strain relations
remaining everywhere linear, so as to make the value of
[|e + Gy) dS
taken over the base as small as possible. The general effect of these frictional forces is
easily traced, at least mm a block of regular shape. Under vertical pressure the solid
tends to expand horizontally, and this the frictional forces on the base must oppose. Thus
supposing the origin at the c.G. of the base, the frictional forces are on the whole
directed towards the origin, or Fx+Gy is negative. Thus the surface integral would add
numerically to the right-hand side of (28), and so its omission makes the calculated value
of (—3kév) too small. The value of k deduced from (28) and the observed value of (— dv)
would consequently be too small also, Another source of error would be the want of
absolute rigidity in the supporting plane, in consequence of which the points of appli-
cation of the large surface forces H on the base would not all lie in the plane z=0.
This error would be minimised by taking the height of the block great.
§ 12. In any aeolotropice solid the bulk-modulus may be determined as follows. Cut
a rectangular block out of the material with its edges J,, l,, J; in any orthogonal
directions. Place it on a smooth unyielding plane with an edge, say 1/;, vertical and
apply symmetrically a total pressure P, over the upper face, measuring the corresponding
reduction (— 6v,) in volume. Repeat the experiment with the edges J, and J, successively
vertical, applying total pressures P, and P,, and determine the corresponding reductions
in volume (—6v,) and (—6v,). Now the origin being at the c.G. of the base, the axis of z
vertically upwards, and the pressure being symmetrically applied, it is clear that
| [SH (ob, + yas) aS
ELASTIC SOLIDS DUE TO GIVEN SYSTEMS OF FORCES. 321
vanishes over both faces. Thus we easily deduce
—6,=1,P;9,, —S.=bP.fo, —8,=hPiea,
where @, fi, % are given by (9). Whence by means of (10) we obtain
— {82, /1,P, + 8v,/1,P, + 80; /1;Ps} = Cr+ fot Go = L/h vceecreccnsccneeenees (30).
If in each case we have the same mean pressure p per unit area of face, this becomes
= (Nh SEOs OME VOLM conn, aonnonocscn ons09b00coK005Na8e (31).
1
Rotating Bodies.
§ 13. Suppose a homogeneous elastic solid to rotate with uniform angular velocity o
about a principal axis of inertia through its c.G. and to be exposed to no forces other
than the “centrifugal forces”. This motion is dynamically possible, i.e. no constraint is
required to preserve the direction of the axis of rotation or to prevent the body travelling
off into space. Taking the axis of rotation for axis of # and denoting the density as
previously by p, we have
Wy= AES op, X=0, andl LSS i =O,
Substituting in (23), we find for any isotropic body
3kdv =|fJorp (y? + 2°) dedydz,
or OU = Os Big arsumecpae cus Set aseaie sae ae ae (32),
where J is the moment of inertia about the axis of rotation. The value of k might of
course be deduced by means of this formula, supposing it possible to measure 6».
In the case of an aeolotropic solid, free from surface forces and rotating about a
principal axis through the c.«G., let us take this axis for that of a, and let the axes of
y and z be the two other principal axes at the c.G. Then denoting the angular velocity
by @, and the increase in volume by 62,, we find from (20)
bu, = [flere (fy? + JZ") dadydz
SCE Fat Ogu eS Rest tee iteacs esr: aces pects (38)
where A’, B’ and C’ are the moments of inertia with respect to the planes yz, za and
xy. Similarly let 6v, and 6v, be the increases in volume when the body rotates with
angular velocities w, and w, about the axes of y and 2 respectively, then
= OHAG OG), Coy=OF (AVG io 1377 oocnosocnBsooucnasounoba.d (34),
Thus we obtain
1 1 1\ dv
= Sa) (ee ee
Vk=q+for.+9=4 \(e +B zx) ot
1 1 1\ & 1 1 1)\ 62, ae
+ = Cl’ B) (F + A’ x) | ccc vcevasecescccce (35),
322 Mr C. CHREE, ON CHANGES IN THE DIMENSIONS OF
If the body be a sphere of radius R and the three angular velocities be equal, this
simplifies to
(Sv, 4 8, + u,)/v _ 2w°p R?/5k slcfalaininisloroisielelcatuviniemteinicieiiersieetelsisvels (36).
§ 14. The form of rotating body for which the present method supplies most
information appears to be a might cylinder, including the right prism. Let the axis of
the cylinder be axis of z, the origin being at the middle point, and let the axes of z
and y be the two principal axes of the cross section. Denote the area of the cross
section by o and its principal radii of gyration by «, and «,, so that
oK;? = [[yrardy, ch | [erdedy duiars stele wisiohiowreblee noire ee eee (37),
where the integrals are taken over the cross section.
The increments 6v,, 6v, and 6v; in the volume v, = 2/c, where 27 is the length, when
the cylinder rotates with angular velocities »,, #, and , about the axes of a, y and z
respectively are, by (20) and (37),
8v,/o2 pv = foe? + gol?/3, — Sv./@29v = eoe.2 + Gol?/3B, — 8vs/@2300 = yes? + foie seeceeeeee (38),
from which & can be found as in (35). For the case of isotropy
Sy, = 2P? (« : =) , dy= wee («? 2 5) one oe (ct eke (39).
Thus in isotropy, when o,=0,=@;=, we have
80; + 60. — 0, = 2 (@l)700/9K .....ssuecssesnenes ocnnseacesegeans (40),
a relation wholly independent of the shape of the cross section, and which in the case of
a very thin disk approximates to the form
OU sj OV OUs eee sare eae see siete se oe Sea eee ees (41).
§ 15. In the case of any right cylinder we may find the mean change in the length,
or what in a thin disk is called the thickness.
For (15) gives the value of
ug = |[|gardyas = II fee Ghyt{0 hI (0 2a aecete meee HOB EAeR EO DeAG0S 90000 (42),
taken throughout the volume. But the axis of z being along the axis of the cylinder,
this is simply 2c6l, where
is the mean, taken over the cross section, of the increments in the half length l. Let
now the cylinder rotate with angular velocity w, about the axis of a, taken as before
along a principal axis of the cross section, then substituting in (15)
A=0 Viy=Ziz=o7p,
ELASTIC SOLIDS DUE TO GIVEN SYSTEMS OF FORCES. 323
we find, calling the mean increment in the length of the half axis é1,,
281, = (w?p/B.)| | [(- Ney? + 2) dadydz,
or 81, [C= wet GP —yaien) | Bigs, -Asagencde te d-psereg 140k oasesa nce 0s sea (44),
|
Similarly if 6/7, and 6/, be the mean increments in the half length of the cylinder for
angular velocities », and , about the second principal axis of the cross section and the
axis of the cylinder respectively, we find
BLL= ofp GP — mani) E SALT Por ee eae oA (45).
5s, T=— 0p (Nake + Nsoky"), E;
In any case supposing o, = ©,=@,=, we find
(EAE EY SES == O51] 215 ee (46),
a very simple relation which for a very thin disk approximates to the form
YRS RENNES ea cs. cc a (47).
§ 16. When the cylinder rotates about its axis of figure its mean length is certainly
reduced when 7, and 7, are both positive. There is however no reason why one at least
of these constants should not be negative in some forms of aeolotropy, for at least some
combinations of orthogonal directions. If 7»; be negative rotation about the axis of #
always increases the mean length, and if m, be negative rotation about the axis of y
always increases the mean length. But when these quantities are positive the mean length
is diminished by rotation about the axis of « when
UH on eee SR naga fe sere ia tase nanaser cereacrer (48),
and by rotation about the axis of y when
E <iWaah Mnig Se Atenetetos cob ebegas ) adds cae (49).
In the case of isotropy 7, =» =7, and » would appear to be essentially positive. In
a circular isotropic cylinder of radius R, assuming uniconstant isotropy, Le. 7 = 1/4, we
find the mean length increased or diminished by rotation about a diameter of the central
normal section according as
l/R> or > 3// 48, i.e. 3/7 approximately,
When an isotropic cylinder rotates round its axis, the changes in the volume and in the
mean length are connected by a very simple relation, the same for all forms of cross
section, viz.
(TOO NSEC ED) SU aR ge oetine a Seen eee (50).
It is also worthy of notice that ultimately in a very thin circular isotropic disk the
reduction in the mean thickness is twice as great when it rotates round its axis as
when it rotates round a diameter, the angular velocity being the same in the two cases.
324 Mr C. CHREE, ON CHANGES IN THE DIMENSIONS OF
§ 17. In the case of any rotating right cylinder we may find the mean change d0 in
the area o of the cross sections by combining the previous data. For v= 2el, so that
Sao SS Bath OU) E wiv aaanclencuae oaeenneneeene een eaee tee (51),
where the mean values refer to any one case of rotation.
For instance, when an isotropic cylinder rotates first about a principal diameter of the
central section, and then about its axis of figure we obtain
Ey eae | it (5 ze (5 uD “|
Saye ae 3h E P+ sgt Be)
= ap (— Ay PO (lp) GAYA wo dicskcaresiocsnceresemeasanaes (52),
: ile
lre=o2 “ Q 2 2
6c,/c = @; p(a,+ 4) («2+ K,7)
Stang 9 (Uia9)))Oeieiata Mig) irc Waljecse sess inate ctectee eee ceaec Geer ene (53).
The last result it will be noticed is independent of the length of the cylinder. Since
every cross section of an isotropic circular cylinder rotating round its axis must remain
circular, we may deduce the mean change in the radii of the cross sections from the
equation
SRF AB ome dew ase casauesons-ved saucenooee tte hee eee (54).
When a cylinder rotates about a diameter of the central section the alteration of a radius
in any given cross section depends on its inclination to the axis of rotation.
§ 18. In the case of rotating rectangular parallelepipeds certain additional results of
interest are easily obtained.
We shall confine our attention to isotropic materials.
Thus suppose the rectangular parallelepiped 2a x 2b x 2c to rotate about the axis 2e,
taken as axis of z. Then we find the mean change 2éa in the dimension 2a, supposed
parallel to «, from the formula
bo |[[4% dvayae= [[furp (et — ny?) dedyde,
whence Ona =—(e04p| (G2 — D2) ls ane oeeroertesionee- soee-oae eeeear ee (55).
Thus this dimension has its mean value increased or diminished according as
Sor Se ee (56).
The tendency to increase in length in a material line perpendicular to the axis of rotation
will thus become reversed when the dimension which is at right angles both to it and to
the axis of rotation is sufficiently increased.
Consider next the rectangular parallelepiped 2a x 2a x 2c, one cross section of which,
supposed parallel to ay, is a square. Any diameter in the central section ay is a
principal axis of inertia, and so may serve for an axis of rotation without the existence
of constraints. Take then for axis of rotation a diameter inclined at an angle 6, to the
ELASTIC SOLIDS DUE TO GIVEN SYSTEMS OF FORCES. 325
axis of w, supposed parallel to an edge 2a, the axis of z being as stated above parallel
to 2c. We then have
X/sin 0, = — Y/cos 6, = w’p (wsin 6,—y cos 6;), Z= w’pz.
Thence we easily deduce for the mean change in the dimension 2a parallel to «—i.e.
inclined at an angle 6, to the axis of rotation—
8a/a = w’p {a? sin? 0, — 9 (c? + a? cos? O,)}/BE .......ceeeeseeeeeeeeees (57).
Thus 6a increases algebraically as 6, increases from 0 to 7/2. The mean alteration in
the dimension 2¢ perpendicular to the axis of rotation is easily shown to be independent
of 6.
Finally consider the cube 2a x 2a x 2a. Here any line through the centre is a principal
axis and may serve as an axis of rotation without the application of constraints. Take for
coordinate axes the three perpendiculars from the centre 0 on the faces, and for axis
of rotation a line whose direction cosines relative to Ox, Oy, Oz are respectively cos 6,,
cos @, and cos@;. Then
X = o’p (x sin’ 6, — y cos 6; cos 8, — z cos 0, cos 5),
and the other components of the bodily forces may be written down from symmetry.
Employing these values for the component forces, it is easy to find the expression for
the mean change in the dimension 2a parallel to Ow, and it may be reduced to the
simple form
$a/a = wpa? (1 — 9 — (1+ 7) C08? A} /BF ..eee ceececceeeesenneeetees (58).
The mean change in a dimension parallel to an edge thus depends solely on the angular
velocity and on the inclination of the edge to the axis of rotation. Attention may be
specially called to the cone of semi-vertical angle
Oi== Contin =a) Cm) hers. elses. ombee ow (59),
whose axis is the perpendicular from the centre on two opposite faces. Its generators have
the property that when they act as axes of rotation the mean dimension parallel to the axis
of the cone is unaltered.
§ 19. To enable a solid to continue rotating about any axis other than a principal
axis through its ¢.G. some constraint must exist. When the axis of rotation is excentric—
i.e. does not pass through the c.G.—there must be pressures between the axle and its
supports balancing the “centrifugal force” of the mass supposed collected at the c.g. This
implies the existence of terms in the surface integrals in (20) and (23). If everything be
symmetrical about a plane through the c.G. perpendicular to the axis of rotation, it is
obvious from symmetry that if we take this axis for that of z, and neglect friction parallel
to z on the axle, the surface force H at the bearmgs will vanish. If further the dia-
meter of the axle be small compared to diameters of the body perpendicular to the axis
of rotation, the coordinates z and y in the surface integrals may be treated as small
quantities, and for a first approximation the surface integrals may be neglected. In such
a case formula (32) gives as before the change of volume in an isotropic body, but the
326 Mr C. CHREE, ON CHANGES IN THE DIMENSIONS OF
moment of inertia round the axis of rotation is of course greater than about a parallel
axis through the cc. Thus if « be the radius of gyration about a parallel to the axis
of rotation through the ca. and y be the perpendicular from the c¢.G. on this axis,
we have
dv = w'pv («2 + F)/3k = Su, {1 + (y/«)*} ....... Peser en sucodedecsnen (60)
where 8 is the change of volume for rotation with the same angular velocity about a
parallel axis through the c.G. Thus while a displacement of the ¢.G. from the axis of
rotation has but little effect so long as it is small compared to «, it is most important
when comparable with «x.
bo
§
the ¢.G. perpendicular to the axis of rotation we in like manner obtain a formula of the
0. In an aeolotropic solid of form symmetrical with respect to the plane through
general form (33) provided we take for our coordinate planes the principal planes of
inertia at the poimt where the plane of symmetry cuts the axis of rotation. When the
principal planes containing the axis of rotation are parallel to principal planes through the
c.G. the effect of a displacement of the c.G@ from the axis of rotation is as easily
traced as in isotropy, but otherwise it must be remembered that the values of the
elastic constants vary with the directions of the axes. It might thus in some cases be
most convenient to take the two coordinate axes, which are perpendicular to the axis of
rotation, parallel to principal axes at the c.G., though this introduce a product of inertia
into the formula deduced from (20).
§ 21. When the radius 7, of the axle, assumed circular, is small compared to the
distance of the ¢.G. from the axis of rotation we can easily find a fairly accurate measure
of the correction to the value of 6v required on account of the hitherto neglected sur-
face integral. Thus for isotropy, let the axis of rotation be axis of z, and let the ©.G.
lie on the axis of y at a distance 7 from the ongin. Also let @ denote the angle
which a radius of the axle makes with the plane yz. We shall suppose the body
symmetrical about the plane wy, and neglect friction on the axle parallel to its length, so
that there is no component parallel to z in the surface forces. The forces exerted at
any point of the axle by a bearing may then be resolved into NV along 7, and 7’ per-
pendicular to it. Thus supposing there to be two bearings, and assuming NV and 7 the
same numerically at —@ as at +6, we must have
4i| Nicos Ord = ar pups)... hes. be ee (61).
0
Also since Fa + Gy =— Nr, 4
the surface integral in (23) becomes
=A | G20 Asx anietss ses hfe oe Aes veee.(62).
To evaluate this integral exactly we require the law of distribution of N over the sur-
face of the axle between @=+7/2. As this is unknown, I have calculated the correction
to év on three hypotheses. The work is easy so it will suffice to state the hypotheses and
quote the results. These are as follows:
ELASTIC SOLIDS DUE TO GIVEN SYSTEMS OF FORCES. 327
Hypothesis Correction to év
1° NV uniform, i.e. independent of 6, — (w*pvyr,/3k) x 7/2,
2° N cc cos 6, — (w*pryr,/3k) x 4/7,
3° N concentrated at end of diameter 0=0, — (w’pvyr,/3k) x 1.
The true formula of correction will probably vary from one shape of body to another,
but the result must lie between those of 1° and 3°, and most likely will in general be
not far from the result of 2°. Taking this as the most likely value we have in place
of (60)
ov = w’pv (« +7 4 ges) [3k Stes Hata See RO ca SeRe on ORE ie (63).
§ 22. The effect on the length of a right cylinder of an excentric position of the axis
of rotation is also easily studied provided it be parallel to the axis of the figure, or else be
in the central cross section and be perpendicular to an axis of symmetry of that
section. It will suffice to give the results for an isotropic material in these two cases,
neglecting the correction arising from the surface integral. This correction may however
easily be approximated to, just as in the case of the change of volume.
Let the c.c. of the cross section be at a distance 7 from the axis of rotation.
Then for the increment 6 in the mean half length we find from (21):
1° when the axis of rotation is parallel to the axis of figure
Of — ap Gesamte TA ete ta seaaes. ccaesaeeceaenh (64),
2° when the axis of rotation lies in the central cross section and is perpendicular to a
plane of symmetry .
Ob [Pep {kPa (2 PAYED saiatee «Sts oon ede = eden nce (65).
The notation will easily be understood from the previous examples.
The effect of the excentric position is in either case to promote shortening of the
mean length.
Gravity at the Earth’s Surface.
§ 23. Let a homogeneous elastic solid of any shape be suspended from a point on
its surface. The centre of gravity must le on the vertical through this point, say at
a depth h below it. Taking the point of suspension for origin, and the axis of z vertically
downwards, and denoting gravity by g so that Z=gp, we find from (20), for an aeolotropic
solid
OU WSS ONG git cee ads aisees Tata Oe octal ss sbe es vac (66);
whence, or from (23), for an isotropic solid
SOS OBES asoace.conecec 0c cOdOCO OEE OnOOee Rep RObGOSe (67).
If on the other hand the solid rest on a smooth horizontal plane—or be supported at one or
Vou. XV. Parr III. 43
328 Mr C. CHREE, ON CHANGES IN THE DIMENSIONS OF
more points in a horizontal plane—let us take this for wy and let the axis of z be drawn
vertically upwards through the c.@. Then putting 7=— gp in (20), and noticing that
| [Heas and | i HydS
must vanish owing to the conditions of statical equilibrium, we find for the change 6v’ in
the volume of an aeolotropic solid
OU — UG cocoeceatecnenereac ates cores neeectienene (68).
For an isotropic solid
OU) Vi — Opis | Bleloaiee couseienencasncwiacecens eens (69).
In these two formulae h’ is the height of the ¢.G. above the horizontal plane of
support. There may be a number of isolated areas of support, as in a girder bridge, pro-
vided all are in one horizontal plane; and in any such case in an isotropic material the
volume is diminished or increased according as the ¢c.G. in the position of equilibrium is
above or below the level of the supports.
If the same material line be the axis of z in the two cases answering to (66) and (68),
and the length of this diameter be d, we find
in the aeolotropic solid (dv — 8v’)/v = gpdgy ......cecceereeneeneeeeeees (70)
» » isotropic COUl=00)) MP pd dio eenaetes-etessacterseenes (71).
The quantity & is essentially positive, and thus in isotropic solids the volume is greater
when the body is suspended and less when it is supported on a smooth plane than it would
be if the body were free from the earth’s attraction. The quantity g, is positive as a
rule in aeolotropic solids, but there is no obvious reason why in some solids it may not
be negative for certain directions of the corresponding axis.
§ 24. To get some idea of the magnitude of this effect in isotropic solids we shall
consider some special cases of bodies which may reasonably be regarded as fairly isotropic.
In steel* we may regard a length modulus of 25 x 10’ centimetres as a fair average for
EB, and may put 7=1/4. Taking these values, and denoting the densities of steel when
suspended and when supported by p, p’ respectively, we find for its density p if unacted
on by the earth’s gravitation
p=p(14+2h/10°), p=p (1 —2h'/10°),
where / and /’ are the lengths occurring in (67) and (69) measured in centimetres. If the
body were a right cylinder its height would equal 2h or 2h’. Thus the cylinder would
require to be 5 metres high before its specific gravities when suspended and when
supported differed from one another by one part in a million. Steel, or iron, is how-
ever the metal in which the effect is least. In such a metal as lead it is very much
greater. Thus if we assign to / in cast lead* a length modulus of 16 x 10° em, and
suppose 7 = 1/4, the difference between the specific gravities when suspended and sup-
ported would amount to one part in a million in a cylinder about a third of a metre in
height, i.e. little over a foot.
* See the table of moduli in Sir W. Thomson’s article on Hlasticity in the Encyclopaedia Britannica.
ELASTIC SOLIDS DUE TO GIVEN SYSTEMS OF FORCES. 329
In sheet lead, according to Sir W. Thomson’s table, this difference of the specific
gravities would arise in a cylinder about 4 inches high. Of course these numerical results
are intended merely to give an idea of the magnitude of the effect, and it must not be
supposed that the elastic data they are based on—more especially the hypothesis of uni-
constant isotropy in sheet lead—possess any great accuracy.
§ 25. In the case of a right cylinder we can also find the alteration in the mean
length due to the action of the earth’s gravitation. Thus supposing the cylinder first
suspended, and then supported on a smooth plane, with its axis of figure, taken as axis of
z, vertical we find from (15) for the mean increments 6/, and 6/,’ in the length J for any
elastic material
Sle ee AT, i= deep) Biss sstileg tht sace: beet ok (72).
Here £, is Young’s modulus for the direction parallel to the axis, and so presumably
is essentially a positive quantity.
If again the cylinder be suspended with its axis horizontal in such a way as to prevent
flexure—for instance, by a large number of strings attached to points along a generator—
and the vertical plane wz contain the c.G., the axis of the cylinder being axis of z, we find
from (15) for the increment 8/, of the mean length
DNA a A 1 ON Ree hoe EE (73),
where /, is the distance of the c.c. below the horizontal plane through the points of
suspension. While if the cylinder rest on a smooth horizontal plane in this position, the
increment 6/,/ in the mean length is given by
SL 1b Sag phi | ENaee Ale ee te ce (74),
where /,' is the height of the c.G. above the supporting plane.
For an isotropic material we have only to replace #, by H and n, by 7 in the
last three formulae.
The general conclusion we are led to is that under the action of gravity any elastic
right cylinder lengthens when suspended with its axis vertical and shortens when suspended
with its axis horizontal, unless in the latter case m7, be negative; but when supported on a
smooth horizontal plane it shortens when its axis is vertical and lengthens, unless 7, be
negative, when its axis is horizontal.
If we suppose the same diameter d vertical in the two cases (73) and (74) we get
(8h) — 81,)/l = nugpd/E, ...... shee Rate esicesiiash oasis (75).
Comparing this with (72) written as
CSR pale eee Ns aesocessune (76),
we see how much more effective gravity is in altermg the length of a long bar, of small
diameter, when its axis is vertical than when it is horizontal. But if the diameter of a
45—2
330 Mr C. CHREE, ON CHANGES IN THE DIMENSIONS OF
long horizontal cylinder be considerable, the effect of gravity on its length is deserving
of attention, especially in materials such as lead or gold, and to a smaller extent in
silver and platinum.
In any right cylinder the mean change in the cross section in the several cases
just treated may be found by combining the results for 6v and 6 by means of the
formula
a eae 5) | mat | | Maree te SAMO a Le RASA (U1)
§ 26. As the plane supporting a solid is never quite smooth, it is desirable to see
what effect the roughness of this plane would have on the previous results. Confining our
attention to isotropy, we require to add to the value of 6v for a cylinder supported with
its axis vertical on the plane z=0, the value of the surface integral
Lr
ap | {Fe + Gy) dady
taken over the supported base, where F and @ are the components parallel to w and y of
the frictional forces.
Let N and 7 be the components of the frictional force at any point along and
perpendicular to the radius vector r from the origin. Then the above integral becomes
1 [iv
3g || Nraendy.
Now the tendency of the supported solid—whose ¢.G. is assumed above the supporting
plane—shortening under gravity is clearly to expand horizontally, and thus the frictional
force is towards the origin, or N is negative. The surface integral is thus negative and
from the corrected formula (69), viz.
Su’ = — gph'v/3k + | | (Nr {SB ftedy's. 285. b.tech Sita (78),
we see that this correction tends to increase numerically the reduction in volume due to
the action of gravity.
The corrected formula (72) under the same conditions is
&L, = — gpl:/28— | | (GiNGE) TA) Wik Dihishecec. dsscssntbensk Sore (79),
where the surface integral is taken over the supported base.
The frictional forces thus tend to reduce numerically the shortening in the cylinder’s
length due to gravity. The corrections in these two cases are less, ceteris paribus, the
smaller the base of the body,
Bodies under the mutual gravitation of their parts.
§ 27. In a gravitating sphere of radius R, volume v and uniform density p we have
X/z = Y/y = 4lz = gp/h,
ELASTIC SOLIDS DUE TO GIVEN SYSTEMS OF FORCES. 331
where g is the acceleration of “gravity” at the surface. Substituting these values in (20)
and remembering (10), we deduce for the change of volume in any elastic sphere
=O) = 70/1) NV zonaddcoa cole eee eee OeER ER SE AEE Dae (80).
Knowing the change of volume we can at once deduce the change of radius. If we
were to apply this result to a sphere of the earth’s size and mass, we should find that
unless we assigned to k a much greater value than in any known material, under normal
conditions, our formula would imply strains much in excess of those to which the mathe-
matical theory of elasticity is legitimately applicable.
§ 28. To determine the effect of a small deviation from the spherical form, let us
consider a homogeneous solid whose surface is given by
= AD) Ciara eres cccee eaters «dco snstresdenioee. aaa (S1),
where R;o; represents a term, or a series of terms, involving surface spherical harmonics
of degree 7, and the ratio of each term to R, or the ratio of the sum of all the terms
of all degrees to R, is supposed so small its square is negligible. For such a body the
gravitational potential is given by
V =— gor /FP + & (Be Rio; (r/R)' = (Qe+1)} ..0 2... eee eceeceeeeeeenee (82)*,
where g represents the mean value of “gravity” at the surface.
Supposing the material elastically homogeneous but of the most general aeolotropic
character given by (1), we find the change of volume from (20) by substituting
A=? ae? Galore Z=p a> F=G= H=0.
The sum of the terms independent of ¢; inside the integral is simply — 2gp Ry.
Thus integrating the terms involving o; by parts we find, using (10) and representing
the element of normal to the surface by dn,
— dv/gp = [[[2R-xardyde
= Ea +1) | [Ren & as |
2 Eo 2 +1) |{[Re; ri dedyd| Ee hone oe (83),
where the volume integrals are taken throughout the entire volume, and the surface
integral over the whole surface (81).
Now || [22x dedydz — I [Ro (rx) 74 sin @drdédd,
and as ry, is independent of r this becomes, neglecting terms of order (R;o;/R)?,
|| [2B-yodedy dz =2R Ifa + 53 (Ryo:/R)} (ry) sin 0d0dd.
i)
* Cf. Professor Darwin, Phil. Trans. 1882, p. 200.
ee)
vs
to
Mr C. CHREE, ON CHANGES IN THE DIMENSIONS OF
But 1 YXo = Sr? (Cex? + fay? + go2* + ays + doze + cury)
=1(e,+f)+ 9%) +sum of surface harmonics of 2nd degree...............(84).
Thus using (10) and remembering that the integral of a surface harmonic over the
surface of the sphere vanishes, we obtain
| | [2R-y,dedyde = 3B (5 t)+ 2Rs i [3 Gio R) (ry) sin aod.
ne : , : Sates dx dx
Again in the surface integral in (83) we may replace Pa by r apo OF 2y,, and
may then put r=R. Also transforming the last volume integral in (83) into polar co-
ordinates, and neglecting terms of order (Rjo;/R)?, we see that the integral vanishes by
the ordinary property of surface harmonics. Thus, combining the several simplifications,
we replace (83) by
— 8v/gp = Rv/5k + 2: || \@- = 4 ei) RB fs a (7-*xy) sm OdOdd.........6+. (S85).
Referring to (84) we see at once from the mee properties of surface harmonics
that the only terms in =(R,o;) which can contribute anything to 6v are those of the
second degree. Again, the most general possible form of R,o, is given by
Ryo./R = 37? (Apt? + Bay? + Coz? + 2Dayz + 2H 2a + WKLY) ..0.csevseveees (86),
where the constants are subject to the one condition
Thus we may replace (85) by
— bv/gp = Rv/5k +1R! [fa. B,, G., D., Es, FQ, ¥, 25 X (€oxfo» Jos, Atos & $b, deha, y, 2) da
where da is the element of surface of a sphere of unit radius.
Now it is easy to prove
[leas a 3 [[ysaes =...= 47/5,
while the integrals of all terms involving an odd power of «, y or z vanish.
Thus using (87) we obtain from (88)
— by = SE +2 += sl bl) + By fi, + Cog, + Dia, + Exh, + Fe} etre (89),
where A., B., C, are subject to (87).
This form of the result may be the most convenient under certain conditions, since
the stress-strain relations in most kinds of aeolotropy are simplified by taking the axes
of coordinates in certain fixed directions, but the physical meaning may be rendered
clearer by a change of axes.
ELASTIC SOLIDS DUE TO GIVEN SYSTEMS OF FORCES. 333
By the properties of quadrie surfaces we may change the directions of the axes,
keeping them orthogonal, so as to transform (86) into
RoGo/R = 47 (A, 2? +- Byy? + C,2?)......-. HOA UD Oe oO ceRerae (90),
where A, + BY +C, =A,.+B,+C0,=0...... S bOSCORCOSS ADE Corer (91).
Thus putting Cn Ane bt (Ane reaa a). SoBe WAR rae cenes Cot RCORCEERe (92),
we have Rigs Rar Ay? (222 ay) By! (a a) en weecceceeetenstenen (93).
Let now e,, f,, g be the extensions, for uniform normal unit tension, in the directions
of the new axes, then we transform (89) into
=e [Weg CAN! (pacman (er =f") hence eRe OLDS
5 Bem v 2 “ paws i ” , geI\) OL
or —ov= PERC =) 102" Coy a eee eee Oe),
Now a positive value of A,’ means an increase of that diameter in whose direction
go is measured and a diminution of all perpendicular diameters, while a positive value
of B,” means an increase of that diameter in whose direction e,/ is measured, a dimi-
nution of that diameter in whose direction /,’ is measured, and an unchanged length in
that diameter in whose direction g,’ is measured. Thus the general result implied in (89)
(94) is that the diminution in volume in the mass due to its own gravitation is
greater or less than in a sphere of equal volume according as the longest diameters in
the nearly spherical body
r— ht Reo.
are directions in the material along which the reduction of length accompanying uniform
normal pressure is above or below the average.
For any isotropic material the reduction in volume has the same value as in a sphere
of equal volume. Thus the reduction in volume of a given isotropic mass due to its
mutual gravitation is in general either a maximum or a minimum when the bounding
surface is spherical.
To determine whether in this case the reduction is a maximum or a minimum we
would require to go at least as far as terms of order (Rjo;/R)*, and it would be necessary
to employ a more exact formula for the potential than (82). Such formulae are unknown
to me save for ellipsoids, in which case we can go to any required degree of accuracy.
As regards harmonic terms of degrees above the second, it seems most likely that for
a given maximum value of R;o; the effect on the change of volume will in general be
less the greater ¢ is. Thus the second harmonic term, unless relatively inconsiderable, may
be anticipated to have usually a predominating influence. When the elastic properties of
the medium, while showing aeolotropy, vary but little in different directions, the terms
in A,” and B,” in (94) may conceivably be of no greater importance than those depending
on the squares of the harmonic terms. It has thus appeared desirable not to assume
isotropy in the following treatment of the ellipsoid even when nearly spherical.
334 Mr C. CHREE, ON CHANGES IN THE DIMENSIONS OF
Travitating Ellipsord.
§ 29. Let a, b, ¢ be the semi-axes of an ellipsoid of uniform density p and volume »,
of a homogeneous aeolotropic elastic material given by (1), and let
* fo du
| = J(@ +u) (bP +u) (e+ ux)
Then denoting by » the gravitational force between two unit masses at unit distance,
we find for the bodily forces*
X=-pAc, Y=-pBy, Z=-C:z,
Aesth d d :
where A = 3upv oe , B=3yupv = , C= 3pypv a ei ciejsiseilnnee Selva ctenoeae (96).
The surface forces everywhere vanish. Thus from (20) we find for the change in
volume of the ellipsoid due to its mutual gravitation
— dv/p = |[teae + P Bf, + 2Cq.) dadydz
= Pa lesAe, + DDT peiCOUp|, seis saclancseb ok esesnsschsaaressena (97).
If g,, g, g; be the values of “gravity” at the ends of the three principal axes of figure
1A —p, bee. co = o,5
thus = (00/0) = 4 \(A87ep ts DE aot: CP39p))-nc--0- ere eeasnccioseabaececirs (98),
or for isotropy — dv/v= = ‘ (BOT DENI CDs) icawacisoatteetecsievosessaraticseeette (99).
The quantities A, B, C, or g,, g., g; may be expressed as elliptic integrals.
When the ellipsoid is nearly spherical, let
bia Le C2 aa he eee ocseas recente (100).
dy
Then expanding at ete. in powers of e, and «, and neglecting powers above the
ec
fourth, we easily find
_ 8upv /l , evte? | de¢+ 2ere?+3es\ )
ar (5 fe clare 56
_ Bdupv (1 , 367+." , 15a + 6e7e." + Se,
B= (5+ —~s0 as Et ke I (101).
onnehe? (5 ef + Ber , Bat + Gere? + Be)
~ a \8 10 56
Now let R be the radius of a sphere equal in volume and mass to the ellipsoid,
and let g be the value of “gravity” at its surface; then
R? = abe = a (1 — e?)' (1 — 4
upr/Ri = g
* See Thomson and Tait’s Natural Philosophy, Vol. 1., Part 1, p. 47.
ELASTIC SOLIDS DUE TO GIVEN SYSTEMS OF FORCES. 335
Substituting the values of A, B, C from (101) in (97), eliminating a, b, ¢ by means
of (100) and (102) and arranging the terms, we find
— dv =tgpRv [et fi + got gs fe? (Go +4 — 2fr) +e? (@o + fo — 2M%)}
— shy (e4 (47f, — 13e — 139) — eee (1le, + 5f, + 590) + es (479 — 13e, — 13f,)}]..-(103).
Employing (10), we may write this in the more convenient form
r ay aes 4
= y= 80" E = a : 2 {a2 (1 — Bhf,) + 2 (1 — 3kg.)}
+ =e {10e4 (1 — 3hf,) + ee." (Bhke, — 1) + 10e.4 (1 — sky) asesaneee (104).
It is easy to show that the terms in e¢; and e, agree with those already obtained
in (94).
For any isotropic material we have the simple result
Rv €, — €26,” + &*) ‘.
— by = SE (1 - SSS Inara: (105),
Thus in an isotropic nearly spherical ellipsoid the reduction in volume is always less
than in a sphere of equal volume, or the sphere is that form of ellipsoid in which the
reduction of volume due to the mutual gravitation of the parts is a maximum. The
smallness however of the terms in (105) depending on the eccentricity seems rather
remarkable.
In an aeolotropic material the terms in e}, ee? and ef which depend on differences
of elastic quality in different directions have obviously the same physical import as the
terms in e,° and «¢*?; i.e. they signify an increased or diminished reduction of volume
relative to that im the sphere according as the longest diameters are directions in which
the contraction under uniform normal pressure is above or below the average.
For a prolate spheroid about the axis 2a, putting «?=e¢?=e in (104), and using
(10), we get
+ y= 8k Stine “(14 57) Ble, - 1)| Arrilgtiay peak (106).
For an oblate spheroid about the axis 2c, putting ¢,=0, and e,=e’ in (104), we find
, Ry é 2 In 10 Is
— oy = 8 -Ste (1457 2) 3h) | ieee seem: jue LO'),
As in (104), R denotes the radius of the sphere of equal volume and g gravity at its
surface.
We notice that 6v’=év when e =e in all isotropic materials. In an aeolotropic
material when the spheroids have their axes of figure in the same direction in the
maternal, ¢ in (106) and g, in (107) are identical. Thus when e’=e the effects of
aeolotropy in the two spheroids are very nearly equal numerically, though of opposite
sign.
Wow, XV. Pare dit 44
336 Mr C. CHREE, ON CHANGES IN THE DIMENSIONS OF
§ 59. In this paper our attention has hitherto been confined to the mean values of
the strains, but we may obviously from the equality of (I.) and (II.) arrive even more
easily at the mean values of the stresses. For instance, to find the mean value of 7
answering to a given system of applied forces, viz. X, Y, Z per unit of volume, and
F, G, H per unit of surface, put /’=g' =a =b'=c'=0 in (IL), and regard e’ as constant.
Then for the corresponding displacements we have
dan (=H = Osea eed. gee (108),
and so from (I.) and (II.), dividing out by e’, we find
IH aedady dz = [| [Xedvdyae + [| Poas wbebrernsdngusetepomeiteer (109),
where the volume integrals are taken throughout the whole volume and the surface integral
over the entire surface, or surfaces, of the solid.
Again, regarding a’ as constant, putting
Sif =F =) SoS) a WD).
and substituting in (I.) the corresponding displacements, viz.
(ce Si A074 07 eaobe snapooeaBeacepossbendaacce sae (110),
we find
[[]Pacayaz = f/f Ve fy) dadyde +4 ||(Ge+Hy) AS = ae (111).
The formulae for the other mean stresses may be written down from symmetry.
The results for the mean stresses are wholly independent of the aeolotropic or
isotropic nature of the medium, They may be verified in the simplest manner by direct
reference to (12) and (13).
The information derivable from the values of the mean strains and_ stresses is
necessarily in general of an imperfect character, as the law of variation of the strains
and stresses throughout the solid is essential to a complete study of an elastic problem.
Stull the mean strains and stresses may indirectly prove of considerable service in veri-
fying the accuracy of mathematical work, and perhaps occasionally in affording a test of the
sufficiency of theories which supply for a definite physical problem a mathematical substitute
as to whose approximate equivalence doubts may be entertained.
[April 22, 1892. By ordinary Statics the bodily and surface forces must satisfy three
equations such as
[|| Xavdyaz + || Fas=0,
and three such as
[{[izy- Yz)dadydz + | ty- Gz) dS=0,
ELASTIC SOLIDS DUE TO GIVEN SYSTEMS OF FORCES. 387
Employing these we can write some of the general formulae in the paper in a variety
of equivalent forms. For instance, we may transform (15) into
Eng = [I LX ( = 95% — ey — 1352) + Y (— nay — nu2Z) + Zz] dxdydz
+f FC + 4 )+ Hey ds,
and may combine (22) and (111) in the form
noir = || rdadydz= [ha —p)Yz+ pZy} dvdyde+ | f {1 —p) Gz+ pHy} ds,
where p is any constant, including 0.]
i> sy :
7; eT Ses vt ~~
ieee orm: AW Sori rire line - ;
} alil vo 2% iyie ej Tiedt ah.
1 i,
wri ea e
7
LE ’ yay Tye
ca ‘ :
preg. vil : Li tp bor (58) wpa i) ani |
ih whi) wit Dee Livv= th 1 bershoyinalher | |} . eo». 7
eo: |. en
LO scapes ih. ete eree,, Cy ee
se ¥ =
X. The Isotropic Elastic Sphere and Spherical Shell. By C. Cures, M.A.,
Fellow of King’s College.
[Read February 13, 1893.]
CONTENTS.
PART I. GIVEN BODILY AND SURFACE FORCES. | §§ 23—37. Mixed radial and transverse displacements:
3 x 7 Determination of arbitrary constants, values
§1. Historical Introduction. of displacements and seenae thin shell,
§ 2. Fundamental Equations. limits to applied forces, effect of degree of
§ 3. Three classes of displacements. harmonies, stress gradient curves, summary
§§ 4—11. Pure radial displacements : of results.
General results, thin shell, limits to applied | §§ 38—40. Solid sphere.
forces, stress-gradient curves, comparison of § 41. Nearly solid shell.
bodily and surface forces.
§§ 1213. Mode of presenting surface forces. PART Il. GIVEN SURFACE DISPLACEMENTS.
§§ 14—15. Solution in arbitrary constants for displace- § 42. Pure radial displacements.
ments of second and third classes, § 43. Pure transverse displacements.
§ 16. Equations determining arbitrary constants. §§ 44—51. Mixed radial and transverse displacements, etc. :
S$ 17—22. Pure transverse displacements: Determination of arbitrary constants, values
General results, thin shell, limits to applied of displacements, thin shell, solid sphere,
forces, effect of degree of harmonics, har- nearly solid shell.
monies of degree 1. § 52. One surface arbitrarily displaced, other free.
Part I.
Equilibrium under given bodily and surface forces.
§ 1.
elastic spherical shell is of great interest as one of the few elastic problems of which a
mathematically exact solution has been obtained. The problem has been solved in several
than physical interest. The
aim of the present solution may best be indicated by a brief reference to previous
solutions.
THE determination of the displacements, strains and stresses in an isotropic
different ways, but with results rather of mathematical
The first treatment of the problem is due to Lamé*, who considered the case when
the surfaces of the shell are acted on by any given forces, but took into account only
one or two simple systems of bodily forces. His solution is in polar coordinates, and is
an elegant if somewhat lengthy piece of analysis. It obtains expressions for the dis-
placements involving arbitrary constants, and the method of determining these from the
* Liouville’s Journal, Tome 19, pp. 51—87, 1854.
Vou. XV. Par LV. 45
340 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
surface conditions is clearly shown. A physicist, however, desirous of applying the solution
in practice would probably find the labour of determining these constants sufficiently
arduous to deter him from his purpose.
A solution better known in this country is that of Lord Kelvin*. It is in some
important respects more complete than Lameé’s, as the -method of treating bodily forces
derivable from a potential is included, and the case of given surface displacements is
also considered. In the opinion of Thomson and Tait+ the use of Cartesian coordinates
in this solution in place of the polar coordinates of Lamé is a great simplification.
This is not an opinion which the author of the present solution can endorse, and
it seems to him that for practical purposes Lord Kelvin’s solution stands very much in
the same position as Lamé’s.
Recently the cases of given surface displacements and given surface forces have been
solved im a way quite unlike either of the preceding by Cerruti}. His results in the
ease of surface displacements are intelligible only to one familiar with what may be
called the “potential methods” of solution origimated by Betti and Lord Kelvin, and
whose best known applications are due to Boussinesqg. Judging by the abstract in the
‘ Beiblatter’ to Wiedemann’s Annalen§ the solution for given surface forees—the original
of which the author has not seen—is of the same character. The mathematical difficulties
in this form of solution are very great, and the results do not seem of such a character
as to lend themselves readily to practical applications.
In 1887 a paper|| was contributed by the author to the Society, containing inter alia
a solution in polar coordinates which led by a more direct route than Lamé’s to equiva-
lent results.
This paper determined explicitly the arbitrary constants for the case of a solid sphere
under given normal surface forces, or with given normal surface displacements, but for
other cases the results laboured under similar disadvantages to Lamé’s, as the labour of
determining the arbitrary constants was left for the reader. This defect it is the primary
object of the present paper to remove. It assumes the mathematical work of the previous
paper, reproducing only so much as is required to render the results clearly intelligible ;
it then determines the arbitrary constants for all cases and furnishes an explicit solution
applicable without serious trouble to any special problem. The opportunity is also taken
of considering in some detail the conclusions to which the solution leads when the shell
is very thin.
The results obtained in this case, being independent of any assumptions as to the
relative magnitudes of the several stresses, seem not unlikely to be of service in testing
the results arrived at by the ordinary treatment of thin shells.
It must of course be borne in mind that there may exist in some other forms of
thin shells phenomena widely different from those shown by a complete spherical shell.
* Royal Society’s Transactions for 1863, p. 583; or t+ Rend. R. Acc. dei Lincei 5, 2 sem. pp. 189—201, 1889 ;
Thomson and Tait’s Natural Philosophy, Part u., pp. also Mem. R, Acc. dei Lincei, pp. 25—44, 1890.
735 et seq. ’ § Bd. xv. pp. 630—1,
+ Natural Philosophy, vol. 1. Part u., Art. 735, || Camb. Phil. Trans. vol. x1v. pp. 250—369,
AND SPHERICAL SHELL. 341
For example, the strains and stresses produced by the flexure of thin plates with straight
or curved edges, especially in the case of narrow strips, or the strains and stresses pro-
duced by surface forces at points on a thin shell where the curvature is unusually great,
for instance near the ends of the axis of a very prolate spheroid, may follow laws which
bear but a slight resemblance to those arrived at here.
§ 2. Employing the ordinary polar coordinates 7, 8, ¢ as in my previous paper, and
denoting the displacements by u, v, w, we have for the components of strain
du
dr?
ty Meteo sen
r TT rsiné dd’ f
1 dv _ildw w
rsin @ Fal rd@ or cont, |
dw _w a is du
dr r rsn@ dd’
Of these the first three are in the terminology of Todhunter and Pearson’s “History”,
stretches, the last three slides, i.e, shearing strains.
The dilatation 6 is given by
5a eT +o cot to Heater AIOE An EL (2).
The stresses, employing Professor Pearson’s notation*, are
= (m=) d+2n&, |
@ =(m—n)d+2n(4 47S), |
ob = (m—n) 8 + 2n (“+ 2eota+ 5 7a) ; |
ae dy vii a | cecevovevecivicscuecisisbcleie (3),
a n (or rr dd)’
Bak (e w 1 7
=n(— 5
% =" > train’ db |
1 /dw 1 dv |
wo ane (F-weot += 7) J
where m and » are Thomson and Tait’s elastic constants.
Of the stresses the last three in (3) are the shearing stresses.
* Todhunter and Pearson’s History, vol. 1. pp. 882—3.
45—2
342 Mr C, CHREE, ON THE ISOTROPIC ELASTIC SPHERE
For shortness let
1
eh 2 sin 1a hae sin 6) — ent
1 (du :
ae aa ier (wr sin 0)| 5. hh canesasoenuiioncadereseenets (4).
© = sin {5 (vr) — Ot
Then for an isotropic solid of uniform density p, acted on by bodily forces derived
from a potential V, the internal equations of . are
a =
(m+n) 7r* sin 05 Waa ees aa pr’ sin 6 — 0,
dg
(m+n) i 1B ais ee sodiamisciaisesise nae’ (5).
2 dB dA aV
CS MAS aR pos eo
§ 3. We shall consider first the case of given surface forces.
If over a bounding spherical surface the components of the applied forces along r, 0, ¢
be respectively F, G, H, then the surface conditions are
where the + sign is taken at the outer, the — sign at the inner boundary.
The displacements constituting the solution of (5) and (6) for a spherical shell may
most conveniently be subdivided into the following three classes :
(i) Pure radial displacements, in which there is no displacement perpendicular to
the radius;
(ii) Pure transverse displacements, in which there is no displacement along the
radius ;
(iii) Mixed radial and transverse displacements.
Crass I. Pure radial displacements.
§ 4. These displacements in practical cases answer to bodily forces derived from a
potential
Vr? + Vir,
where V and V’ are constants, and to uniform normal surface forces, say
m= R over r=a,
7 =R' over r=)
AND SPHERICAL SHELL. 343
Supposing a>b, and V, V’, R, R’ to be positive quantities, the applied forces have
the directions and magnitudes shewn in fig. 1, where 0 is the centre of the sphere
OB=b, OA=a.
Fig 1
The potential Vr? is such as would arise from mutual gravitation in the shell, or from
a term in the centrifugal force independent of surface harmonics, if the shell were rotating
uniformly about a diameter. The only displacement is along the radius and is of the form
i | /
wa grYy +172, — Es eV ee ew (8),
R aab Th D a Tinga S09 gang es OCROCORE Sp Herc
where Y, and Z_, are arbitrary constants to be determined by the surface conditions (7).
Employing the value of * given in (3) and noticing that & reduces to a we obtain
two simple equations for the determination of Y, and Z,. It is hardly necessary to
record the values of these constants. When substituted in (8) they give
meee! ade, r {apt aspen Oth ato Lie: Oy ;
an > mpn + (8m —n) (a@—6) b*) Ee yetee + 5 (m+n) b)pV4 Pade
pe 1 r—a3hs ,. m+n ef m—-na—b \
~2m+n' 4Inag—b {R = 5 (m+ mem BO barre ab pve oe)
The value of the dilatation is
B. ths 'p Lae 3 spr, oOm+n : Z|
a m+n" "(3m — n) (a? — “yy eR - es LGaEan)\ Gal D1 Raia AG ape
nae E
m+nr Bi)
aie s du P ee
The principal strains are ae along r, and two equal strains u/r along any two directions
orthogonal to one another and to 7, We may suppose @ and ¢@ these two directions, and
may regard the corresponding principal stresses as #@ and %. They are given by
pa u
6 = $6=(m—n)d +2n-,
r
and may be found at once from (9) and (10). The other principal stress > is of more
importance for the theory of thin shells, so it is desirable to express it in a form suitable
for applications of this kind. This object is secured by the formula
—~ @r—} Ba—r., sm+n {at (a? — 7°) (7° — b*) — BF (7? — B) (a —1*)}
"a — BF ra —B 5 (m+n) > (a? — b8)
_m—n a (a — 7) (r* alias (r =) \ Che >)
m+n (a? — 6)
pV
pV (lilly:
544 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
; é . : du :
The algebraically greatest strain at any point may be either u/r or — according to
dr
the nature of the applied forces. The stress-difference is the positive value of
§ 5. When the shell is very thin we may conveniently put
a—b=h, a-r=é,
so that h denotes the thickness of the shell, and & the distance of any point from the
outer surface. Retaining the lowest and next lowest powers of h/a and &/a in the
coefficients of the several terms, we easily deduce from the previous formulae the approxi-
mate results:
m+n [(-2e SES Ase (1-242 gman s\ =e
u=
4n (3m —n) mtn a h m+na/) h
1 5bmt+nh m—né\ , m—n ae a
+215 m+n at ance =) a'pV (1- m+n py i: (18),
as [i(1+ a) R= (1-2 *) r+2(1-3 a" pam £) ap!
~3m=—7 Wh 2m+na m+na
--"= h' 3m—n E
m+n a m+n a
=) ssi (14),
~ h=—é Ela phat) pr ome ae) 2h 4 E apV
aa (1425 R+i (1 oe ee a? (1 Pag Me a)
_m—n E(h—&) (i+5: 1 li 7 *) a-pV" ...(18),
m+n ae +3 a
AEgets (1+£) e- ve (1-™=*)es(1-5 pupa nS é) aipV
p 2 a
2m+na m+na
i m—-nh 3m—n =
-3 1-S + a pV" ....(16).
m+na m+n
§ 6. If we denote Young’s modulus by EB, the bulk modulus by k and Poisson’s ratio
by 7, then
E=n(3m—n)/m, k=m—n/3, 4=(m—n)/2m.
Using these, and retaining only lowest powers, we easily find from the results
(13)—(16)
ulr = oB i,” wa Gua ed 2 ocsaa tbe Sale vam ete teow ac bere meetin (17),
du_—=S du da,
| eae ee oar tec (18),
gen ten ee ae (19),
AND SPHERICAL SHELL, 345
=e 7 1D} dpndendescmecedbdondStedagbanoagecbencnticormp adr: (20),
La
a 9
S=a om NN aatioch OD nO QOD ROC COCDCOCDCeED Bee DeeC DE RCCEC He TCeT (21),
where IPS 1 I a AMY O10 Sonic cco ooponcbceouedoouseabD08 (CP).
These values of the strains and stresses may, under certain restrictions explained below,
be called the “first approximations ”.
The quantity F' is obviously, to the present degree of approximation, the resultant per
unit area of surface of the entire radial force exerted by combined surface and _ bodily
forces on the shell.
The necessary restrictions to the use of the results as first approximations will be
easily grasped by considering the case when there are no bodily forces. In this case we
must clearly have £ = R ? small quantity in order that (17) may be a legitimate first
approximation from (13); in other words if R and R’ be of the same sign—i.e. both
tensions or both pressures,—they must not be so nearly equal that their difference bears
to their sum a ratio of the order borne by the thickness of the shell to its radius. The
general conclusion is that the results (17)—(21) are not to be employed as first approxi-
mations when F is so small compared to the individual bodily and surface forces of
which it is composed as to bear to them a ratio of the order h/a.
§ 7. We shall first consider the case when F is of the same order as its greatest
components, and consequently (17)—(21) are satisfactory first approximations. The strains
are then all approximately constant at every point of the thickness, and the same is true
of the principal stresses # and 4, whose directions are parallel to the surface. Also the
radial stress, while rapidly varying along the thickness is, to a first approximation
negligible compared to the other stresses. The important strains and stresses are in
fact due to the stretching or shortening of the “fibres” parallel to the surface, which
accompanies the increase or diminution of radius produced by the application of #. What
the exact mode of application of / may be, whether it consist solely of bodily or solely
of surface forces, or partly of both, and whether, if composed of surface forces, it be
applied over the outer or the inner surface, is to a first approximation of no consequence.
As regards the absolute magnitudes of the strains and stresses in this case, we see from
(17)—(21), that they bear to the strains and stresses which a longitudinal traction of
intensity F would produce in a long bar of the material ratios of the order a:h. This
is a very important consideration, as it leads at once to a restriction in the value per-
missible to F': viz. that the ratio of F to the greatest traction permissible in a long bar
of the material must be at most of the order h/a of small quantities. This is obvious
at once on the stress-difference theory of rupture from the form of (21). It also follows
at once from (17) and (18) from the mathematical condition that the strains must be
small,
346 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
It also may in general be deduced on the greatest strain theory of rupture from
(17) and (18), since either w/r or = must be positive. An exception to the latter proof
would however arise if 7 were very small and #’ directed inwards.
§ 8. We have next the case when F is so small compared to its components that
(17)—(21) cease to be satisfactory approximations, and we must fall back on the more
general results (13)—(16). If we suppose that the bodily forces per unit of surface are
small compared to the surface forces, or more generally that the resultants of the bodily
and surface forces are separately very small, then, with the exception of * in so far as it
depends on the bodily forces, all the strains and stresses are to a first approximation
constant throughout the thickness. The fact that * is now of the same order as the
other stresses is also important.
The limits allowable in the strains or stresses depend on the material, or on mathe-
matical restrictions independent of the nature of the applied forces, and so these quantities
may be as large in the present case as in the previous. The conclusion to be derived
from a consideration of these limits in the present case is that the separate forces R, R’ ete-
may now be comparable in magnitude with the greatest traction permissible in a long bar
of the material. In the present case the alteration of the radius is small and the con-
sequent stretching but trifling, but the direct action of the applied load on its immediate
neighbourhood is important.
§ 9. One general conclusion of considerable physical interest is obvious on inspection
of (13) and (16). The terms in &/a inside all the brackets are positive, and thus the
values of u—and so obviously of u/r—and of # or $$ are invariably numerically greatest
over the inner surface of the shell.
§ 10. The variation in the value of the stresses #, $$ with the distance from the
surfaces is seldom of much consequence, but the variation of * is interesting in itself
and important in the theory of thin shells. We shall consider it in the several cases
when there are only surface forces over one of the two surfaces, and when there are only
bodily forces.
In this and subsequent occasions certain curves called here “stress-gradient curves”
will be found useful. In these the abscissa measures the distance from a surface of the
shell, and the ordinate the corresponding value of the stress under consideration. In
none of the cases occurring here is there any change in the sign of the stress as the
distance from the surface alters, so for convenience the curves are all drawn on the
positive side of the axis of abscissae. The same curve thus applies whether a surface
force be a tension or a pressure. The rate at which the stress alters with the distance
from a surface is measured by the tangent of the inclination to the axis of abscissae of
the tangent to the stress-gradient curve. The numerical value of this tangent is here
termed the “stress-gradient”.
AND SPHERICAL SHELL. 347
Take for instance the case of a force R’ over the inner surface. Then by (15) the
first approximation, viz.
mr = (E/h) RY,
a straight line passing through the origin when the
abscissa measures the distance from the outer or unstressed surface. The “stress-gradient ”
is thus to a first approximation uniform, precisely like the temperature gradient in the
steady state of heat conduction through an infinite plate. To this degree of approxima-
tion each thin layer of the shell bears, as it were, its fair share of the applied surface
force. Similar results clearly hold in the case of a normal force R over the outer surface,
because h—€ is now the distance from the inner or unstressed surface.
?
gives for the “stress-gradient curve’
Taking into account the second approximations we see that in the case of both R
and hk’ the stress-gradient is steepest at the inner surface of the shell, and that the
gradient continually diminishes as we approach the outer surface. In the accompanying
figures 2 and 3 the thick lines BKE, DHA are the gradient curves in these two cases
according to the second approximations, while the dotted straight lines answer to the first
approximations.
Force R. Force R’.
Surface forces.
E
K
B A B A
Fig2 Figs
In both figures B represents the inner, A the outer surface, and BA the thickness.
In fig. 2 the foree R—represented in magnitude by AH#—acts on the outer surface;
in fig. 3 the force R’—represented in magnitude by BD—acts on the inner surface. In
both cases the dotted lmes are parallel to the tangents to the second approximation
curves at the point where €=h/2—or what we may call the “mid-thickness”.
The cases when bodily forces act may also be represented by stress-gradient curves.
Thus fig. 4 applies to the case of bodily forces derived from a potential Vr*, and fig. 5
to bodily forces derived from V'r; in both figures B represents the inner, A the outer
surface. The dotted curves in both figures refer to the first approximations. They are
parabolas whose vertices answer to the mid-thickness, and whose axes are perpendicular
to the axis of abscissae.
The thick line curves BDA answer to the second approximations. In fig. 4 the
points where the dotted and thick line curves intersect answers to the mid-thickness.
Force from Force from
Vr2 gas
Bodily forces. |
D
B
Fig 4 Fig5
Won, SOV. 1BAan IW AG
348 Mr ©. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
In both cases the gradients are steepest at the two surfaces, where the ordinates are
zero, and the gradient at the inner surface B is according to the second approximations
slightly greater than that at the outer A. When the shell is very thin the difference
between the ordinates of the dotted and thick line curves is much exaggerated in the
figures.
§ 11. Before quitting the subject of uniform radial forces a few remarks on the
relative magnitudes of the effects of bodily and surface forces may be of service. Let us
confine our attention to the terms in V and R, because the same conclusions hold in the
case of V’ and R’.
The bodily force is to a first approximation, ze. treating 7 as constant, 2pVa per unit
of volume, or 2oVah per unit of surface of the shell. Now from (17)—(22) we see that
according to the first approximation all the strains, and likewise the stresses @, 44, arising
from the bodily force bear to those arising from the surface force precisely the ratio
2pVah : R that the bodily force measured per unit of surface bears to the surface force.
As appears, however, from (15) the radial stress arising from the bodily force bears to that
arising from the surface force a ratio of the order (2p Vah) (h/a) : R.
If then a radial force act over one only of the two surfaces of a thin shell, the strains
it produces, and the stresses whose directions are perpendicular to the radius, are precisely
of the same order of magnitude as those produced by a bodily force the same in direction
at every point of the thickness, whose total amount per unit of surface is the same; the
radial stress however due to the surface force is, except in the immediate neighbourhood of
the unstressed surface, very much larger than that due to the bodily force.
§ 12. Before considering the two other classes of displacements it is necessary to
explain the form under which the surface forces are given. In a complicated problem
like the present, in order to avoid cumbrous mathematical analysis, care must be taken
to let the solution follow its natural channel. The following method of treatment is
very forcibly suggested by the form of the general solution.
Let 7;, T; represent surface spherical harmonics of degree 7, including constant coeff-
cients. The case when 7 is fractional does not seem excluded from our general solution,
but when, as in the present instance, the spherical surfaces are complete 7 will be a
positive integer. Then if © and ® be the tangential components of the forces applied
at one of the surfaces, say +=a, in the directions 6, @ at the point considered, we are
to present © and @ in the respective forms
MLE NAA (0 Ue
@=> EB + 9 4 wijeacis SAM etlnay a aeeose ene enna (23),
1 dT’, dT;
o=5 lanaag a | eer ee reer e renee seer eseesseseseere (24).
The summation is with respect to 7 The surface forces are practically split into
two sets, one derivable from a “potential function” =Z;, the other from a “stream
function” >'T;.
AND SPHERICAL SHELL. 349
Tt will, I believe, be found that in most practical cases the tangential surface forces
fall naturally into this shape, but if any difficulty should be experienced in giving them
this form recourse may be had to the following results. Multiply (23) by sin@ and
differentiate with respect to @, then add to (24) differentiated with respect to ¢. This
eliminates the T harmonics. Then employing the equation
: Z Ie a fn. a; il GENS
me A Be (Pe, v ps 25
tC +1) Vit og ap (sn? b) ta dd? (Disyy Epo RedtBe deme em (25),
satisfied by a surface harmonic Y; of the ith degree, we find
‘ 1
s 1 |
{[2@4+1) Ti] snd Fe 70 (Osin @)+ Tat Dean ohtnaiacis secteeebions (26).
Next multiply (24) by sin @, then differentiate with respect to 6 and subtract from (23)
differentiated with respect to ¢. This eliminates the 7’ functions and leads with the help
of (25) to
ae 1 dO ae
SG+) 8-359 |- dd + fg(Psin 8) | Bo rene noes erect (27).
Expanding the right-hand sides of (26) and (27) in the ordinary way, and equating
harmonics of the same degrees on the two sides of the equations, we have at once the
values of all the 7’ and T functions.
The radial surface forces are supposed presented in the form of surface harmonics and
are denoted by = R;.
§ 13. For some purposes it might have been more advantageous to group the radial
surface forces along with that part of the tangential surface forces expressed by the T
functions, deducing both from a potential
S[(r'/a) Q; 47-7 ai? Qa],
where Q;, Qi. are surface harmonics of the ith degree, and r is put equal a after
differentiation. The relations between Q;, Q-;, and R;, 7; are simply
Bee Ee ae. (28).
T, =Qi+ Qin )
§ 14. In dealing with the solution of the equations (5) in terms of surface har-
nonics it will suffice to take as a type the terms which contain harmonics of a single
degree. Thus suppose the bodily forces derivable from the potential
V+ V5,
where V;, V_;, are surface harmonics of the ith degree.
The typical terms in the displacements are then those given in p. 268 of my previous
paper. Slightly altermg the notation, we may write
\
Sar (-~ Vi 4 Y;\ 40% (= SP ty ae ee (29),
md +2 m+n /
350 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
ae ee
8) Soeafe Bee pV im — 2n
i
(3 ima 2m
(2 (28+ 3) mtn ~ 2(21+3)n
vs. ie pra (@@+1)m-+2n,,
‘ 2(21-1) m+n 2(21—1)n
vit +r>Z,
j an 18D, yeses--(80),
_d ett ( pVs . @+8)mt2n—) 1 ie r* (pV_sa , ((—2)m—2n
i [- 2(27+3) (m+n (i+1)n ghia ia * 2(2i-1) m+n * in Ja
t+1 in| + a +n 0 = Leta ieee
SUL, cd gett ((+3)m+2n Pik
w= 0a (7 sO eet (i+1l)n vibe ai
rt (pV. = (i—2)m—2n
+3 @i—1)| men in
dats
V1! = Z| — £ [ks +X.) ...(82).
Here Y;, Yin, 4%, Zin, X:, Xi. are surface harmonics of degree « whose form
depends on the surface forces, and in the case of the first four harmonics also on V; and
V__,. The letters may be regarded as including arbitrary constants to be determined by
the surface conditions. In my previous paper dashed letters Y;’ etc. stood in place of
Y__, ete.; also X;,/sin@ stood in place of - aX: and w; stood for os Thus the
sin @ do dé
present X; has not precisely the same meaning as that letter bore previously. The
present notation has the advantage of replacing two letters—connected through an equa-
tion—by a single letter; but it in no respect adds to or takes from the solution as
first enunciated.
§ 15. In order to apply the surface conditions (6) we require the typical terms in
the expressions for rr, 76 and rs. Referring to (3) we easily deduce from (29)—(82) the
following values:
r=—
is : aaNet p
{(9; +. 3), A ay
acy | (i+ 3)m +0 +2 Ln} eh:
+ {(®-i-—3)m+n} ¥,| +2(¢-1) nr Z;
1 pie x e ° —pa te
at |-! (22—1)m—(#@ +7 —-1)n} ee {(@+3¢-—1)m+n} ¥i,| —2 (0+ 2) nr*3Z_e,
Per (33),
=_@a t+1 4 pV: _ti+2)m—n iV 2(i-1)n ., ;
eae | %+3" m+n ~ (i +1) (27 TaD ye ee a nae
atk Vii. (@-1)m—n 2; ee + eee
iio 5 rio
2-1 m+n 7(21—1) se aire
nr 87 +4]
1
” sind i [(@ — 1) r*7X; — (i + 2) r+ X¥__] ...(84),
AND SPHERICAL SHELL. 351
ri VY, + nF,
r
aL ani pV; t(¢+2)m—n
* sind dd| 2+3”” m+n ~ @+1) 21 +8)
j3PV-in (@-1)m-n roy, +25 2+ 2) 97 «|
2(¢—l1)n
z
Pudge’ cage oe 1) i+1
— 1a aT —1)r7X;—(¢4+ 2) r7*?X_,] ...(35).
It should be noticed that im the typical terms in the displacements and the stresses,
the terms in V_,,, Yj. ete. may be deduced from those in V;, Y; ete. by simply
writmg (—7—1) for (+72) throughout all coefficients and indices; the converse mode of
deduction is of course equally correct. This fact is an important aid to simplifying the
algebraical work of evaluating the arbitrary constants.
§ 16. The surface values of the stresses (33), (84) and (35) are to be equated to
the given surface forces. Thus over r=a we must have
Be oe ee an,
Se SAB: intl Ge? Hw serie abe”
and over r=)
dT, a Is aT? 5 tale eh dT;
~ dé 'snddb’™ snddd do’
where R;, R; etc. are surface harmonics as explained above.
These six equations ae lead to the following six :—
(Calta: CN (el UD) Gy feat, ra nil 1a ota qupdobe nogedooudoesocaL (36),
(COSA GEN (GSS) oa GE We aon soe aohbHbodnoocoeubencee (37),
- . Maf¥i +2 (1) na2Z,4 EF DMEM ay, 204 2) nae,
t+ 8) mee See OLA CN 0 Gees ee
Be; 2+3 omen . 21-1 Mae +n BeH(Eh))
_t(i+2)m—n ip ne ACSIA eS oe 2(¢+ 2) aia:
meaner aa a
=T;+ tel nat pVs a ye PVE RT WONT oe 6 oe ae eee on (39),
95-3 mtn Ww-1 m+n
(F®-71-—3)m+n
ties) oy 65 GAY ar oe me
OS YE — 2G 422) nb ZF
2+3 21-1
_ pr, 2643) i f Ate i=l)n, sali (2i ame o tt ]) nop a (40),
_ 7/4 x ° nbi ie n vg jm Be ROP PRRR e oe c ici (41)
352 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
These six equations clearly constitute two independent sets; the first set, comprising
(36) and (37), determines the two unknowns X; and X_;,; the second set, comprising
(38), (39), (40) and (41), determines the four unknowns Y;, Z;, Yi, Zi1. The equa-
tions are to be regarded as simple equations, in which the right-hand sides are known
quantities.
There are no terms in V; or V_;, on the mght of (36) and (87), and so the values
of X; and X_;, are independent alike of the bodily forces and of the surface forces de-
rivable from a potential 7;.
Again there are no terms in X; or X_;, in the expressions (29) and (30) for 8
and uw; thus the displacements depending on X; and X_;, do not contribute to the
dilatation and have no radial component. They constitute what were termed above “pure
transverse displacements”. Owing to their great simplicity it is convenient to regard them
as next in order to the pure radial displacements.
Cuass II. Pure transverse displacements.
§ 17. From (36) and (87)
X; =(e?T, —bYT,) = {¢-l nero), )
: : cqanaify, eacesen suseuene 42).
X_« = (ab)? (6° T; -—aOT,) + {G+ 2)n(e@r- b#4)}) ey
The corresponding displacements are by (31) and (32)
it 1 d rt (eHeT; onic bit2 T;) a (aby (OT; =e T,) :
aig n (a — b#*1) sin @ dd | T= pels 142 a s|| GoasonEcaac (43),
if 2 ; F
= — 0 (gin — 64) dO [same expression as in square brackets in value of v]............ (44).
For such displacements 6, as already stated, is zero and the only stresses existent
are 78, 76 and #. The two former are given by
Ay he
a+! — ft sin 0-db
7 = [ri7 (ait; —b'T,) — 1 (ab)? (6'T; - Pca 0) | a (45),
= 1 d ; é é egy 1 fA :
= — Sn BA do [same expression as in square brackets in value of 7#]...... (46).
Having regard to (3) and (25) we may throw the value of # into the form
ro 2
= — aa Ke +1)+2 | [same expression as in square brackets in (43)]...(47).
The case when T;, T,’ are zonal harmonics merits special attention on account of its
great simplicity; for it v and 7 are everywhere zero.
AND SPHERICAL SHELL. 353
In the case of a thin shell we find from (43) and (44) as approximate values with
our previous notation
Me ee ppb od Dg h—€\ ie NeN S| f
v= G— 1) G+ 2) nh an) da [e + man ) T,- (1 => i 7) 77 | MoTstejefe\elafalaiajo\e/a (48),
=- aE =I = [same expression as in square brackets in (48)]...(49),
2 Slit Gli DEN a. hl gh &\ 3
9 = 9 a |e (1+) t +5 (1-2 5 )™ npumonbondace neoBanabBBooe He (50),
ro = — £ 9 [same expression as in square brackets in (GU) Nese cocdeee ere ne (51),
epithe lees sat Bll ire » & || / *) ges ( ™ =) A 2G
a @=1) G4 2Qyh [i@+ +2 gp |[(1! Ls 1 i T; Freres (52).
Owing to the similarity in form we need consider only one of the two displacements
and one of the two stresses * and 7. We may most conveniently select w and +4, because
in the case when T; and T/ are zonal harmonics vy and vanish.
Attention must be paid to the directions in which the surface forces are measured.
At the outer surface the positive direction along ¢ is that in which ¢ increases, but
at the inner surface the positive direction is that in which ¢@ diminishes. Thus the
applied forces at the two surfaces are in the same or in opposite directions at corre-
sponding points,—i.e. points on the same radius vector,—according as
dT; d dT,’
do *° “a6
are of opposite signs or of the same sign.
§ 18. There are two principal cases, of a character precisely analogous to the two
that presented themselves in the case of pure radial displacements. In the first case
dT, dT,
dé dé
is of the same order as the greater of the two = and oe in the second case the
former quantity is small compared to the latter. In the first case the statical resultant
of the forces applied at corresponding points on the two surfaces is of the same order
of magnitude as the greater of the forces applied at these points. In the second case
the forces at the two surfaces are approximately equal and opposite. In the first case we
get as satisfactory first approximations
a d
w=— G = 1) G 7 2) ih dé (T;- T;) Sie] e's/e{n\u sfexv}e(vinyoln\alniniala\hieiPint=/u/eielals]eleinsoie.eie)e(e\«, sisfaloie (53),
ihe a W608 2 dd? pp 54
66/1 = G— aXe a 2) nh |: (a + y+ S| [ 7 i | ATIOND OOAODODAS Omee Jagone (5 ):
354 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
Thus the displacements, and the shearing strain and stress whose axes 6, @ are parallel
to the surfaces, have nearly constant values throughout the thickness; also this strain
and stress bear to the other strains and stresses 7$/n, 7 ete. ratios of the order a: h
and so are relatively very large. In the case of these and similar statements it must
be remembered that the magnitude of surface harmonics varies over the surface, so that
terms which at most places are far the most important are zero, and may be vanish-
ingly small compared to the other terms, at certain points or along certain curves. In
order to avoid the prolixity that the continual reference to such special loci would entail,
it will be assumed in what follows that the reader keeps the necessity of such limitations
continually in view. He should notice that if either T;—T, or its differentials with
respect to a variable it contains be everywhere very small, while 'T; and T/ themselves
have their maxima values considerable, the harmonics must be of the same form and
not merely of the same degree. Also near loci where the principal terms in a displace-
ment vanish, the other terms may largely predominate, but the displacement all the
same will be but small compared to the values it possesses where the principal terms
are largest.
: ; d ,
To return to our consideration of the case when do (ti Ti) is not small, we see
that the conclusion it leads to is that when in the neighbourhood of a point on the
surface there is everywhere a considerable resultant tangential force—the forces tending to
pull round the surface in the same direction——there is a large displacement in this
direction, and the strains and stresses whose directions are parallel to the surface tend
to become large. The magnitude of these strains and stresses imposes an obvious
limit to the magnitude of the resultant of the applied forces. Noticing that a shearing
strain o is equivalent to an extension o/2 and a compression —o/2 along the directions
bisecting its axes, we should deduce from (54), by means either of the greatest strain
theory or of the mathematical condition that the strains must be small, the conclusion
that the ratio of the resultant of the tangential forces at corresponding points on the
two surfaces to the greatest traction permissible in a long bar of the material may be
at most of the order h/a of small quantities.
§ 19. We now pass to the case when S(t) bears to = a ratio of the order
hk: a for all values of 6 and ¢, 7.e. when the tangential forces over the two surfaces are
derived from the same harmonics and are at corresponding points nearly equal and opposite.
It is easily seen from (48)—(52) that all the strains and stresses are now to a first
approximation constant along the thickness. The stresses are now also all of the same
order of magnitude, and the same is true of course of the strains. The order of magni-
tude is the same as for the stresses and strains in a long bar of the material subjected
to a longitudinal traction of similar magnitude to the foree on one of the surfaces of
the shell; and thus this foree may now be of the same order as the greatest traction
permissible in a long bar.
§ 20, As yet nothing has been said as to the influence of the degree of the harmonic
from which the surface forces are derived; but this is of considerable interest and claims
AND SPHERICAL SHELL. 355
attention. From (48) and (49) we see that for given maxima values of the surface
dé
so fall off very rapidly as the degree of the harmonic increases. The formulae (50) and
(51) do not contain « explicitly; thus 7, and the corresponding strains depend to the
present degree of approximation only on the magnitude of the applied forces. A general
forces—.e. of etc.—the displacements vary approximately as 7~* when 7 is large, and
law applicable to # is not so easily laid down.
In a general way, when 7 is large we may regard the ratio of the maxima values of
dT;
dé
values of @ and the corresponding strain vary for a given magnitude in the surface forces
to those of 'T; as being of the order 7:1. We thus conclude that when 7 is large the
inversely as 7. A large value in 7 implies a rapid fluctuation in the magnitude and sign—
ae. in the direction relative to 6 and ¢—of the resultant of the forces applied over a
surface, the area throughout which this resultant retains one sign becoming more and more
restricted in the direction parallel to @ as 7@ increases. This consideration explains the
rapid diminution in the displacements as 7 increases. Take for simplicity the case when T;
is a zonal harmonic, when the surface force is everywhere perpendicular to the axis of the
harmonic and has a constant value round the perimeter of any small circle whose plane is
perpendicular to this axis. When 7=2 the surface forces vanish only at what we may call
the “poles” and the “equator”. The forces over one of the two hemispheres tend to twist
the sphere round the axis of the harmonic in one direction, and the forces on the opposite
hemisphere have an equal tendency in the opposite direction. It is thus obvious that as
we leave the equator, where the displacement will be nil, and travel towards one of the
poles along a meridian, the action of the forces over the successive zones into which
we may suppose the surface divided by “parallels of latitude” will all conspire, so that each
zone will be turned through a small angle relative to the preceding zone in the direction
of the forces. To find where the displacement is a maximum we notice that
Pi eae Bae ye eA)
dé
so that w is a maximum in latitude 45°. The angular displacement w/asin @ increases, as
we have said, right up to the poles, but after latitude 45° the linear displacement falls off
owing to the diminution in the radii of the parallels of latitude.
Now if we take for comparison 7= 4, we get
ro & wW sin 8 cos @ (7 cos? 6 — 3),
so that the direction of the surface forces and of the displacement changes sign not only at
the equator but also in the latitudes sin a Bi or a little under 41°. As we travel from
the equator to a pole the rotations of the successive elementary zones are in the same
direction only till we reach the latitude sin RiVGe or about 221°, where w/sin@ is a
maximum, and the latitude where the displacement w is a maximum is only about 21°.
There is thus much less room when 7=4 than when 1=2 for the cumulative effect
of the rotations of the elementary zones to produce a large displacement; and obviously
Vou. XV. Parr IV. 47
356 Mr 0. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
as t@ increases this is more and more the case, because the parallels of latitude where
the surface forces and the displacement vanish and change sign become increasingly
numerous.
A general idea of the reason why the stress #6 and the corresponding strain
diminish as 7 increases seems also easily attainable. The strain consists in a shearing of
the parallels of latitude on the same surface of the shell relatively to one another.
Now suppose a long flat bar of uniform breadth and thickness held at both ends to be
acted on in its plane by a series of forces of intensity +P on one half of its length
and —P on the other all perpendicular to the length. Then it is easily proved that
the maximum shearing force over a cross section diminishes rapidly as / diminishes though
P remain the same. This is of course intended only for a very rough illustration of what
happens, as the conditions it supposes differ widely from those of the actual case.
As regards 7», 78, since at the surfaces they must equal the applied forces, it is
obvious a priori that the magnitude of their principal terms can not depend on the
degree of the harmonic.
§ 21. The stress #, as we have seen, has under ordinary conditions a nearly constant
value throughout the thickness, but the variations of the other stresses along the thickness
are always rapid unless the forces at corresponding points on the two surfaces are nearly
equal and opposite. To consider the law of these variations, let ©, © denote the total
components parallel to @ and ¢ of the forces over the outer surface,—these forces being
assumed of course to come from one or a series of the T functions—and let ©’ and ®’
be the corresponding quantities for the inner surface. Then from (50) and (51) we find
as our second approximations
It is certainly noteworthy that the law of variation of these stresses along the
thickness is, to so close an approximation, the same for all forees applied over one only of
the surfaces, whatever be the degree or degrees of the harmonic term or terms from which
they are derived. A similar conclusion as to the variation of the displacements along
the thickness follows from (48) and (49), but the amplitude of the displacements depends
on the degrees of the harmonics as well as on the absolute magnitudes of the surface
forces.
Comparing (55) and (56) with (15), we see that the law of variation of 7 or
7% along the thickness of a thin shell for a tangential force over either surface is _pre-
cisely the same as the law of variation of rr in the case of a uniform normal force over
the same surface. Thus the stress gradient curve 2, § 10, will apply to the case of
tangential forces derived from stream functions over the outer surface, and the curve 3 to
the case of tangential forces over the inner surface.
AND SPHERICAL SHELL. 357
§ 22. Before quitting the subject of pure tangential displacements it is necessary to
point out that, in general, surface forces derived from a harmonic of degree 1 must be ex-
cluded from our solution. The reason will appear from a consideration of the simplest case
>
that of the zonal harmonic P,.
Thus put T,=®,P,, T=,P,, where ®,, ®,’ are constants. Thence, since
= = =sin 6,
we have ré = D, sin@ over r=a, (57
Ea Ge 7 SobLrPareHorocubmedcnacacoedasoorrd 57).
The forces over either one of the surfaces clearly all tend’to turn the shell in the same
direction round 6=0, the numerical magnitudes of the resultant couples being $7a*@, for
the outer, and §7b°@,' for the inner surface. Unless these couples be equal and opposite
there will not be equilibrium. We shall first show that when there is equilibrium our
solution applies.
For equilibrium we must have
T,/T, = D,//®, = (a/b),
Substituting in (44), we see that the coefficient of 7 takes the form : and so
appears indeterminate. The corresponding terms however in (45), (46) and (47) con-
tribute nothing to the stresses and consequently nothing to the strains, and so this term
has nothing whatever to do with the elastic problem. A displacement wa«rsin@ is in
fact a rigid body rotation round 6=0, and the magnitude of such a displacement is fixed
by other than elastic conditions.
We need thus consider only the second term in (44), or may take
= Gia Si OM teenectiate nee ueeieas castionest testes (58).
w=
This is the complete answer to the elastic solid problem in the present case.
We have clearly, however, not obtained a complete explanation of the elastic solid
aspects of the case 1=1.
It is obvious that the resultant couples over the two surfaces need not in a shell
always balance one another, while, if the sphere is solid, equilibrium under forces of this
kind over the one surface is impossible. When the applied forces are not in equilibrium
motion will ensue, but elastic strains and stresses will exist during the motion. Their
investigation requires account to be taken of the “reversed effective forces”. When this
is done it will, I believe, be found that when the initial circumstances are completely
given the displacements, strains and stresses at any subsequent time, supposing the limits
of perfect elasticity not to be exceeded, are as determinate as in any case of equilibrium.
The problem is an interesting one, but its present consideration would lead us too
far afield.
47—2
358 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
Crass III] Mixed radial and transverse displacements.
§ 23. The displacements are those represented in the formulae (30), (81), (82) by the
terms in V;, V_;, and the four harmonies Y;, Z;, Y_;,, Zi whose values are determined
by the equations (38)—(41). Thus the displacements of this class depend on the bodily
forces, the normal surface forces and that part of the tangential surface forces which is
derived from the 7, or potential, functions.
Let us consider the determinant whose terms are the coefficients of the four unknowns
in equations (38), (39), (40), (41), each divided by n. Calling this II we have
| (@—1—3)m+n
omy (F+4+3i-—1l)m+n
a’, 2(¢—1)a*, a“, —2(i+2)a7*
(20+ 3)n (21—1)n
i(@i+2)m—n oe 2(- 1) i at cream = Nmint 2(¢+2) =
~ @+1)(2t4+3)n ’ v : i(21—1)n : t+1
e (F-71-3)m+n i (2+ 3i-1)m+n
2 (i —1) b>,
bi, 2 (6 42)b-+8
(27+ 3)n ; (2i-—1)n
_t@ + 2) m—n,, 2(¢— VY) yi9 _@-Im-n ey 2 (t+ 2) Ae
((4+1)(22+3)n ? a ‘ v(2i—1)n ‘ iad
Denote the coefficients of the members of the first row in the expanded determinant
by the letters Il,, I, ete, the coefficients of the members of the second row by In,
IL.. ete. and so on. Also for shortness let
emt Ke +3) 4e+i- i} (a'IL,, + bI,) + +1) (Gn + pnt.) ee (\
P ene [{e i +3) = + e4i— 1} (a'TL,, + b'T) + + 1) (a’In + vtt.)] = Wy vesese (61),
p(m+n)* K
w43 2+3)™+e+i-1|
j (ai Thy + DM) + G+ 1) (aa + vt.) = aig iuni(62),
p | (m+n)7
21+3
(24 +3) 42 44-1! TL, + BM, + +1) GTI, + BM, | =o «...-.(68),
n s
p(m+ny"
m1 \ 2i — 1) = —(?+i- »} (a7, +6710) +1" Ty, +b-M1,) |= wm, ...(64),
AT
ae = om \2i ai) = —(#+i-1 ) (a1, +b“ T,,) +1 (a-“ 11, 40-7 Tg tis
p(m+ny” +n)y?
pim+ny" |} (1) "—(@+i-a} Maine
Sir {
(2i — 1)" — *—(+i-1)} (a*"'0,, FO, ) 41-71, +6, a (67).
AND SPHERICAL SHELL, 359
Then
nlY; = ARAL, + Ty + Ri + Ty Tg + Very + Vigyoy! ..-..0eeeceeee (68),
VA ase AOS el Dh ee AID 5 TS Vor cs Ve eect) ona eenaneRenEe (69),
Neely aS WAN te ADs ID ote Ware We te” Goacedsonnonour (70),
mIZ_, = Ry, + Ty, + Res, + FT +e Vig te Vigyory cocecerece sees (7)
§ 24, This constitutes from a purely mathematical standpoint a complete solution of
the problem, but to render it of practical value we must evaluate the determinants.
We find
TI = 4(¢—1) (i + 2) (ab) TT + {2 (4 + 1 (22-1) (27 +.8)} .......00e SpodGoe (72),
where
TI = (ab) | {ce + 40 +3) ~ — (20+ 1) \@e +1) - +204 i} (a — be) (qri+s — pris)
—((—1)1(@ +1) (4 + 2) (27 — 1) (27 + 8) (m/ny (aby (@ — »| sod((/8)))2
4(¢—1)(¢+ 2) a7
v(4+1) (22-1)
| [eee 1) a+ 21+ 1} a? (a — hb)
+%(t+ 2)(21—-1)(m/n) be — v)| sieissiosatssels (74),
_ 24:2) a7 974 m ! a? ee } ie ees
Da= eer Darey ae+ys +241} 45 +2) — 1h (ait — Bits)
+(¢—-1) (i+ @i+3)™ la 1)=— 1 be (a? | eS aaa (75),
2 =46=1) G42) a7 b> | “A . Oe ph epee
= iG+1FQi+3) (20 + 47 + 3) rs (20+1)} (a b™ 8)
+(¢—1) (+1) (2143) (m/n) a (@— v)| ch eee (76),
ll —2(i—1l)a" 03
uP (G+1)(Qi—1) (+3)
| {ce +404: 3) G+ De l fogs ~1)— =1) b? (a? — p)
+1642 @i-1) HG +2) m1 fens (—09| me ae
ae (@ —1)(@+ 2) ab"
A i(a+ 1)(2—1)
| {ei 1) = 0) aE 1} @ (a — b>)
—(¢+1)(+2) (27 DFE (@-0)| Mista wertatisn ater (78),
360 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
2(i+ 2) a3 5 ‘
De= G5 DOE »er+ 1)" + ais tht —i-3)™ 41) (a2+8 — p2i+8)
(Dek 3) \(e+3i— 1)" 4 i} be (a2 -05] ba sees (79),
—4(i-1) 4203 /(. : tte BAB
ll,,= Eee ch mt {er+ai+a)” — (2+ »} (a2*+3 — D243)
—(@-1) 7 (24+ 3)" a+ (q? — & | Beare ae (80),
—2(¢-1)a—" b> m
oa 7 (¢+1)(2i—1) (21+3) {er ae ae +h x
{@ S51) “+ 1} (a — b1)
6-1 )G-:2) (ty me —i+s)™4 i} a (@?— | dete (81),
II,, = I, with @ and b interchanged \
II, = IL, »” ” ”
IIs = Il,; » » »
IT, = Il 4 ” ” »
i IL, : ‘ ‘ A gh hed ise Oe eee (82).
1g = Il. ” ” ”
II, = TI. » »” »
Vil = i ” » ”
The last 8 relations are obvious, since the third and fourth rows of the complete
determinant II may be deduced from the first and second respectively by writing } for
a. An inspection of the determinant also shows that we may deduce II, from II, I,, from
II,., I. from I, and II,, from I. by substituting (-i—1) for (+7). It is thus in reality
necessary to calculate only 4 of the 16 minors.
We also find
_ 4(¢-—1)(¢+ 2) (ab)** pn x“ Oe
“= S6 fi) Gi Crease Ke +)" 4241} 2
{ei +3) = — i} (a — B?-) (q2'+8 — J2/43)
+7 (i+ 2) (27 — 1) (27+ 3) 2 ~(~ - i) (ab) (a — vy | sancatioeases (83),
2(i+ aylair+ 1242141}
G@+l)(i—1)(ai+3)
{i +2)" — i} p (ab) (qi — Beit) (243 — BPt8) eee. (84),
AND SPHERICAL SHELL. 361
—4(6=1) 6 +2) (21 can
ak GEG 2) 1h p (ab)*(a—) (ari# — B28) .(85),
L=2@ =I CF2)m(,. 9) 8 (42 — 2) (q2itl — Arti
= (ENING (+2) — 1} pa) (a? — b*) (a Qe cians seoarsviisensd (OO);
,_—*@=1) @+ 2) (21 +1) (,. ell ee ee eres a
a @- eQfiy {i —1) et i} pcb) 4 (@ 8) (a —b2-) ......:..(87),
a mae a ~ a \( m1) ah p (ab) (a? =D) (a BH) ccc (88),
,_4(¢—1) (4.4 2) (ab)-*-4 pn APSR ORR TID. ee )
“i 4G +1P (2-1) (+3) m+n | {es a Spe 4
(24 — 1) m+ 1} (ati — pe) (grits — pri+s)
l
—(i— 1) (6+ 1)(2i—1) (27 + 8) = (= i= 1) (aby (a? — Hy] SIRs ose (89),
; 2(i—1) RR RL ‘
= ®(—1) (i+ 3) ie +41+3) ak (20+ } o
(G-1)"4 i} p (ab) (qi — Bi) (qi — Bt), (90).
§ 25. Substituting the values just found for II, ... a... ete. in (68), (69), (70) and
(71), we obtain the values of Y;, Z;, Y_;,, Z_i1; and inserting these in equations (29),
(30), (31) and (32) we have the typical terms in the values of the dilatation and
displacements explicitly determined. The solution so obtained, it must be remembered,
includes only what we have denoted, § 23, “mixed radial and transverse displacements ”.
It answers both to bodily and surface forces; the types of the former are derived from
the potential (see § 14)
Vit OV;
the latter have for their types (see § 16):
over r=a,
aT; caries 1 aT;
do’ sind dd’
362 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
The dilatation and displacements are as follows, II being given by (73):
t7+1)(214+3)2 ei—1 __ ei—1
nls = R; eo {(@e +1))= "4 2i+1) a (a — 0? )
+7 (¢ + 2) (20-1) (m/n) P- (@ -»}
_t(r— Dat
{(@e +4643)" — (i+ 1)) (zits — p2its)
ay
+(¢— 1) (@+1) (27+8) (m/n) ev} |
+ R; [coefficient obtained from that of R; by interchanging a and b]
{(@e P41)= + 2i+1) a? (a — b>)
Deon
—(@+1)(@ +2) (21-1) (m/n) PH (ee - mh
i(i+1)(2i-1)
Oe
a {(ce + 4¢ +3) ™ — (25 + 1) (ai+3 — B2i+3)
n
—(t —1)7¢(2¢ +8) (m/n) a (a2 - »)} |
+ 7; [coefficient obtained from that of 7; by interchanging a and }]
= m 2-1 _ fri-1 2%+3 _ Arts
aon Ki E {(@e +02 "421 +1) (a2 — Be) (qeits — Brits)
+4 (6 +1) (6 +2) (2i— 1) (21+ 8) (m/n) (ab) (2 — vy}
— i (2i— 1) (21 +1) \( +2)™ — i} r (ab) (@ = B) (a8 = ie)|
m
+pViii |- (i+ 1)? (27 + 1) (27 + 38) \( - De a 1} r* (ab)-* (a? — b) (a4 — 5)
_@+1) ee
(ab)
=| (cae +46 + 3) (ee 1)) (a2 — br) (a?+3 — B43)
—(i=1) i (+1) (Qi — 1) (26 + 8) (m/n) (ad) (a2 — vy] (91),
2nIlu
S [- wre (i m -2) |(@e+ny+2i+ 1) a?(a*1 —b*) 44(i +2) (2-1) - r(a—bh
+5 am {( {( (+1) +2 +1) (i+ 2)™ -1) (az+ — p2i+8)
+(i-1) (+1) (24 3)™(@- 1) ™—1) 0% (0h
AND SPHERICAL SHELL, 363
+i(@+) "+ 2) aR 5 {(e +4143) —(2i + 1) (et? — 08%)
+@-1) G41) (26+) a (a - mh
t+17-
74+2a5
51 (2i-+ 46 +3) — (21 + 1)) (@-1 ye 1) Be (a2? — 9)
+i (6 +2)(2i—1) = (G+ 2) 1) a (a? — vy]
+R; [coefficient obtained from that of R; by interchanging a and }]
eT, E (§+1) & ig 2) aaa (2P+ 1) +204 1) a (a — 5)
—(i+1)(i+ 2)(21-1) = be (a2 -»}
t@+1) ro ES m : m 4 4
TED ayn (2? +0 F +242) (@-4-B F +) roe
mm
—(@-1)i(@i+3)™ (@ +3i-1)" +1) (eB)
+i@+)) (7 (+1) +2) sap {(@e +4143) - @i+ 2) (eH
2G —a)ya 8) — a (a — bh
mm
= Alga aes 5 (Qe +4i+8) 7 —@i+1) (@+9i- F +1) B@— be)
4+2 abd
m
—(¢+1) (¢+2)(2i-1) = (@- t— a) aad 1) TE (GF —w}]
+T; [coefficient obtained from that of 7; by interchanging a and 6]
mm
; tt = ie £5 Fonsi ANE é
+ ipV; bow {(@ +1)™ +2) +1) ( +1)" - 1) (a — bei) (qzits — baits)
m (m
+¢i(t+ 1) (@+2)(2¢—1) la 2) (ab) (a? -1y|
Bos We 5 . il) ee een vee
a Oa +1)" 42141! {é+2) = i} ayn (2 +1 _ Bit1) (q2its — p2i+3)
+i@isp {ean +2bler 2)~ - (a? — b*) (a*+8 — $23)
-i@+1)Q@i-1)™ ie = 1 py
Vou. XV. Parr IV. 48
up Di
364 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
5 t+1
er are |- G+HD@i+n(i*- 2){¢- + i} ae (a? —B) (a —
me
+7 (¢+1) (22+ 3) — M \G—1) B+ } eae (a? — 8) (a4 — pit)
rr
a7 (ab)***
me
(@ +4143) - (21+ 1) Gr eg. 1) (az — pei-2) (qaits — Beis)
ne
=Ga1)e641)(i48) m (i — 2) (ab) (@- -¥)|
m
-i45 31a + 4a +3) nate ah {a C=) + i} Cpa my (oy | --.(92),
+7 (i+2)(20-1) ea + jaye (P+ DF +2841) (16422 —1) ome de)
m
+@-DE+Y A+" (@ 17-1) 6 (@— oy}
me
-\¢ 2) 2} a 7 {(@e +4i +3)" —(2i+ 1)) (azi+9 — pe+2)
+(¢—1) (64+ 1) (2¢4 3)" at (q2— w)}
i ears a , m F m uta ks.
+3 a5 (@ +4143)" —(i+1))((@-DE-1) b@ — 51)
+4 (¢+2)(2i—1) = (i (i + 2) ~ = 1) a @-)}]
a [ covticient obtained from that of 7 by interchanging a and |
= E \( + 3) = a (aby a {(@i+ 1)— +2i+ 1) a? (a — $2")
m \-% a+1 r-
= G41)G+2) 21-1) ZO (@— byl an (2+) 2+ 2i + 1) x
G -i-3)"+41) (az+3 — p2its) — (-1i@i+a™ (~ + Sis +1) bt (q? -»»}
—(i+1){@-2)™— 2} ap {(@# +4143) ™—(2i+1)) (a? — be)
— G-1)i(264+8) = ar (0)
1 yi
ad , m . 2 é Aedes
+79 pp (244i + 8)2-@r+y) (431-1241) — bi)
— (i+ 1) (4 + 2) (2 -1)* (= 1-3)" +1) (a — 0)
AND SPHERICAL SHELL. 365
— cocficient obtained from that of = by interchanging a and |
dV; pit Es ha m BO; .
a0 (dey dé |- (aby {(@i+ 1) he + 20+ 1) G - a — 1) (a? — pr) (grt — peits)
+4 (¢42)(2i— vale +3)" + 21) (abe (@ = *r}
a {2 +1)" 4214 1} \( +2)" — (ait — be) (qzits — p2its)
l Fe
Ber 41) \¢ — 2} {Gi +2) -1} or (a? — b?) (a?+3 — pr4s)
+0? (i=1)— AG G+ 2-175 Sane - (a? — b) (a4 we)
BV See: ie +4; ea eae ep oma
= BS a Bae ase 1 — p-)
— (i +1) (284 3)™ ra ~1) "4 eee anes = (a? — B*) (wi — be)
yt
* (ab |
m
(Qi + 40+ 3) ™— (i+ »))( (= 2)" +1) (a — be) (a9 — Be)
—G@-1)G+1)Qi+3)— (@ — 2) 4214 2) (ab)"> (a? = vy
E ne Rete m
ty (a+ 4+ 8) P- i+) {e-y "+ 1}
pi
(a7 — 21) (aH = ven | emere (93).
aa (aby
The value of w is obtained from that of v by replacing Jf by pcs In any
dé sin 6 dd"
one of the quantities 6, uw, v, w, so far as they depend on the surface forces, terms in
r— may be obtained from those in 7**1, and terms in 7‘ from those in 7’, by simply
writing (—i—1) for (+7) in all indices and coefficients. The same substitution deduces
the coefficient of V_j;, in each case from that of V;. The quantity TI on the left of the
equations will be found to transform into itself, 7.e. to remain unchanged, when (—i—1) is
written for 7.
The solution just written down may at first sight seem rather cumbrous. It must be
remembered however that it contains the answer to innumerable special problems, and that
in very few practical applications will there be found anything like so general a system of
apphed forces as that treated here. Having regard to the actual facts, the comparative
brevity of the solution is in reality somewhat remarkable.
§ 26. From these typical terms in the displacements the typical terms in the stresses
may be found by means of the general formulae (3). Three only of the stresses, viz. 7, 76
and 76, are given explicitly below. They possess greater inherent interest than the other
three, more especially in the case of thin shells. The method by which they were actually
calculated was by substituting in (33), (84) and (35) the values deduced from (68), (69),
48—2
366 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
(70) and (71) for Y;, ete. The expressions are as follows, Tl being as before given
by (73):—
1
a
R; aoe (25-4 vf {2+ 14284 i} f
== A —{( ty 1) a (a at be) (rts — +8) + aw (= =s Bt) (ats ee b*+8)}
—(i—1) ¢(¢+ 1) (¢ + 2) (22 — 1) (27 + 38) (m/ny? a (a? — b*) (7? — BY) (4.1) a4 + ir}
+i(t+ 1) (i+ 2)(24 1) {eae +40 +3)" — (264 Dh pe
{b? (a 3 7) (rn = b+) —@ (7° ME b?) (an = pitt)}
y-i-3
+(i—1)i(@¢+1)(284+38)— 7 {Qi +1)— “4 2iet} 2! x
i —1 fite
{as (@ = 7?) Gn = b2+1) — Gr (7° x b?) (a 25 ran]
+ — a R; [coefficient obtained from that of R; inside square bracket by interchanging
a and b]
+4) TE cay | {i+ #i+3)™—(i+1)} *
\@ elon 12 of 1} (a — B+) (a? — 7°) (7241 — BH)
+(@i+1 {+ Bi- 1) + 1} {oe ~i-3) "+ i} be (a? — B) (7? — B) (a4 — 724)
™m
+ 2 (2% + 1) Cr — 1) (i (a Bt 1) = + i} {qt (a ent 7) (en et b+) _ ben (r° — b*) (a ‘= =]
HG) we! [coefficient obtained from that of J; imside square bracket by inter-
ar
changing a i b]
1! a
+ pV; i? \(+2)— 1} oe =
=({er+ 1)= + Qi+ i} (azi+3 — pei+8)
x {a (a* == r’) (it 1% b+) ae ea (r* = b?) (a me, pitty}
+(i+1)(t+2)(2i- 1)= (ab)"> (a? — b*) {b? (a? — 7°) (7? — b*) — a? (7? — 8) (a — ay]
able 2{i—1) =I 2 ; Le 2-1 _ G1
+ pVia i+ {G-) E+ i} a pe {2 +4i4+3)™— (24+ yf (a — p44)
x {a? (7? = b?) (ain a pitt) =; & (a = T°) (Coosa = b4)}
+(i—1)1(21+3) = (a* — b*) {b> (7? — B*) (a2?! — 4) — a (a2 — 7’) (7 wy] ava (94),
AND SPHERICAL SHELL. 367
na=" | {ee +1) 42141 {i+ 2)@—1} x
Cn (a (ae — 18) (8 — BSH) — Bn (92) (et pH
+{@r+4+3)7—-@i+D} {@-1) 2-3} x
cele aeem
+ (2641) jiG+2)2-1} (1B ES 7 (a — B) (7? — BY) (a4 — - #4)
n ain ait b3
dk; ; -
aa coefficient obtained from that ane = by interchanging a and b
mm
Gi 9*38 " m ;
+70 %4+1 aby | {ee + 44 +3) — (21 +n}cer il) = +2i+1} x
1G or 1) 2 (E> a8 6) (a**s — b#+8) + 1a2 (Ge = 6) (78 ek b+8)}
—(¢-1) i (+1) (6+ 2) (2i— 1) (2+ 8) (m/nP BA (@? — 8) (PF? —B) {64 1) r+ + ta}
4i(¢+1) (i+ 2)(2i- 1)= {ea +464 8) — (2+ yh be x
{a? (7? =e b?) (a ak =n) —s & (a? ae 7) (1 = b?+1)}
4(i-1)i(i+ 1) (2i+3)™ {(2ir+ 1) 42141) a? x
{p24 (7° can b?) (a — aH) pond qo (a —_ T°) Conait = me]
GLE I ae
+00 % 41 (eb)
| cocficient obtained from that of a inside square bracket by inter-
changing a and |
dV; . ile atone are ee 8 “
+P 6 iMG + 2)= -1} (ay {ee +1) ait 2+ 1 (a+3 — 578) x
eee (a? —_ rT) Gh Le, 51) pee Gr (r° a 6?) ("1 = 7)
+7%(i+2)(2i-1) ~ (ab) (a? — b?) {a? (7? — B*) (a4 — 7) — B (a? — 7°) me)
d os
+p G+ I) {@- 1+ 1 oF = =| {ee +4143) ™— (21 +0} (a — b>) x
{b (a pues 7) ase = b) -@ (r? = 6?) (ah — qaahyy
+ (iI) G+ 1) 243) ~ w= b) (B (72 — BY) (G9 — 1")
= a (a? — 1) (4 = vm) arte 9 0 ued (95).
368 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
i
sin Odd’
the case of the displacements, the substitution of (~7—1) for (+7) deduces terms in 7~!
and r~** depending on the surface forces from those in 7‘! and r‘— respectively, and also
The expression for * is obtained from that for * by replacing 5 by As in
the terms containing V_;, from those containing Vj.
§ 27. The complete expressions for the displacements wu, v of the third class are found
by summation with respect to i of the typical terms given by (92) and (93), and a similar
summation is of course required of the typical terms given in (94) and (95) for * and 79.
The complete expressions for w and ré are derived from the complete expressions for v and
WSS se xa!
oy ae Oe Se
in @ dp dé
regarded as i=2 and i=. The case 7=0 would answer to forces, such as uniform normal
r® by the substitution of | The limits of the summation had better be
tractions, whose values are independent of the angular coordinates; and the correct solution
is in reality derivable from (92). We already, however, have considered it, treating the
displacements so produced as of a separate class, and have given the solution in (9). It is
in fact easily verified that if m (92) we put 7=0, and replace R;, R;/, Vi by R, RB’, V’
respectively, we obtain the corresponding terms in (9). The terms in V in (9) are not
represented in (92). The potential from which the bodily forces answering to the solution
(92) are derived satisfies Laplace’s equation V’=0, or answers to forces other than the
mutual gravitation of the shell. But the potential Vr? answering to (9) includes mutual
gravitation and “centrifugal force”, neither of which satisfies Laplace’s equation.
The case ¢~=1 must in general be excluded from the solution for the reasons stated in
§ 22 in the analogous case in pure transverse displacements. In any particular case where
forces involving harmonies of the first degree are distributed over the two surfaces of a shell
in such a way that the entire system of forces is in statical equilibrium the solution (91),
(92), ete. will give correctly the elastic displacements.
§ 28. The forms under which the stresses 7, 7, 7d are presented may seem at first
sight rather peculiar. They have been adopted with a view principally to two ends, viz. to
afford a ready means of testing the accuracy by reference to the surface conditions, and
to facilitate application to the case of thin shells. The coefficients are all constructed
on a uniform and very simple plan, Take for instance the values of 7» depending on
Rk; and T;. In the case of R; the expression inside the square bracket must by the surface
conditions vanish when r=b, and a glance shows the occurrence of r—6 as a factor in every
term. The terms in the last 4 lines contain in addition the factor a—r and so vanish
likewise over r=a. The first 3 lines inside the square bracket on the other hand when a
is substituted for 7 fall at once into (2i+1) II, and so the surface condition **=R; over
r=u is seen to be satisfied. The first terms are those which are of most importance near
the surface where the corresponding stress is applied, and in the case of a thin shell these
terms are of a higher order of magnitude than the subsequent terms which vanish over both
surfaces. The expression for 7 in terms of 7; has to vanish over both surfaces, and so is
arranged .to show the factors a—r and r—b in each term.
AND SPHERICAL SHELL. 369
The terms which vanish at both surfaces can be thrown into a variety of equivalent
forms, some more convenient for one purpose, some for another. Use may be made for
instance of the identities:
qr (a? — 7°) (tH b+) — fa (re — oD) (ah — pitty
= r{a 2i—1 (e-—r 2) (pt) — fi) — hia (7 — b*) (a — 7)}
22 7 = Bra) (a%+ p= party 7 (Gs — je) (v1 a b#t2)h
In what precedes we have always described b as the radius of the inner surface and
we shall continue to do so. But from the form of (92), (93), (94), (95) we may clearly
in these equations regard a as the radius of the inner surface if we take the undashed
letters R;, 7; to denote the forces applied over that surface. When the outer surface is free
of force the reader may find it a saving of time to take this view.
§ 29. We pass now to the consideration of the form taken by the displacements
and stresses in a thin shell, The expressions given below for rr, 7 and 7 were calculated
directly from (94) and (95); and the values of u, v, w might similarly be derived from
(92) and (93). As a matter of fact, however, the displacements were found by inserting in
equations (68)—(71) the approximate values found for II, IL, ete. by expanding the
expressions (72)—(90) in powers of h/a, where h is the thickness of the shell. To save
- space these approximate values of II etc. are not recorded here.
Denoting Young’s modulus by £ and Poisson’s ratio by », and putting as before
a—b=h, a—r=, we find, retaining the lowest and next lowest powers of h/a and é/a,
the following results* :
AS (207+21—1)m—n sen “|
ae E (G—1)G+2)@m—n) "E a
piGe (20 + 21 — 1) m— n&
nh [; ( — 1) (+2) (8m — aay BOE Dry ;|
re a 1 nh re a 1 — 2h/a nh
Sent) r= NG+2) # ' ste OD la DG+2)'B ;|
4 gh Ms sies +1)m—n} th mw Era (a+ 2 —n} h
(i—1)(3m —n) Ha 2(—1)(8m—n) a
i
— gil Vain ei eis. mE (@+1P{(G-l)m+nh :
i | WEEN GW=n, PO Ha? 2642)@m=ny alot (96),
_ ak; @ 1 n2é—h|_ dkRi @ 1—2h/a _ Rah
~ dO 2nh sss a d@ 2nh|(@-1)@4+2) HL a
dT; a 1 _1h-€) dla 1 — 2h/a 1é
dO nh|(@—1) +2) 2 a dé nh |(i—1) (i+ 2)
4 of ? dV; 1 see te ((+2)m—nh
2n dé 3Bm—-n a t-1 3m—-n a
ag fa ea|f ul eae (—1)m+nh (97)
Son de Nie 3am —n a V+2 8n—n Va) $
* The reader must bear in mind that these results answer
only to the displacements of our Class (iii), ie. to the
system of applied forces given near the beginning of § 25
above.
370 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
e=R~+|1- al i [t+ CoD Esa mo 2h F)
m+n m+n
m+n ah
Bm a 6m a
=a (gt 1) (1 t+) mtnhk , ee
m+n @ 6m a 6m
+ 7/8641) —— eC 1 a ee ee ed
i a (@+ 2)m—n&(h—€)[, _ (@+6t+2)m—-mh— — E
— ato V-7%2 =
gels m+n a? 3 (8m —n) mt Ve
40-p Vig G+ 1)" (@—1)m+n E(h—&) f)
m+n a
(i244 —-3)m+(tt+1)nh-2€ |. aé
j1- 3(8m—n) ar (i+ 3) = siebaje Satisie seer (98),
= dR m {(20? + 21-1) m—n} E(h—- &) 1 wees + 21 +1) mn — 2n*h
Mt dO (m+n) (3m — n) ah 3m {(202+2i-1)m—n} a
nm 3 (50? + 51 — 3) m?— (027 +748)mnt+nr&
3m {(22? + 22-1) m—n} a
_ aR m (20? + 21-1) m—n} E(h— &) ie 3 (30? + 81-1) m+? +t—4) mn — rh
dé (m+n) (38m—n) ah 3m {(2i? + 21 —1) m—n} a
RSME 5t — 3) m? —(v? +0 + 8) mn tn? E
3m {(20? + 27 -— 1) m—n} a
dT; h—-€| 5 aes ee uy El, _(@+t+2)m+ nh-E
ad a = s. ——,
dé ih m+n + a8 h m+n a
aig Vi, (E42) m—n} (i+ 1) mtn} EU-8 ,
P de (m+n) (3m—n) (he
a a Pe E
[2-5 3 {Qi+ 1)m+n} =i hs 2)£
1p Va 4 lG Hho icy OAR
i @+1) (m+n) (3m —n) a
((+1)(Bim—n) h-2E é
E PU CHEE re +43) 5] some (99).
The value of w may be got from that of v, and the value of 7 from that of 7, by
vere 1. Ud d
substituting amo dé for da°
§ 30. Noticing that En--# we find, retaining only the algebraically lowest powers
of h/a,
aa, tw Ee Mince) ateeae ne, Oy ase eee (100),
sake t
AND SPHERICAL SHELL. 371
where Hi ee — te tiaras piV ag (Util) mts pee ane. Atcissesvtlesedeesieetoet (102),
and k as before is the bulk modulus.
Obviously F; is the total radial force per unit of surface, at the element considered,
arising from all the bodily and surface forces which contain harmonics of degree 7.
With the exception, as explained below, of cases in which 7 is very large, (100) and
(101) will be satisfactory first approximations unless #; be small compared to the individual
forces R;, R;, etc., of which it is composed. These results are the exact equivalents of the
results (18) and (19) for uniform normal forces.
§ 31. Before examiming more minutely these and similar results, it is convenient to
form some idea of the magnitude of the strains and stresses. The actual determination of
the greatest strain and the stress-difference is complicated by the fact that the directions
of the principal strains and stresses at a point will not in general coincide with the funda-
mental directions 7, @, ¢, and also by the fact that the magnitudes of all the terms involved
fluctuate over the surface. Exact determinations are apparently possible only for particular
cases treated individually. Without actually calculating the greatest strain it is, however,
fairly obvious that it will in general be a quantity of the same order of magnitude as the
greater of the two expressions u/7 and 20 whose sum constitutes the stretch along @.
This consideration enables us to reach some important conclusions for the cases when
all the forces act on the surfaces. Let
R;- £; = JN.
@ 2p ie
TBE OO NA ayy oe on eee ee (103),
1 d ‘j ' =,
and ashe Sale Linhares
so that F;, @;, ®; are the components along r, @, @ of the resultant of the forces on both
surfaces derived from harmonics of degree 7. Then, retaining only the algebraically lowest
power of h/a, we find
a (22+27—1)m—n a@ 4t(%+1)
Leal nh 2 (i—1) (i+ 2) (3m — n) i lice 2nh (4 —1) (i+ 2)’
-1ldv PF; a 1 dO, a 1
r dd d@? 2nh (i—1) (i+ 2) + a0 mh (@—1) (@ +2)’ ceccora. nor SU)
iol aie 1 1 d®;a 1
psn 0 dé sin? @ d@® 2nh(i—1)G+2)' sn db nh (i—1) (i + 2),
These quantities must in general not exceed the order of magnitude permissible to
strains in the material, and this condition clearly cannot be satisfied all over the surface
unless F;, @;, ®; and their resultant be kept so small that their ratios to the greatest
longitudinal traction permissible in a long bar of the material be, at most, small quantities
of the order h/a. This condition will of course be satisfied for the components along r, 6, d
if it is satisfied for their resultant.
Wor, OY, IBA AYE 49
Sy)
“JI
to
Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
The condition that the resultant must be small must clearly also hold though bodily
forces act in addition; and, as the resultant of bodily forces per unit of surface will usually
be very small in the case of a really thin shell, even when their direction is the same all
along the thickness, this condition will in general be sufficient. The condition will, how-
ever, cease to be sufficient if the bodily forces are so intense that their resultant per
unit of surface bears a ratio of the order h/a to the greatest traction permissible in a
long bar of the material. This follows from the fact that the principal terms in the
displacements and strains depending on V; and V_;, do not cut out when
ia’ p V; = (a + 1) map [Poke —=3{));
Unless the bodily forces be of unusual intensity we may for a first approximation neglect
the terms containmg h and & im the coefficients of V; and V_;, in (96) and (97); but
if the resultant of all the applied forces along the thickness be small compared to the
resultant for one only of the surfaces, we must retain all the terms im these expressions
depending on surface forces. In such a case the individual forces R; etc. over either of the
two surfaces may be of the same order of magnitude as the greatest traction permissible in
a long bar of the material.
§ 32. One of the most striking features of (96) and (97) is brought out by a com-
parison of the terms in R; and J7;, regarding these as quantities of the same order of
magnitude. According to the first approximation the term in wu depending on &; is of
the same order of magnitude as that depending on 7;, and the terms in wv depending
on R; and T; are likewise of the same order of magnitude. These latter terms are in
fact precisely equal if R;=27;. Similar results follow a comparison of the principal terms
in R; and T;.
From these considerations we see that the magnitude of the maxima values of a
displacement whether radial or tangential depends rather on the magnitude than the
direction of those of the applied forces which vary harmonically. It should, however, be
: 3 E dR; dR; : : : :
noticed that, since for instance —," and —— vanish when R; is a maximum, the tangential
dé dd
displacements due to the normal surface forces derived from a particular harmonic vanish
where the radial displacements are a maximum. Also the radial displacements due to the
tangential surface forces derived from a particular harmonic will have their maxima values
at points where these forces themselves and the tangential displacements vanish. _
§ 338. We have next to consider the nature of the terms in h/a and &/a inside the
square brackets in the expressions (96) and (97) for the displacements. Supposing that the
resultant per unit of surface of the applied forces is a quantity of the same order as
the resultant of the forces applied over one of the surfaces, these terms—at least when
i is not very large—are to be regarded as of secondary importance. Being linear in &, these
terms have necessarily their mean values at the mid surface. Again the coefficient of & is
in every case positive. Thus to a second approximation the displacements numerically con-
sidered, when they vary with &, have their maxima values at the inner surface, their mean
values at the mid surface.
AND SPHERICAL SHELL. 373
The fact that the radial displacements arising from tangential surface forces are, even
to a second approximation, the same at all points along the thickness is worthy of notice. It
shows that while, as we have seen, the radial displacement arising from tangential surface
forces is similar in order of magnitude to that arising from equal radial forces, the radial
strain in the former case is small compared to that in the latter.
It will be noticed that when surface forces alone act, even if the total components
F;, ©;, ®; for the two surfaces absolutely vanish, the values of u, v and w—and con-
sequently of all the strains whose directions are parallel to the surface—are approximately
constant all along the thickness. The values of these strains are in general of a higher
: . du *76 7b .
order of magnitude than those of the three strains = = and =, but this ceases to be
the case when the forces at corresponding points on the two surfaces are nearly equal
and opposite.
§ 34. We have next to consider the influence of the degree of the harmonic on the
values of the displacements. When 7 is large we shall regard at as of the order if; etc. ;
dé
i, = and oe as of given magnitude.
From (96) we see that the radial displacements arising from radial surface forces have
neither their “principal” nor their “secondary” terms much affected by the value of 7:
but when z is large the radial displacements depending on tangential surface forces have
their “principal” terms varying inversely and their “secondary” terms directly as 7. This
latter law applies also to the tangential displacements arising from radial surface forces.
The influence of the degree of the harmonic on the tangential displacements arising from
tangential surface forces is even more important, for when 7 is large the magnitude of the
“principal” terms varies inversely as 7%. We notice that in the case of surface forces
the “secondary” terms in the tangential displacements when 7 is large bear to the
“principal” terms ratios of the order 7*h/a, and that the same law applies to the radial
displacements derived from tangential forces. Thus, except for the radial displacements
derived from radial forces, the importance of the “secondary” terms relative to the “ prin-
cipal” increases very rapidly with the degree of the harmonic from which the surface
forces are derived. In fact when 7 is very large 7*h/a ceases to be small and the “secondary”
terms may be of as great or even greater importance than the “principal”. In such a case
we ought not to rely on (96) and (97), but must have recourse to (92) and (93) to ensure
that we do not neglect terms of the same order as we have retained.
and we shall regard R;, R
dV;
In the case of bodily forces when 7 is large, if we treat 7V;, "8 ° aV_;. and
— as of given magnitude, we see that the “principal” terms in w are nearly inde-
pendent of 7, while the “principal” terms in v vary inversely as 7. The “secondary”
terms in both w and vw increase rapidly in importance relatively to the “principal” terms
as 7 increases.
49—2
374 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
§ 35. We have next to consider the stresses. Of these the three @, 44, 6 have
“principal” terms independent of &. Thus unless the resultant force over the thickness
of the shell be small compared to the resultants for the two surfaces separately, or else
i be so large that terms in &/a become important, these stresses have nearly constant
values throughout the thickness.) The “ principal” terms in these stresses may easily be
derived from the displacements, the relation (101) being employed in the formulae for @
and & unless F; be small. These stresses unless F;, ©;, ®; be small are of a higher order of
magnitude than =, @ and ~; but they are of less interest in the theory of thin shells,
and further, owing to the variety of the differential coefficients they contain, they can
hardly be considered satisfactorily except by treating each individual case by itself. It is
thus sufficient to point out that the conclusions to be derived from them, through the
maximum stress-difference they supply, as to the magnitudes permissible in the applied
forces, are of the same character as we arrived at by considering the strains.
We now pass to the stresses 7, 7, 74, and since the two latter are exactly similar
in form we need not consider 74 separately. We shall as before speak of the terms contain-
ing the algebraically least powers of h/a as the “first approximation”, but in almost every
ease it must be borne in mind that when 7 is so large that 2?h/a ceases to be small
the “secondary” terms may be of as great or even greater importance.
In the special case when there are no bodily forces and when the surface forces
at corresponding points on the two surfaces are exactly equal and opposite, the “prin-
cipal” terms in rr depending on the radial forces, and the “principal” terms in 7 and 7
depending on the tangential forces are constant throughout the thickness. In the same
case the principal terms in rr depending on the tangential forces, and the principal terms
in *# and *% depending on the radial forces vanish. Thus all three stresses rr, 7, 7
show a remarkable approach to constancy along the thickness.
In general, however, when the forces at corresponding points on the two surfaces
give a moderate resultant, the rate of variation of 7, 7 and 7# along the thickness is
very rapid. The law of variation when forces of one type only—ie. either radial forces
alone, or tangential forces alone—act over one only of the two surfaces, is conveniently
shown as in previous cases by stress-gradient curves. The only novelty is that two
curves are now required for each type of forces, one, the “radial” curve, representing
the variation of 7+ with &, the other, the “tangential” curve, the variation of 7 and 74.
As regards both types of surface forces, we see that to a first approximation the
stress of the same type as the applied force—r being a radial, * and 7¢ tangential
stresses relative to the surface—has for its gradient curve a straight line whose zero
ordinate answers to the unstressed surface. Also the gradient, to this degree of approxi-
mation, depends only on the local magnitude of the force and not on the degree of
the harmonic it comes from. The stress-gradient curves of the opposite type to the
applied surface forces are to a first approximation parabolas, the maximum ordinates
answering to the mid-thickness, the zero ordinates to the two surfaces.
In the case of the bodily forces arising either from V; or V_j, the radial and
tangential stress-gradient curves are to a first approximation parabolas symmetrical about
AND SPHERICAL SHELL. 375
the maxima ordinates, which answer to the mid-thickness, and with zero ordinates an-
swering to the two surfaces of the shell.
§ 36. When we take into account the “secondary” terms, and notice that m—n
is positive in all known materials and 7 is not less than 2, we find that in the case
of radial surface forces the radial stress-gradient curve lies below or above the straight
line given by the first approximation according as the forces act over the outer or the
inner surface. These curves are shown in figs. 6 and 7, the dotted line referring to
the first, the thick lime to the second approximation.
R; Radial Forces. R’
Radial curves.
D
H
B A
Fig 6 Fig 7
As in previous curves B refers to the inner, A to the outer surface. In both
the thick line curves the gradient is steepest at the outer surface. This it will be
remembered is the opposite of what happens when the radial forces are of constant
magnitude over the surface (see § 10).
When the radial forces act over the outer surface the tangential stress gradient
curve given by the second approximation lies, as shown by fig. 8, above the parabola given
by the first approximation; but when the forces act over the inner surface the second
approximation curve, as shown by fig. 9, lies above the parabola given by the first
approximation only near the inner surface.
R; Radial Forces.
Tangential curves.
B A B “A
Figs Fig 9
The mode of distinguishing the first and second approximation curves is the same
as before.
The radial* and tangential gradient curves answering to the tangential surface forces
Tangential Forces. Tangential Forces.
Ty Radial curves. 1, T; Tangential curves, T';
Fig10 Fig ll Fig 12 Fig 13
* In Fig. 10 the thick line curve will lie completely below the dotted curve if i<4.
376 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
are shown in figs) 10—13. The notation and mode of representation is the same as in
the other curves. The tangential gradient curves, as in the case of Class (ii) displace-
ments, are of the same general form as the gradient curves 2 and 3 for uniform radial
forces.
The radial and tangential* gradient curves for the bodily forces are similarly repre-
sented in figs. 14—17.
In both the radial curves the stress gradient according to the second approximation
curves is steeper at the inner surface and less steep at the outer surface than accord-
ing to the first approximation, or dotted line, curves.
Bopity Forces.
V; Radial Curves. Vas Vi Tangential Curves. Vut-y
Fig 14 Fig 16 Fig 16 Figi7
In the case of each curve it is to be kept in view that what is shown is the
relative magnitude of a single stress at different distances from the surface along a
single radius vector. The law of variation as & varies in the value say of 7r in terms
of R; is the same for all radii vectores, but the absolute value and the sign of * vary
with the values of @ and ¢.
Again the maxima values of the radial and tangential stresses arising from one
and the same type of surface forces are of different orders of magnitude in h/a. Thus
the “principal” term in the approximation to the stress opposite in type to the applied
surface force is only of the same order of magnitude as the “secondary” terms in the
approximation to the stress of the same type as the applied force. In other words the
stress opposite in type to the applied surface force is to a first approximation negligible
compared to the stress of the same type. It should also be noticed that the “ principal”
terms in the stresses arising from the bodily forces will be of the same order of mag-
nitude as the “secondary” terms in a stress arising from a surface force of its own type
only when the bodily forees per unit of surface are of the same order of magnitude
as the surface forces.
In the preceding remarks on the gradient curves we have assumed “secondary” terms
small compared to those containing algebraically lower powers of h/a. As 7 increases,
however, the “secondary” terms in those stresses that are of the same type as the
applied surface forces rapidly increase in relative importance, and they cease to be small
compared to the “principal” terms when i*h/a ceases to be small. Moreover when 7
becomes very big the stress opposite in type to the applied surface force ceases to be
* In Fig. 16 the thick line curve will lie above the dotted curve close to B if i<5,
AND SPHERICAL SHELL. 377
small relative to the stress that is of the same type. Thus for a complete investigation
of what happens in any instance when 7*h/a is not small recourse should be had to the
general formulae (94) and (95).
An approximation to what happens when 7 is very large in the case of both displace-
ments and stresses may be found by retaining only the highest powers of 7 in (92), (98),
(94) and (95). Thus, for instance, on the left of these equations we may take II as given
by the following simplified form of (73):
Ul = 474 (m |n)? (ab) {(a— = pt ) (a?*8 as b+) = 72 (ab) (a — b?)?}.
The course then to be adopted depends on how big 7 and h/a actually are. Until this
is known we are rather in the dark as to the relative importance of the two terms in
the above expression for I, or of the several terms in the coefficients of R; etc. on the
right of equations (92)—(95),
§ 37. Before quitting the subject of thin shells it may be well to give a brief
summary of the results we have established for all forms of applied forces, whether the
displacements they lead to be of the first, second or third class. As previously a denotes
the radius, h the thickness of the shell, and h/a is very small. Our conclusions are as
follows :
(1°) The resultant per unit of surface of all the forces applied along a radius—whether
these be bodily or surface forces, or both combined—must be small compared to the greatest
longitudinal traction* permissible in a long bar of the material. The ratio borne by the
former quantity to the latter may be at most of the order h/a of small quantities.
If, however, the surface forces at corresponding points on the two surfaces be nearly
equal and opposite, the resultant of either set may be of the same order of magnitude as
the limiting longitudinal traction in the bar.
(2°) If the resultant of the forces applied along a radius do not vary very rapidly
in magnitude or direction relative to 7, 0, ¢—i.e. if there be no surface harmonies of high
degrees with large numerical coefficients—and if this resultant be not small compared to the
resultant of the forces applied over one only of the surfaces, then approximate values to the
radial strain and dilatation at all points in the shell are
du_ = na
dr BR”
las
= 36 a
where F is the radial component per unit of surface of all the applied forces acting along
the radius through the point considered, while » is Poisson’s ratio, HZ Young’s modulus
and k the bulk modulus.
(3°) Under the same conditions as in 2°, the stresses 7, 70, 7, usually assumed
negligible in theories of thin shells, are in reality small compared to the other stresses,
* Measured of course per unit of cross section,
378 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
to which they bear ratios of the order h/a of small quantities. In this case the stretch-
ing of the shell is the important factor in the values of the principal strains and
stresses,
(4) If there be no very intense bodily forces, and if the surface forces at corre-
sponding points all over the two surfaces be nearly equal and opposite, the stresses *,
*, ~ lose their inferiority relative to the other stresses. This also happens in any case
when the magnitude, or direction relative to 7, 6, ¢, of the applied forces varies rapidly
from point to point of the surface.
(5°) If a force of given type—radial or tangential—whose rate of variation with the
angular coordinates is not very excessive, be applied over one only of the surfaces, the
stress of the corresponding type has to a first approximation a straight lie for its gradient
curve, and the stress of opposite type—tangential or radial—unless it absolutely vanishes
has for its gradient curve according to the first approximation a parabola whose vertex
and maximum ordinate answer to the mid-thickness.
(6°) The displacements, strains and stresses arising from a bodily force are in general*
of the same order of magnitude as those arising from a surface force when the two
forces measured per unit of surface are of equal magnitudes. In practice this means
that in a very thin shell the effects of bodily forces must be very small unless these forces
be of extremely great intensity.
Solid Sphere.
§ 38. The displacements in the solid sphere may be derived from the corresponding
results for the shell by omitting all terms containing b raised to a positive power. We
shall represent all three classes of displacements simultaneously. With our previous notation
answering to
bodily forces from the potential r?>V + =r'V;,
r=R+ lies
ad dT; 1 a, :
a= >| + ano ae | Pan ea ety ca. (105),
surface forces |
ie [1 d%_av,
. pa sn@dh dé
we get
rR Le pv ee
38m—n 5m+n |Bm—n
u
er nl
1
pos 4 tees eee sy se SP lp a 65 Lgl inl
+ Serasweecaepn |PMfir rem n} ar {(i+1)m—n}r j
- {2 iy ’ — } ges —s , , —s a
+ R; hear i (i+2)m—n} a (i + 1) (mi — 2n) a
taied- 1) 7, We — 2n) ve Le {(i2-i—8) m+n} I] savareecaas (LOO)!
=| a i-l1 (rire
* There are exceptions amongst the strains and stresses; compare for instance terms in 7, and V; in (99).
AND SPHERICAL SHELL. 379
1 1 aT; r
~ nsin 0 ale 1 dd ral
+ San (ere are ma are Iya [Pao (ari Demme —[648)m—n oH}
atte Fo (+8) m+ 20} =
— {iG see -i-3)m4n) 7} ee (107),
w=— > Be E ao Fi
Gra T - m— (Qi +1) a ee F 75 ti © (G+ 2) m=) cer — (+8) m =n} oa
af = 6 ae i d {t (i+ 2) (eet) i= ut —{(@+3) m+ 2n} I
de as is (i {+ 3) m + 2n} == aS - {(@—t—3) m+ n ap (108).
The summations run from i=2 to t=. The value 7=1 is incompatible with the pre-
servation of equilibrium.
§ 39. It must be carefully noticed that though we may thus deduce the displace-
ments for a solid sphere from those for a shell, the strains and stresses due to given forces
over the outer surface are not the same in a solid sphere as in a shell whose outer
boundary is the same, however small the radius of the inner surface may be. In the
solid sphere we omit in the displacements all terms vanishing with 6b, and deduce the
strains and stresses from the terms left; but in a shell a displacement b'r~*, while
itself negligible however small + may be, will supply a strain varying as (b/r)*. Such
a strain will be very small except near the inner surface, but close to that surface it
may be very large. Thus the strains and stresses near the centre of the solid sphere and
near the inner surface of the nearly solid shell may be, and in fact generally are, widely
different*.
§ 40. In the case of purely surface forces derived from a potential (r*/a’*)Q;, as in
§ 13, the results (106), (107) and (108) take the remarkably simple forms
an \ Gel ok
o> Te 2n (i—1) a
ah ( eHer
Ln 7 leon ot BS. Skettis duets ce testacm cones (109).
sa 1 d r'fa a
~ rsin 6 dd eee :
In this case the dilatation 6 obviously vanishes, as Q; is a surface harmonic.
* For an explanation of this seeming discontinuity see the Society’s Proceedings, Vol. v1. pp. 285-6, 1892.
WO, .OV, IBA IY . 50
380 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
Nearly solid shell.
§ 41. There is considerable interest attaching to the action of forces applied over
the inner surface of a nearly solid shell, i.e. a shell for which b/a is very small. The
method of treating this case will perhaps be sufficiently illustrated by the deduction of
the radial displacement answering to the purely radial force R;'. To find this we employ
(92), retaining in the coefficient of each power of r only the lowest power of 6. The
result is of course only a first approximation, neglecting higher powers of b/a than those
retained. It is
Qn (Qi? + 44+ 3) m — (Qi +1) n} {(2t2 + 1) m+ (20 + 1) n} u/R,
=i(i+1)(¢ + 2) (2i— 1) m (im — 2x) (br) a4
~ — f [(a4 + 228 — # — 204+ 3) m? + 2mn — v7] rag
+ {(22 + 4¢ +3) m— (22+ 1) x} pawl {@?—1)m—n} ie —7t{(i+1) m+ 2n} ee
t 4+2 pit? re
Near the inner surface, ie. when r is of the order b, we may obviously neglect the
terms in r+! and 7 compared to those in r~* and r**, and so get the approxi-
mation
R; Rite Mn BE Bie
aenneee = — {(42 -1)m—n} ——7$ = Ile soys'eecigs E
u OniQi?-+1)m+ i+ 1m} Bac 1)m—n} aH if +1) m+ 2n} = | (111)
This result, it will be observed, may be derived from the term in R; in (106) by sub-
stituting 6 for a and writing (—i—1) for (+7) in all indices and coefficients. The same
substitution applies in the case of any displacement for any surface force. Thus if we
want the displacements, strains or stresses near the inner surface of a nearly solid shell
arising from forces applied over that surface, we have only to transform the corresponding
results for a solid sphere, acted on over its surface by forces following the same law,
by replacing a by b, and i by —(é+1) in all indices and coefficients. When 7 is large
u diminishes with extreme rapidity as r increases so long as (111) remains a satisfactory
first approximation. A similar result holds for the other displacements and for the strains
and stresses.
The formula (111) applies only when r is of order b, On the other hand when r
becomes of the order a the terms retained in‘ (111) are negligible, and the terms in
r~ and r in (110) then constitute the first approximation. In this case it will suffice
to point out the physical consequences.
Regarding r in (110) as of order a we obviously have u/r of the order (b/a)'*', and the
same result holds for all the strains and stresses due to R; or to tangential forces derived
from a potential 7’. In the corresponding case of tangential surface forces derived from a
“stream function” T the rate of diminution in the strains and stresses as 7 increases
when r is of order a is measured by (b/a)'**, Thus in all cases the strains and stresses
due to surface forces derived from: surface harmonies of high degrees are comparatively
AND SPHERICAL SHELL. 381
insignificant except close to the inner surface. At very moderate distances from this
surface the strains and stresses will be almost entirely due to those forces which are
constant or which vary but slowly over the surface. Regarding the strains and stresses
as propagated outwards from the surface, the effects transmitted from adjacent parts of
the surface where the applied forces are oppositely directed tend to neutralise one another,
and thus the action of the medium is to obliterate the effects of any want of uniformity in
the distribution of the surface forces. This damping out of the effects of the forces derived
from the high harmonics relative to the effects of the constant forces does not however, it
should be noticed, increase with the distance, after this has reached the limit at which
the terms in r** and ri in (110) constitute a satisfactory approximation.
Part II.
Equilibrium under given surface displacements.
§ 42. The previous solution may also be applied to a shell whose surfaces are subjected
to given displacements. These displacements must of course be of such a character as not
to strain the shell beyond the limits permissible in the material. All rigid body dis-
placements may be excluded. As the case of given surface displacements seems of much
less physical interest than that of given surface forces it calls for less fulness of treat-
ment.
The displacements may most conveniently be considered under the three classes of
Part I.
Cuass (i). Pure radial displacements.
The two constants of the solution
(i= VANES LYTIA_S oncosondboossnoDondEednagccebcbebeonono.60%60% (1)
are to be determined from the data
u=U over r=a,)
CS ee ee (2),
where U and U’ are constants.
The solution obviously becomes
u={r(aU — BU") + abr (aU —bU)} = (a2 —B)......eeeretenessestierenreetes (3),
= OGM USA) (GP 109) -seenpennadeees60985606 35506 uobeD iaodonond (4)
or = {(8m —n) (aU —b?U’) — 4na*b*r- (aU — bU)} + (a5 — DB)... eee eee e eee (5),
60 = $6 = {(8m —n) (a?U — BU’) + 2narb*r= (aU’ — bU)} + (a2 — BS)... eee eee (6).
For a thin shell, putting a—b=h, a—r=€, we get the approximate values:
b= Tes (i+8+0k-*8 db Radédoduqncaso sects adocte nee ncdepeasanedcd (7),
h a h\ a
382 Mr C., CHREE, ON THE ISOTROPIC ELASTIC SPHERE
h h- U'
rr =p fine m(1 +7) - 4n : —# ; {(m-+n) (1-4) 44n ‘| salaatereburaen (8),
w=a=7 {m—m(1 +7) +208 FT fmm (1-2) an Maat (9).
Two important conclusions as to the necessary limits to be assigned to the surface
displacements in thin shells are easily deduced. From (7) we have the approximate
results
u/r=u(1 + &/a)/a= S a (1 + “E) + Us (1 + — Reronmormanonoad ((0))).
a ah
de ae he
= Tae aaa (1 )
d ;
Now u/r and = are strains, and thus U/a, U’/a and (U— U’)/h must be small
quantities of the order permissible to strains in the material. The last limitation, which
is fairly obvious a priort, must be kept in view in judging of the accuracy of approxi-
mations. It shows that terms in U— U’ may be of less importance than terms in Uh/a.
If U’=U the strains and stresses have their values very nearly constant along the
thickness, the approximate values of the stresses being
Fr = 2 (m —n) ee
#@ = $6 = 2mU/a
Crass (ii). Pure transverse displacements.
§ 43. Here we have to determine the X;, X_;, of (31) and (32) Part I. from the
conditions
see 2ae% ver r=a
~ siné dd’ a ae =4, re
Ber 1 dt, watt Y seit eee eee weer rere see eeeseseeeeees oO),
~sin@ dd’ —« = |
where T;, T; are surface harmonics of degree 7.
We easily find
i+
= = 6 ii fr (PT ,— FT) + (2) z (aT; — vn} + (ain — wey | Rea (14),
w=- = A [same expression as inside square brackets in (14)] ......ceseeeeeeeee (15).
In a thin shell approximate values are
nie ope 4 8) its F(1-*—8) JERS ery ee eee ae (16),
AND SPHERICAL SHELL. 383
dT; 1 h oh- dT; 1 h 3
Aln=— Fee (145 3"), = 5(1—= + 38) OR Re eae i vos Shaiis s,s SO (17),
d? ee 1h-&/1+ ”
@|u=— fii 17, Soe 6° | h £ ( q
ae cer E(, A= 28)
(i G+1)7/+2% oa mt a ee ae (18).
The value of v may be got from that of w and the value of 7 from that of 7 by
writing — = 77 5 for == The reason for writing down the value of w rather than that of
v is Pee w alone exists when T; and T, are zonal harmonics,
Since r/n and #¢/n are strains we see that the displacement at either surface divided
by the radius, and the difference in the displacements at corresponding points on the two
surfaces in the same direction divided by the thickness, must be quantities not exceeding
in order of magnitude the limits permissible to strains in the material, When the dis-
placements are equal over the two surfaces, all the strains and stresses have to a first
approximation constant values along the thickness.
Crass (iii). Mixed radial and transverse displacements.
§ 44. Here we determine the Y;, Y_;., Z;, Zi. of the formulae (30)—(32) Part L—
in which V;, V_;, are now supposed zero—from the conditions
u=U; pets cee ON over r=]
eae ab) = Saino dp mee ay
eda ar 1 ary salle: weiss ain a cetsaiotle tuesse aacentes i
Uli Sear) Uren vay =
where U;, U/, 7;, T/ are surface harmonics of degree 1.
These conditions give
im — 2n sh (@+1)m+2n 7. a7 =U
~3@i+8)n @HY,+a7Z,;- = <r =a a*Y_;,+4a oes scoveels wieielsarclwiecetrteie (20),
(i+ 3)m-+2n ae ing,
TGs) @isrs)ynt +; :
vine ie ce a aed 7) 21
+ i (2 =) 7, a aes Fe at 1 a a = LG cecvcvccccccccccees ( );
am — 2n ers cigits (i+ 1) m+ 2n PY. ey =U! 29
= 2(@i+3)ne Y;+ 07> Z; 2(21—1)n ea Beat ECE CECE CCEDECE (22),
(@+3)m+2n pop t i 7.
SGV esyne a
A iE La oc a. (23).
|
21 (21-1) n t+
384 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
The method of treating these equations employed in my original paper* for the
ease of normal displacements seems the simplest way of solving the above. For short-
ness let
U;- tT. ae U; +(% + 1) TT = B;,)
UZ = ol, = A, UZ ts G a 1) Te = Bis ee er i (24).
Then putting
= yum + Bie 1) nh {( t iti Im + em +1) x} —2+1 (72-1 _ fei— +3 _ foit+s
~ a (+1) (28-1) (+3) v2 De rage te
m\?
he 2 f2\2 95
ee (at Phys... ROO MS (25),
we easily find
: (¢+. 1)m+(2i+1)n at — pt Lape
ILY;= a(@i— In (aby- ; fait? A, b'*2A,}
. = (a= Biase Be — ba BA is abyctieenees.. eh et (26),
mre: ON CEE CE an) iS ae
i 2+1 tah) aE i(2i—1)n ~ (ab)h ee)
{im + (21+ 1) n} {+ 1) m + (2141) n} at — Bs 2
“5 7(@+ 1) (22-1) (20 +38) vw? (ab) pe side})
m ER Bethe
+($) (a? — b*) (a BB; — b-B; | ee ee (27),
Y__,= = (a? — b*) (a’*A; — b'*°A;’)
Gs Gt pe (atts — B28) (HB, — BBY) oceececeee (28),
i+1 {um + (26 + 1) m} {G+ 1) m+ (2041) 0} a=, gd ai a:
= a [Ser saeene (aye EAs Oe
+(5) (a? — b*) (a'A; — b'A/)
m™m um + (20+ 1)n qn +1 —i+1 —i+1 s
Te Geran — O(a, — 0 By) ln aresese asa (29).
The substitution of (—7i—1) for (+7) in all indices and coefficients transforms II into
itself, and deduces the values of Y_;, and Z_;, from those of Y; and Z; respectively.
§ 45. Substituting the above values of Y;, Z;, Yi4, Z 4 in equations (30) and
(31) Part L, and writing U;—i7; for A; ete., we find
t+1 ({om+ (2¢+1)n} (4+ 1) m+ (24 +1) n} 7 — BP) (7 — B 9)
sir 7(t+1) (2i—1) (20+ 3) x? (ab)
- (3) (a?— b*) (7? — w»}
* Camb. Trans., Vol. x1v., pp. 305, 306.
Ilu=(U; - in) (2)
AND SPHERICAL SHELL. 385
m (¢+1)m+(2i+1)n ane aa iy Bh SAU OPEB, Oe oe. \'|
7 On (26-1) (2i+1)n (2 pa: te) (7 Sa ncn Wk oe)
i b\ i+2
+( us—ity)(?) [coefficient obtained from that of U;—77; inside square brackets by
interchanging a and 6]
5 z {im + (2% +1) n} {i +1) m+ (2i+ 1) mn} (a8 — 5248) (72— SS)
rer) [a oo ae (ab)
m im+(2i4+1)n MG cal _ BB) (7th — =)
SOR IN RE = Sal (a? — 7°) (9°) — BD ) — a? (7? — 8) (& [penin)
+ {U/+(¢4+1) T7;} (*) [coefficient obtained from that of U;+(i+1) 7; inside square
brackets: by interchanging’ 7a “and Pl) srtescesdothn--..0-<-n-seoeseostccaseeses: (30),
= NO ahs ea {um + (21 +1) n} (i+ 1) m+ (2i+1) n}
Ho =— 5 gg Ue- 12) (2) | t(¢+1)(20—1)(224+3)n2 -
21-1 __ bei 143 _ eits f 2
ea (2-6) 2-0}
(o-= i+1 (a —r 2) (7 2-1 b? 2t+1) = 7 2t+1 (Gg — b?) (a v+1 __ ps “«))|
<! (5) (+ 1) m+ (2i+ 1) n
on) (ery
i+2
a: 5 eit) (?) | coefticient obtained from that of 2 (U:- if) inside square
brackets by interchanging a and |
i @i : 1, (2) | Slam + (20 + 1) n} (+1) m+ (27 +1) n}
Pope dost +1) 2s (“) { 444+ D2—-Da+3e
Ge + b+) ( pia pes b>) ( 2 r\2-1
(ab) \2n (
-) : (a = b?) (r* Pat »))
a
m im+ (2+ I)n 1
Qn (6 +1) (21 4+3)n OP
(« (°° a b?) (an = pitty b (a? af 1) (Gane = we)
1
by? 2 23
= Ra = {U/+(+1) 7; (5) coefficient obtained from that of O(Ui+ G+ 1)7;} inside
square brackets by interchanging a and | Meee Cpeieelen secs eae sie ssa oeee sa (31).
The sek Ge be: dednecd) trom that “ala. by; eeplicnie e -by = rh
e value of w may be deduced from that of v by replacing ao >Y alld e
substitution of (—i—1) for (+7) in all indices and coefficients derives coefficients of
U;+(@+1)7; and U/+(i+1)T/ from those of U;—iT; and Uj —iT/ respectively.
386 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
§ 46. The form of the results suggests the deduction of the surface displacements
from two potential functions after the manner indicated for the surface forces in § 13.
Thus if the sum of these functions, for the surface =a, be
(r"/a*>) Q; ae (asa) (Meso 2
where Q;, Q_;, are surface harmonics of degree 7, we should put
Ui= © [erifa) Q + Oa) Qa,
am 1d
r dé
ae ie 1
sn@dé rsindd
where @ is substituted for r after differentiation. The relations between U;, 7; and Q;,
[Same Expression], == swvaccececnsavtsressvieveee (32),
d
e [same expression]
Q_;, take the simple forms
{U;+ (+1) T}/(2i +1) =Q;,
[T,—sTAOR HS SOs, Aun ar eae (38).
The expressions for the displacements are obviously much simplified if either Q; or
Q_; is zero, and the form in which (30) and (31) are presented was chosen partly with
a view to bring this out. Other reasons for selecting this form were that it affords a
ready means of testing the accuracy of the results and that it lends itself readily to
applications to thin shells.
§ 47. The arrangement in (30) and (31) is analogous to that adopted in (94) and
(95) for the stresses 7, 7@, 7. Thus in (30) in the coefficients of both U;—77; and
U;+(¢+1) 7; the expressions inside the { } brackets obviously vanish over 7=b and
take the value II over r=a, while the last lines of these coefficients clearly vanish over
both surfaces of the shell,
For the thin shell, putting a—b=h, a—r=€, as before, we easily deduce the
following approximate results:
_h— , h— 1 h—-
oT pf (1+e)+ ui (1- me aa) (Gps TB ) mm Fe) Ko) eee (34),
-¢ A dT, h—& &)\ dTé h-€&
g (Ui — Ui) 5 BAe (1+ \+ ore (1--=*) a ame on Re (35),
Sout Oe = i
— |(m+n) (142) —4n ad oat (m+n) (1-1) +4n ‘|
T;4 (i+1) h—&\ Ty t(i+1) E BY
a ae (m- 2n ay )= 7 9 (m- 2n') Piieacaeveige teem (36),
—_1dU;m{ gm—nh—é 1 dU; m m—né&
YG a nh \" a ao y(b-2 m i
as h_gh-€\_ 147; hy hg
45 Gg (L481 2 BE) cas sneesteeren( BT),
AND SPHERICAL SHELL. 387
en dt i n~, m E(h— &)
e = edb |i (U' U;’)
t dédd 2n ah
A= E(y 08) gay (1) Ro8E
+7,->* (142% +T, ae ; ) A Setter 58)s
The values of w and 7 may be found from those of v and 7 respectively by sub-
“ier td d
stituting SE for da’
The limitations in the magnitudes permissible to the displacements over either surface,
and to the difference between the displacements at corresponding points on the two
surfaces, are precisely similar to those established in the two previous classes of dis-
placements.
§ 48. When the surface displacements have no tangential component 7 does not appear
in the coefficients in (34) and (35), and the coefficients of U in (7) and U; in (34) are
identical. Thus to the present degree of approximation if radial displacements U and U’
be applied over the surfaces of a thin shell according to any law whatsoever—consistent of
course with the limitations as to the magnitudes of the strains—we have
= I)= TT’ h—
vat SF (148) 40 E (1 =) ed ie ara aca (39),
Oh pe a TB (DB)
y= g(U-U) = Ap. PobbonenaccanbogBbecccuaccoscenpauboasoa0C (40),
BO Oe ee) Fs
TOG Timers aes ORT ec eee ec cere vr ecccvee cer ceeceseccesas (41).
The coefficients in the expressions for the stresses *, 7, 74, @ do not in this case
contain 7 either, and the coefficients of U and U’ in (8) are the same as those of JU;
and U;/ in (36); thus the expressions for these stresses may be found by putting 7;=7/=0
in (36), (87) and (38) and replacing U; by U and U; by U’. It must be remembered
however that if (U—U’)/U be very small, terms involving higher powers of h/a than
those retained in (40) and (41) may be of equal or greater importance. A similar limitation
would apply to the expressions deduced for 7 and 6g.
2 : dT; dT;
§ 49. We notice that the coefficients of de and qa
contain 2, and by referring to (16) and (17) it will be seen that the same factors, e.g.
in both (35) and (37) do not
h ,h—-é | h é.
1 tsi bares and 1 eae in (17) and (37),
occur in the two cases, Now the total components parallel to 0, @ of the tangential
displacements on the two surfaces are given by
Vou. XV. Parr IV. 51
388 Mr €. CHREE, ON THE ISOTROPIC ELASTIC SPHERE
Tass at; 1 aT, wes|_t TA a |
== | snOdp |’ © ~|sin@ dp 0 |’
eT, er me oe (42)
pe spore 1S eee et ee eee
~ “| d@ sn@ dd |’? “~“|smOdp = dé
Thus we obviously have, for the most general tangential displacements consistent with
the limits permissible in the magnitudes of the strains, the approximate results
= VF (Fare 8) uned nal ot suede (43),
w= WF (1 48) +7 (1-"— 4) PAM IO hee (44),
wal n Se (1-+38) vc etda tee ee (45),
= ay (i+e-38—t =n =e) ye, jul sha He (46).
As before, it should be noticed that when the difference between the displacements at
corresponding points is very small compared to the displacement for one of the surfaces,
terms containing higher powers of h/a may have to be retained.
§ 50. Let us suppose that one only of the two surfaces is displaced, say the outer.
We then see from (39), (43) and (44), that the way in which u/U, v/V, and w/W vary
with & is precisely the same. Thus to the present degree of approximation we see that
the same “displacement-gradient curve”—i.e. a curve whose abscissae measure the distance
from a surface of the shell and whose ordinates give the corresponding magnitude of a
particular displacement—would apply in all cases when there is no radial surface displace-
ment, or when there is no tangential surface displacement. to the displacement which is of
the same type as the given surface displacement.
A similar result obviously applies in the case of displacements applied over the inner
surface only. The curve is in the case of either surface a straight line according to the
first approximation, whose zero ordinate answers to the undisplaced surface. The curves
according to the second approximations are of the forms of those in fig. 2 or fig. 3, § 10,
according as the outer or inner surface is that displaced. The gradients in both cases are
steepest at the inner surface,
From (34) and (35) we see that the gradient curves for the displacements which are
opposite in type to the given surface displacement are to a first approximation parabolas
symmetrical about their maximum ordinate, which answers to the mid surface, and with
zero ordinates answering to the two surfaces of the shell.
AND SPHERICAL SHELL. 3 389
There is one important distinction between the displacements which are of the same
type as the given surface displacement and those which are of the opposite type. The
magnitude of the former depends, to the present degree of approximation, only on the local
magnitude of the applied displacement, but the latter increase somewhat rapidly with the
degree of the harmonic from which the displacements are derived. This is obvious when
eo J; ; é
i is large, as we are then to regard dé and ae as of orders 77; and 7U;; thus for a
given magnitude of a the corresponding term in w in (34) varies as 7, and for a given
magnitude of U; the corresponding term in v in (35) varies as 7. When 7 is small the
displacements opposite in type to the given surface displacement bear to those of the same
type a ratio of the order h/a, and so to a first approximation may be neglected; but as 7
increases their relative importance increases, and they may not be neglected even to a
first approximation when th/a ceases to be small.
If we suppose 7 so small, or the shell so thin, that th/a is negligible, we have to a first
approximation for simultaneous displacements U, V, W over the outer surface only
wf =a) Ve an] We EI) scree not achat ea sat enpiad an (47).
This signifies that the resultant displacement at any point of the thickness is parallel to
the applied surface displacement, and proportional in magnitude to the distance from the
inner surface. A corresponding result holds under like conditions for displacements over the
inner surface only.
When th/a ceases to be small it would be wise to employ the exact results (14), (15),
(30) and (31) to ensure that terms are not omitted equal in magnitude to those retained
in the above approximations. This is especially the case when the difference of the dis-
placements at corresponding points on the two surfaces is small compared to the displace-
ment over either surface.
It must also be borne in mind that taking the displacements over a surface zero is
equivalent to supposing that surface held by the surface forces requisite to prevent dis-
placement. Thus the cases treated above where the displacements are given over one
surface only, and the other surface is supposed undisplaced, answer to a totally different
set of matters from that arising when the one surface is displaced in an assigned arbitrary
way and the other is left free of forces. This latter case seems not unlikely to be the
more interesting of the two in practice and we shall briefly consider it presently.
§ 51. Before doing so, however, it may be as well to point out that the solution for
a solid sphere subjected to given arbitrary surface displacements may be deduced from
that for a shell precisely as in the case of given forces. To get the displacements for
the solid sphere we have only to put b=0 in (3), (14), (15), (30) and (31), noticing in the
two latter equations the occurrence of b~**! as a factor in II.
In the case of a nearly solid shell approximate solutions may be deduced by retaining
only the lowest powers of b/a in the coefficients of the several powers of r. This would
be very easily done for the first two classes of displacements as given by (3), (14) and (15).
51—2
390 Mr C. CHREE, ON THE ISOTROPIC ELASTIC SPHERE AND SPHERICAL SHELL.
The formulae (30) and (31) for the third class are not so convenient for this purpose, and
it might be found simpler to substitute in the formulae (30)—(32) Part I. the values
found for Y;, Z;, Yi, Zi. in (26)—(29) by retaining only lowest powers of b/a. Little
interest seems to attach to these results except in so far as they show that when the
inner surface of a nearly solid shell is arbitrarily displaced, the outer surface remaining
fixed, those displacements, strains, and stresses, which depend on the surface displacements
deduced from high harmonics, fall off at first very rapidly in relative importance as the
distance from the immer surface increases, so that at a considerable distance from this
surface the effects of irregularities in the distribution of the surface displacements have
largely disappeared.
One surface arbitrarily displaced, the other free.
§ 52. We need only indicate the method of treating this problem. Take for instance
the case when the surface r=a is subjected to displacements of the third class, given say
by the first equation of (19), the surface r=b being free of all forces. Then we may treat
the problem independently by determining Y;, Z;, Y_i., Zi. from equations (20) and (21)
§ 44 combined with (40) and (41) of Part I. In the latter two equations we are to suppose
the right hand sides to be zero. The solution in this case might also be deduced by
taking (30) and (31) as they stand, but regarding U/, 7/ as unknown quantities to be
found by equating to zero the values of 7 and 7, or 7, supplied by this solution over
np
Here we shall only determine the solution for a thin shell. Suppose r=a the surface
subjected to given displacements, 7 =a—h the free surface. Then, using the second method
indicated above, it is easy to deduce the approximations :
_F TEN Te ee
“= 7(1+27—* ®) SIN CoS Derren hs An ecile (48),
ut £) due
y= v (1-5) Sh secon ia sactog tae ae a (49),
_z E 1 dUé 54
w= W( oi) aadame aie, it ae edivion de dank renal (50)
Here U, V, W are the total components along 7, 6, ¢ of the given arbitrary displace-
ments on the outer surface, and 7; is the term containing surface harmonics of degree 7 in
the potential from which arise the tangential displacements occurring under class (iii), First
approximations to the stresses #, 4, 6 may be derived from these results. The complete
difference between these results and those obtained for the case of one surface fixed and
the other subjected to given displacements should be noticed.
If the outer were the free surface and the inner that displaced, the only change
required in (48), (49), (50) would be the substitution of (-h+€&) for & taking U etc. as
now the displacements over the inner surface.
XI. On the Kinematics of a Plane, and in particular on Three-bar Motion:
and on a Curve-tracing Mechanism. By Professor Cayuey. (Plates vi., vit.)
THE first part of the present paper, On the Kinematics of a Plane, and on Three-
bar Motion, is purely theoretical: the second part contains a brief description of a Curve-
tracing Mechanism, which has been at my suggestion constructed by Prof. Ewing for the
Engineering Laboratory, Cambridge.
Part I.
1. The theory of the motion of a plane when two given points thereof describe given
curves has been considered by Mr S. Roberts in his paper, “On the motion of a plane
under given conditions,’ Proc. Lond. Math. Soc. t. 11. (1871), pp. 286—318, and he has
shown if for the given curves the order, class, number of nodes, and of cusps, are
(m, n, 6, «) and (m’, n’, &, x’) respectively (n =m? —m— 26 — 3x, n’ =m? —m — 28 — 3x’),
then for the curve described by any fixed point of the plane:
order = 2mm’,
class = 2(mm’ + mn’ + nm’),
number of nodes = mm’ (2mm’ — m — m’) + 2 (m& + m’6),
number of cusps = 2 (mk' + m'k),
but he remarks that these formule require modification when the directrices or either of
them pass through the circular points at infinity. And he has considered the case where
the two directrices become one and the same curve.
2. It will be convenient to speak of the line joining the two given points as the
link; the two given points, say B and D, are then the extremities of the link; and
I take the length of the link to be =c, and the two directrices to be b and d; we
have thus the lnk c= BD moving in suchwise that its extremity B describes the curve b
of the order m, and its extremity D the curve d of the order m’: in Mr Roberts’ problem
the locus is that described by a point P rigidly connected with the link, or say by
a point P the vertex of the triangle PBD.
3. The points B, D describe of course the directrices b, d respectively: taking on
b a point B, at pleasure, then if B be at B, the corresponding positions of D are the
intersections of d by the circle centre B, and radius ¢, viz. there are thus 2m’ positions
of D: and similarly taking on d a point D, at pleasure, then if D be at D, the cor-
392 Pror. CAYLEY, ON THE KINEMATICS OF A PLANE,
responding positions of B are the intersections of b by the circle centre D, and radius ce,
viz. there are thus 2m’ positions of B. The motion thus establishes a (2m, 2m’) corre-
spondence between the points of the directrices 6 and d, viz. to a given point on b
there correspond 2m’ points on d, and to a given point on d there correspond 2m points
on b. Of course for a given point on either directrix the corresponding points on the
other directrix may be any or all of them imaginary; and thus it may very well be
that for either directrix not the whole curve but only a part or detached parts thereof
will be actually described in the course of the motion. In saying that a part is described,
we mean described by a continuous motion; say that the point B (the point D remaining
always on a part of d) is capable of describing continuously a part of b; it may very
well happen that the point B (the point D remaining always on a different part of d)
is capable of describing continuously a different part of b, but that it is not possible
for B to pass from the one to the other of these parts of 6 without removing D from
the one part and placing it on the other part of d, and thus that we have on b detached
parts each of them continuously described by B; and similarly we may have on d
detached parts each of them continuously described by D.
4. But dropping for the moment the question of reality, to a given position of B
on 6 there correspond as was mentioned 2m’ positions of D on d, or say 2m’ positions
of the link c: in the entire motion of the link it must assume each of these 2m’
positions, and for each of them the point B comes to assume the position in question
on b; the directrix b is thus described 2m’ times, that is the locus described by B, will
be the directrix b repeated 2m’ times, or say a curve of the order mx 2m’, =2mm’,
Similarly the locus described by D will be the directrix d repeated 2m times, or say
a curve of the order m’ x 2m, = 2mm’.
5. In general if B.D, be any position of the link and if B moves from B, along
6 in a determinate sense, then D will move from D, along d in a determinate sense ;
and if B moves from B, along b in the opposite sense, then also D will move from D,
along d in the opposite sense. Or what is the same thing we may have B moving in
a determinate sense through B,, and D moving in a determinate sense through D,, and
reversing the sense of B’s motion we reverse also the sense of D’s motion. But there
are certain critical positions of the link, viz. we have a critical position when the link
is a normal at B, to the directrix 6, or a normal at D, to the directrix d. Say first
the link is a normal at B, to the directrix >. The infinitesimal
element at B, may be regarded as a straight line at right angles to
the link; hence if for a moment D, is regarded as a fixed point
the link may rotate in either direction round D,, that is B may
move from 8, along b in either of the two opposite senses, say B, Be
ur
D,
is a “two-way point.” But if on d we take on opposite sides of
D, the consecutive points D,’ and D,’, say D,'D, cuts D,B, at an acute angle and D,”D,
cuts it at an obtuse angle, then D,’ will be nearer to 6 than was D,, and thus the
circle centre D, and radius ¢ will cut b in two real points B/ and B,” near to and
AND IN PARTICULAR ETC. ; 393
on opposite sides of B,; or as D moves to D,, B will move from B, indifferently to
By or BY’. Contrariwise D,” is further from b than was D,, and thus the circle centre
D,” and radius c, will not meet 6 in any real point near to B,, and hence D is incapable
of moving from D, in the sense D,D,”. Or what is the same thing the described portion
of d, which includes a point D,’ will termmate at D,, or say D, is a “summit” on the
directrix d. We have thus a summit on d, corresponding to the two-way point on b.
And of course in like manner if the link is a normal at D, to the directrix d, then
D, is a two-way point on d, and the corresponding point B, is a summit on b,
6. If the link is at the same time a normal at B, to b and at D, to d, then each
of the points B,, D, is a two-way point and also a summit; or more accurately each of
them is a two-way point and also a pair of coincident summits.
But the case requires further investigation. Considering the position B,D, as given,
we may take the axis of «# coincident with this line, and the origin O in suchwise
@) B, D, R S
x
that OB,, OD, are each positive and OD, >OB,; say we have OD,=6, OB,=8, and there-
fore 6—8=c. The equation of the curve b in the neighbourhood of B, is y?=2p(«— 8),
where p is the radius of curvature at B,, assumed to be positive when the curve is
convex to O, or what is the same thing when the centre of curvature R lies to the
right of B, (OR—OB,=+); and similarly the equation of d in the neighbourhood of D,
is y?=2o0(#—8) where o is the radius of curvature at D, assumed to be positive when
the curve is convex to 0 or what is the same thing when the centre of curvature S lies
to the right of D, (OS— OD,=+).
Consider now (2, y,) the coordinates of a point on 6 in the neighbourhood of B,,
y:’ = 2p (a,— 8), and taking B at this point, let (z, y.) be the coordinates of the corre-
sponding point D on d in the neighbourhood of D,, y.*= 2a (x,—8). We have
C= (a, — 22) + (Y1 — Yo),
— Yr seni Ys
and here te ce ®=8+5",
Ye loziy leas , . OYs2
whence a= B+ By’ » @%, = Bd+5 Be ie By: , apap OF
p 2 p WA (oy o
The equation thus becomes
(G-py + @- 342 B-B) +y—wrae,
‘ —§ hate 5 —
that is yy (1 + —) = 2YiY2 + YL (1 a °=£) = 0,
a quadric equation between y, and y,. Evidently if we had taken D a point on d,
coordinates (a, y2) in the neighbourhood of D, and had sought for the coordinates (2, %)
of the corresponding point B on 6b in the neighbourhood of B,, we should have found
the same equation between y, and y.
394 Pror. CAYLEY, ON THE KINEMATICS OF A PLANE,
7. The equation will have real roots if
Lalli pe he ae! Je
viz. p, o the same sign, this is po >(p +8—58)(¢ + §— 8),
but p, o opposite signs, then po <(p +B —5)(¢+6— 8).
These conditions may be written
(OR — OB,) (OS — OD,) — (OS — OB,) (OR — OD,) > or < 0,
that is (OS — OR) (OD, — OB,) > or < O.
But we have OD,—OB,=+, and therefore, p, o the same sign, the condition of reality
is OS>OR, ie. S to the right of R; but p, o opposite signs, the condition of reality is
OS <OR, i.e. S to the left of R. Observe that S lying to the left of R, we cannot
have p=—, o=+4, and that the second alternative thus is p=+, o=-—, then OS< OR, or
S lies to the left of R.
The condition was investigated as above in order to exhibit more clearly the geo-
metrical signification, but of course the original form or say the equation
1- (1+ 8=9) (14°58) 4
gives at once °=8 8 +0-B-p)=+.
8. Writing the quadric equation in the form
yt (1 _ 4) — 2yYyo+ (1 + ‘) y= 0,
we have (1 = A) Yi = i an Jz (c+0-p)h Yo3
the two values of y : y, will have the same sign or opposite signs according as
pS nid +5 have the same sign or opposite signs, and in the case where these
have the same sign, then this is also the sign of each of the two values of y% : y
Or what is the same thing if 1 =. and 1 +5 are each of them positive, then the two
values of y, : y are each of them positive; if 1 = and 1 +< are each of them negative
then the two values of y : y, are each of them negative; and if L=7 and 1+< have
opposite signs then the two values of y, : y. have opposite signs. Considering the different
cases p, g=++, +-, —-, we find
p, 7=++4, then values of y, : y, are ++ or —-, according as DR, BS are ++ or —-.
p, c7=t+- - ‘ “ Z } DR, SB ks .
p, c=—— 7 : = ; 3 RD, SB ; "5
AND IN PARTICULAR ETC. 395
and in each case values of y% : y, are + — if the two distances referred to have opposite
signs: DR=-+ means that RF is to the right of, or beyond, D, and so in other cases.
9. The different cases, two real roots as above, are
0 #B wake D R Ss hiye=+t)
p, on++] :
R
= s = Aiy=a=++
: - seat
ee ae s R
» +
i Seer
R s
A A? Yy=t +
OS a R s
B D ” aE
Obviously the cases p, ¢=—-—, correspond exactly to the cases p, c=+,+; the only
difference is that the concavities, instead of the convexities, of the two curves are turned
towards the point 0.
10. If the two roots of the quadratic equation are imaginary, then B,D, is a con-
jugate or isolated position of the link, and B,, D, are isolated points on the curves
b and d respectively.
11. If the roots are real, then the three cases y, : y=++, —— and +-, may be
delineated as in the annexed figures, viz. taking in each case y, as positive, that is
imagining B to move upwards from B, through an infinitesimal are of b, then D moves
from D, through either of two infinitesimal arcs of d, both upwards, both downwards, or
the one upwards and the other downwards, as shown in the figures
Y> Y=ett Yo: Y= I? Yo=+
\D”
B |p’ BI Bi D
B, D, B, ; D, B, Dy
D D
De
and where it is to be observed that reversing the sense of the motion of B from B,
we reverse also the senses of the motion of D from D,: moreover that considering D
as moving through an infinitesimal are of d from D, we have the like relations thereto
of the two infinitesimal arcs of b described by B from B,. Thus the points B, and D,
are singular points of like character.
If y : %=++, we may say that B, (or D,) is a for-forwards point; if y, : y=—-,
then that B, (or D,) is a back-backwards point; and if y, : y,=+, then that B, (or D,)
is a back-forwards point.
Vout. XV. Parr IV. 52
396 Pror. CAYLEY, ON THE KINEMATICS OF A PLANE,
12. The separating case between two imaginary roots and two real roots is that of
two equal real roots: the condition for this is §+oc=8+>p, that is OS=OR, or the two
centres of curvature are coincident; the characters of the poimts B, and D, would in
this case depend on the aberrancies of curvature of the curves b and d at these points
respectively. If each of the curves is a circle, then the curves are concentric circles, and
the link BD moves in suchwise that its direction passes always through the common
centre of the two circles—or say so that BD is always a radius of the annulus formed
by the two circles—and for any position of BD, the two extremities B, D are related
to each other in like manner with the points B, and D,. Thus in this case there are
no singular points B, and D, to be considered.
13. In the case where the curves b, d are circles we have three-bar motion: say the
figure is as here shown; I take in it b,d for the radii of the two 3
c
circles respectively and a for the distance of their centres; viz. we D
have the lnk BD=c, pivoted at its extremities to the arms or
radi AB=b, and ED=d, which rotate about the fixed centres 6 d
A, E at a distance from each other=a. Here a, b, c, d are
each of them positive; a, 6, d may have any values, but then a@ E
F i A
ec is at most=a+6+d, and if a>b+d then c is at least
=a—b—d; but if a=or<b+d, then ec may be =0, viz. it may have any value from
0 to a+b+d. And in either case there will be critical values of c. The cases are very
numerous. To make an exhaustive enumeration, we may assume d at most = 6, and in
each of the two cases d<b and d=b, considering the centre of the circle d as moving
from the right of the centre of the circle b towards this centre, we may in the first
instance divide as follows:
d<b | d=b
© d exterior to © B, | © d exterior to © b,
touches it externally, | » touches it externally,
cuts it, | » cuts it,
, touches it internally, » 1S concentric and thus coinci-
» les within it, | dent with it;
» 18 concentric with it,
and then, in each of these cases, give to the length ¢ of the link its different admissible
values.
14. Considering the case d<b, then we have (see Plate VI.), exterior series, the
figures 1, 1—2, 2, 2—3, 3, 3—4, 4, viz.
AND IN PARTICULAR ETC. 397
fig. 1, c=a—b-—d,
1—2, ,, intermediate,
2, c=a—b+d,
2—3, ,, intermediate,
3, c=a+b—d,
3—4, ,, intermediate,
4, ,=at+b+d.
15. In figure 1, the curves described by the extremities B and D respectively are
each of them a mere point.
In figure 1—2, we have a+d>b+c and a+b>d+c. Hence in the course of the
motion the arms b, ¢ come into a right line, giving a position B,D, of the link, where
B, is a two-way point on b and Dj a summit on d; or rather there are two such
positions symmetrically situate on opposite sides of the axis Av. And again in the course
of the motion the arms d, ¢ come into a right line, giving a position B/D, where D,
is a two-way point on d and B, a summit on D; or rather there are two such positions
symmetrically situate on opposite sides of the axis Az. Only an are of the circle b is
described, viz. the are adjacent to d included between the two summits B, on b; and
in like manner only an are of the circle d is described, viz. the are adjacent to b
included between the two summits D, on d. The described portions on 6 and d re-
spectively are to be regarded each of them as a double line or indefinitely thin bent
oval: and it is to be observed that for a given position of B (or D) there are two
positions of the link BD, each of these positions being assumed by the link in the
course of its motion.
16. In figure 2 the two positions B,D, of the link come to coincide together in
a single axial position BD, but we still have the other two positions B,D, of the link,
where B, is a summit on }, and D, a two-way point on d. As regards BD, this is
the configuration p, c=—-, R, B, 8, D:y : ys=+, and thus each of the axial points
B, D is a back-and-forwards point. Thus only the are B’B, of the circle b is described
by the point B, but the whole circumference of the circle d is described by the point D.
If we further examine the motion it will appear that as B moves from the axial point B
say to the upper summit B, and returns to B, then D starting from the axial point
D may describe (and that in either sense, viz. y=+, then we have y,=+) the entire
circumference of d, returning to the axial point D; and similarly as B moves from the
axial point B to the lower summit By and returns to B, then D starting as before
from the axial point D may describe (and that in either sense, viz. y,=—, then we have
Y2= +) the entire circumference of d, returning to the axial point D. It is thus not the
entire are B,/B, but each of the half-ares BB, which corresponds, and that in either of
two ways, to the circumference of d.
17. In figure 2—3, there are four critical positions B,D, (forming two pairs, those
of the same pair situate symmetrically on opposite sides of the axis Aw), where as before
52—2
398 Pror. CAYLEY, ON THE KINEMATICS OF A PLANE,
By is a summit on 0, and D, a two-way point on d. The described portions of b are
the detached ares B,'B,) between the two upper summits, and B,’B,’ between the two
lower summits: the described portion of d is the whole circumference. In fact attending
to one of the ares on 0b, say the upper are B,'B, as B moves from one of the summits,
say the left-hand summit B,’, and then returns to the left-hand summit B,, then D, starting
from the corresponding two-way point D,, may describe, and that in either sense, the
entire circumference of d, returning to the same point D,; and similarly as B describes
the lower are B,B,’, starting from and returning to a summit, then D, starting from the
corresponding two-way point D,, may describe, and that in either sense, the entire cir-
cumference of d, returning to the same two-way point D,.
18. In figure 3, two of the positions B,D, have come to coincide together in the
axial position BD, but we still have the other two positions B,D,, where B, is a summit
on b, and D, a two-way point on d. As regards the axial points B, D, this is the
configuration p, o=++; B, R, D, 8S; y: w=+H, viz. each of the points B, D is a back-
and-forwards point. The two detached arcs BB, of b have united themselves into a
single are B,’B,, which is the described portion of b; the described portion of d is as
before the entire circumference. It is to be observed (as in fig. 2) that properly it is
not the entire are B,/B, but each of the half-ares BB, which corresponds to the entire
circumference of d.
19. The figure 3—4 closely corresponds to fig, 1—2, the only difference being that
the ares B,/B, and D,'D,’ which are the described portions of 6 and d respectively (instead
of being the nearer portions, or those with their convexities facing each other) are the
further portions, or those with their concavities facing each other, of the two circles
respectively.
Finally in fig. 4, the described portions of the two circles reduce themselves to the
axial points B and D respectively.
20. Still assuming d <b, and passing over the case of external contact, we come to
that in which the circles intersect each other; but this case has to be subdivided: since
the circles intersect we have 6+d ><a, consistently herewith we may have
b, d each<a, A, E each outside the lens common to the two circles,
b=a, d<a, A outside, # on boundary of the lens,
b>a, d<a, A outside, # inside the lens,
b>a, d=a, A on boundary of, Z# inside the lens,
b, d each>a, A, E, each inside the lens;
and in each case we have to consider the different admissible values of c. I omit the
discussion of al] these cases.
21. Still assuming d<b, and passing over the case of internal contact, we come to
that of the circle d included within the circle b: we have here again a subdivision of
AND IN PARTICULAR ETC. 399
cases; viz. we may have d> A, that is A inside d, d=, that is A on the circumference
of d, or d<a, that is A outside d. The critical values of ¢ arranged in order of in-
creasing magnitude in these three cases respectively are
d>a d=a | d<a
b—d—a, p= Vil. b-—d-—a,
b-d+a, b, b+d-a,
b+d-—-a, b, | b—d+a,
b+d+a, 642d, | b+d+a.
I attend only to the first case; we have here (see Plate VII.), interior series, the
figures 1, 1—2, 2, 23, 3, 8—4, 4, viz.
fig. 1 c=b—d-—a,
1—2 ,, intermediate,
2 c=b—d+a,
2—3 ,, intermediate,
3 c=b+4+d-a,
3—4 ,, intermediate,
4 c=b+d+a.
22. In figure 1 the curves described by the points B,D are each of them a mere
point. In figure 1—2, we have two critical positions B,D, situate symmetrically on
opposite sides of the axis, B, being a summit on b, and D, a two-way poimt on d,
and moreover two critical positions B,D,’ situate symmetrically on opposite sides of the axis,
B, being a two-way point on 6, and D,’ a summit on d. The described portion of b
is the arc B,B,, and the described portion of d is the are D,'D,’, these two arcs being
thus the nearer portions of the two circles respectively.
23. In figure 2, the four critical positions coalesce all of them in the axial position
BD; the described portions are thus the entire circumferences of the two circles re-
spectively. This is a remarkable case. The configuration is p,o=++; B, D, R, 8S;
yh: Ys=++. Imagine D to move from the axial point D im a given sense round the
circle d, say with uniform velocity, then B moves from the axial pomt B in the same
sense but with either of two velocities round the circle b; one of these velocities is at
first small but ultimately increases rapidly, the other is at first large but ultimately
decreases rapidly, so that the two revolutions of B from the axial pomt B round the
entire circumference to the axial point B correspond each of them to the revolution of
D from the axial poimt D round the entire circumference to the axial point D. And
similarly if we imagine B to move in a given sense from the axial point B round the
circle b, say with uniform velocity, then D moves from the axial point D in the same
sense but with either of two velocities round the circle d: one of these velocities is at
first small but ultimately increases rapidly, the other is at first large but ultimately
400 Pror,. CAYLEY, ON THE KINEMATICS OF A PLANE,
decreases rapidly, so that the two revolutions from the axial poimt D round the entire
circumference of d to the axial point D correspond each of them to the revolution from
the axial point B round the entire circumference of b to the axial point B.
24. In figure 2—3 there are no critical positions, the described portions of the circles
b, d are the entire circumferences of the two circles respectively, these being described
in the same sense, by the points B and D respectively. It is to be observed that to
a given position of B on b, there correspond two positions of D on d, or say two positions
of the link, but the link does not in the course of its motion pass from one of these
positions to the other; the motions are separate from each other, and may be regarded
as belonging to different configurations of the system. And of course in like manner to
a given position of D on d, there correspond two positions of B on b, or say two positions
of the link: we have thus the same two separate motions.
25. In figure 8 the critical axial position BD of the link makes its appearance, the
described portions are still the entire circumferences of the two circles respectively. As
the point D is here to the left of the point B we must take the origin O to the right
of B, and reverse the direction of the axis Ox; the configuration is thus p,o =+-,
B, 8S, R, D; y%,:Yy,.=—-—. Everything is the same as in fig. 2 except (the signs of y% : y%
being, as just mentioned, — —) that the motions in the circles b and d instead of being
in the same sense are in opposite sense, viz. as D moves from the axial pomt D in
a given sense round the circle d to the axial point D say with uniform velocity, then
B moves from the axial point B round the circle b im the opposite sense, and with either
of two velocities; and similarly as B moves from the axial point B in a given sense
round the circle } say with uniform velocity, then D moves from the axial point D round
the circle d in the opposite sense, and with either of two velocities.
26. In figure 83—4 we have again the two critical positions B,D, symmetrically situate
on opposite sides of the axis, B, a summit on b, D, a two-way point on d: and also
the two critical positions B,D,’ symmetrically situate on opposite sides of the axis, B, a two-
way point on b, D, a summit on d. The described portion of b is the are BB, and
the described portion of d the are D,'D,’, these ares being thus the further portions of
the two circles respectively.
Finally, in figure 4 the described portions reduce themselves to the two points B, D
respectively.
27. The several forms for d=b can be at once obtained from those for d<b; the
only difference is that several intermediate forms disappear, and the entire series of divisions
is thus not quite so numerous.
AND ON A CURVE-TRACING MECHANISM. 401
Part II.
1. The curve-tracing mechanism was devised with special reference to the curves
of three-bar motion, viz. the object proposed was that of tracing the curve described by
a point K of the link BD, the extremities whereof B and D describe given circles re-
spectively, or more generally by a point K, the vertex of a triangle KBD, whereof the
other vertices B and D describe given circles respectively, and that in suchwise that the
points B and D might be free to describe the two entire circumferences respectively: but
the principle applies to other motions, and I explain it in a general way as follows.
2. Imagine the cranked link BD, composed of the bars B@ and D6, rigidly attached
BB to the top and D8 to the bottom of the cylindrical disk K (this same letter K is
used to denote the axis of the disk), and where BS and Dé may be either parallel or
inclined to each other at any given angle, so that referrmg the points B, H, D to a hori-
D
ese zai
2 K 8 Cranked link with disk: elevation.
O B K 1) &
Cy) Arm of Pentagraph : plan.
zontal plane BK D is either a right line, or else K is the vertex of a triangle the other
vertices whereof are B and D. The disk K, with the attached bars B8 and Dé, moves
in a horizontal plane: and if the motion of the pomt B be regulated im any manner by
a mechanism lying wholly below B and supported by the bed of the entire mechanism,
and similarly if the motion of the point D be regulated in any manner by a mechanism
lying wholly above D and supported by a bridge of sufficient length (resting on the
bed of the entire mechanism), then the disk A moves in its own horizontal plane un-
impeded by other parts of the mechanism: and if we fit the disk K so as to move
smoothly within a circular aperture in the arm of a pentagraph, then the pencil of the
pentagraph will trace out on a sheet of paper the curve described by the poimt K on
the axis of the disk, or say by the point K of the beam BKD, Of course for the
three-bar motion, all that is required is that the point B shall describe a circle, viz. it
must be pivoted on to an arm AB, which is itself pivoted at A to the bed: and that
the point D shall describe a circle, viz. it must be pivoted on to an arm DZ#, which is
itself pivoted at H to the bridge. Special arrangements are required to enable the variation
. of the several lengths AB, BK, KD, DE and ED, and the mechanism thus unavoidably
assumes a form which appears complicated for the object intended to be thereby effected.
402 Pror. CAYLEY, ON THE KINEMATICS OF A PLANE ETC.
3. The form of Pentagraph which I use consists of a parallelogram ABCD, pivoted
together at the points A, B, C, D, the bars AD and BC being above AD and BC. There
is a cradle G, rotating about a fixed centre, and which carries between guides the arm
AD, which has a sliding motion, so that the lengths GD and GA may be made to have
K D Gobeces L
YU A B iP
any given ratio to each other. Above the bar DC and sliding along it we have the arm
KL (where K is the circular aperture which fits on to the disk K of the cranked link):
and above AB and sliding along it we have the arm MP which carries the pencil P:
of course in order that the pentagraph may be in adjustment the points AK, G, P must be
in lined. .
XII Examples of the application of Newton's polygon to the theory of singular
points of algebraic functions. By H. F. Baxer, M.A., Fellow of St John’s
College.
INTRODUCTION.
APART from its interest in the theory of plane curves, the theory of the multiple
points is a convenient preliminary to the study of algebraic functions. We may of
course suppose every algebraic curve to be beforehand transformed into one possessing
only ordinary double points. But this transformation is one which it is not in general
possible to carry out practically.
Cayley’s rules for any singularity whatever have been amply justified in many sub-
sequent papers. But in all these a good deal of calculation is necessary to obtain the
series used and the final result. We naturally seek to find a method for evaluating a
multiple point which shall appeal more directly to the explicitly given coefficients of the
curve upon which these series depend. The following paper gives some rules which are
effective in a very large number of cases—founded upon a consideration of Newton's paral-
lelogram. The deficiency of a curve and the equivalent number for any multiple point
is determined by counting the number of unit points within a certain polygon which can
be immediately constructed from the equation of the curve. I have sought to give typical
examples used in other papers as illustrations of other methods and shew the application
of the present rules to them.
For convenience the paper is separated into six parts. In the first part it is shewn
that Abel’s determination of the deficiency of a curve admits of an immediate graphical
interpretation. In the second part that this graphical result is in accord with the theory
of Abelian integrals—the deficiency being defined by the number of integrals of the first
kind that are linearly imdependent and the explicit form of these integrals determined.
Cayley’s rules appear thus as following from Riemann’s number associated with the con-
nectivity of his surface. The general values of the coefficients of the curve thus far
accepted are in Part III subjected to certain restrictions of frequent occurrence and a
graphical rule given for the necessary correction. These rules are applied in Part IV
to various examples; among them is a consideration of Weierstrass’ normal form of curve
of which the corresponding Riemann surface has a branch point at infinity in which all
the sheets are included. And it is proved that the number of orders of integral algebraic
Vou. XV. Part IV. 53
404 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
functions that are not integrally expressible is the same as the number of double points
of the normal curve.
This part may serve as the beginning of a commentary on Kronecker’s paper
(Crelle, 91).
Part V is devoted to shewing that the quadratic transformation applied by Cramer
and Noether is in direct connection with the graphical rules previously given. A par-
ticular monomial transformation obtainable by a succession of such quadratic transforma-
tions is employed in Part VI, to the example given by Noether in exposition of his own
method, and to transform any curve to one whose only singularities are at infinity.
The main result of Part II, found in August 1892, was given, not quite correctly,
in the Mathematical Tripos, Part 1. of this year. This result enables us in all cases to
specify immediately an upper limit to the deficiency of any given curve and a lower limit
to the equivalent numbers of any of its multiple points. Cayley’s
rules of course apply to all possible cases—the rules given here for
the exact values of the deficiency etc. may fail for particular values
of the coefficients of the curve. In the simple case of a curve
wherein all the terms are present, say for instance the quartic curve
the deficiency 3 is the same as the number of unit points entirely
within the triangle ABC in the diagram which represents all the terms
= a of the curve in Newton’s manner.
In case the constant term and the terms z, y, be absent, im which case there is
a double point at the origin and the deficiency is 2, we have the
second figure, having as before a number of interior points equal to the
deficiency. The same is true when the terms in ay, y* are absent.
This illustrates the general rule obtained here.
It may be remarked that Part I. is added for the sake of
completeness: and the results of it not assumed in what follows.
October, 1893.
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 405
PART I.
On Abel’s expression for the least number of sequent intersections of a curve with a variable
curve.
In the Phil. Trans. of the Royal Society [1881] Mr Rowe has given an exposition of
Abel’s great paper (Collected Works, 1881, page 145) upon the sums of integrals related
to a plane curve. Part of this paper is occupied with the determination of what
Prof. Cayley, in an appendix to Mr Rowe's paper, proves to be, in general, the deficiency
of the fundamental curve. The subtlety of the method employed by Abel in this part
of his paper will justify the following diagrammatic interpretation of the algebra employed.
It would not be wonderful indeed if some such method were in the mind of Abel. I
have preferred to give by the way enough account of Abel’s method to make the advantage
of the present representation obvious.
If we have a curve
X(Y=Y"+ Pray" + Pay" + ot Pros
and any associated curve, this latter can in all cases so far as its intersections with x (y)
are concerned be taken in the form
BY) =dnay™ + +40,
wherein Gn4, Qn—2,-+», Yo are integral functions of # of at present unassigned order, whose
coefficients are to be regarded as variable and independent,
then, denoting by 4%, y,.--Yn the n roots of y(y)=0 for any value of w#, the
expression
E=6(y). (yo)... O(Y,)
gives the abscissae of the (finite) intersections of these curves, and the number of these
intersections is equal to the degree of H in a If then one of the roots of y(y), when ex-
panded in descending powers (supposed positive), begin with the term in 27, and @y denote
the highest power of w in @6y when a is written for y, this degree of # may be denoted
by S6y.
In what follows we desire to determine how many of the intersections of yy and 6y
are determined by the others. It is clear in fact that as many points of @y, upon yy, can
be determined as there are assignable constants in @y, and that the remaining intersections
of @y with yy are determined by the values assigned to these coefficients in 6y, and these
remaining intersections alter in a definite way when the coefficients of @y are altered.
Since now there are in @y effectively
Qn +14+GQ2.+14+...+G%+1-1
= Xq¢+n-—1 coefficients,
where q means the degree of q in 2,
53—2
406 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
it follows that the intersections of @y with yy which are determined by the others are in
number
Ly —Xq-—n+1.
In what follows we seek by a proper choice of the terms and degrees in @y to make this
expression as small as possible.
Suppose that the initial terms in the expansions of y,, y,...y,, consist of
Mp, terms of the forms A,v%, A,v%, ... , Ane 22,
na, terms of the forms y= Ba, By, ..., B,, ie
&e.
where Nypy + Np, +... =,
0, >0,>03>....
Then when we substitute in Oy, y=2%, there will be in general one term wherein the
resulting power of « is highest.
Denote this term by a! y? where [p,] is another notation for the highest power of «
in g,. In the same way the term which gives the highest power of 2, when in Oy x is
Pi :
written for y, is denoted by al?! y?2, and so on.
Abel proved that it is possible to arrange the degrees and the coefficients in Oy
so that
p: 1s one of the indices n—1, n—2, ..., n—Np,,
p2 is one of the indices n—my,—1, n —mMy,—2, ..., N— Mp — Np, ,
and he works out the least value of the number of ‘sequent’ intersections of yy and Oy
on this hypothesis, We shall follow him.
Imagine that we have a plane of rectangular coordinate axes, the positive quadrant
of which is ruled with lines parallel to the axes at unit distances apart, and let every
term of @y be represented on this chart, the term z*y* being represented by the point
whose abscissa is h and whose ordinate is k. Thus we shall have g,,+1 terms on a
line parallel to the axis of # at distance n—1 from it, representing the terms in @y
which were written g,,y", and so on, Of these points we shall only here be con-
cerned with those, on the various lines parallel to the axis of z, which are furthest from
the axis of y. The power of « arising from any term in @y when a? is written for y
is easily constructed graphically by drawing through the point of the chart that represents
that term of @y a line whose positive direction makes with the negative direction of
the axis of y the angle (for the present assumed to be between 0 and 5) tanoc. The
distance of the point in which this line meets the axis of # from the origin is the power
of x arising from the term of @y considered: and to say that the term «Jy gives the
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 407
highest power of # when y is written #71, is to say that, if through the point of the chart
whose coordinates are [p,], p:, there be drawn a line whose positive direction makes with
the negative direction of the axis of y the angle tana, (between 0 and 5), and a
parallel line be drawn through every other representative point on the chart the first
drawn line will meet the axis of z further from the origin than all the points in which
the other lines meet the axis of # Let the point [p,], p, be called R,, [p.], p. be called
R,, and so on, and let the line parallel to the axis of x at distance / be called J;. Then
as we have said Abel shews that the point A, may be taken to be on one of the lines
Ina, Ins; ++» Un—ny,. These lines we shall call the first set—and so for each of the following
sets. Suppose now that a line o, is drawn through &,, and a line o, drawn through
R,, and so on. [By a line o, we mean a line making an angle tan~'c, with the axis of y,
as previously explained.| These lines form with the two axes of coordinates a closed
polygon, and it is obvious that the expression of the characteristic property of the points
R,, R,,... is that all the lines R,R,, R.R;,... shall lie within this polygon. This is the
expression of Abel’s conditions
Ce Oni eae
[pr] = [p1] ate = Tk (Px Pak Px):
7, being the tangent of the angle which R,R,,, makes with the negative direction of the
axis of y, and is obviously sufficient to ensure that the term corresponding to R, gives a
higher power of « than either of the terms corresponding to R,, R;,... for y= 2%, and that
the term corresponding to R, gives a higher power of « than the terms corresponding to
R,, R;,... for y= 2%, and so on.
Consider now R,. We have to ensure that for y= this shall give not only a
higher power of x than the term corresponding to R,, but shall also give a higher power
of «x than every other term in the set to which &, belongs. Abel shews that the analytic
condition for this can be reduced to the two following criteria:
(1) that the term R, for y=" gives a higher power of # than y=«*r gives in each
of the terms of the following set (r+1) only;
(2) that y=2’" gives for the term R, a higher power of 2 than it gives for any of the
terms of the previous set (r —1).
And it is easy to see that these conditions are sufficient. For suppose the first
satisfied. Imagine lines drawn through all the points of the set (r+1) parallel to c,
and a line parallel to these drawn through R,. By hypothesis this last line meets the
axis of « at a point further from the origin than any of the other lines do. If now all
these lines be turned to a greater inclination with the negative direction of the axis of y,
into the direction ¢,,, each about the pomt through which it was drawn, this statement
will remain true—namely, the line drawn through the point R, parallel to o,, is further
from the origin than the parallel lines drawn through the points of the set (r+1).
408 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
Therefore the line through the point #,, parallel to o,, which by hypothesis is further
from the origin than the parallel line through &, is also further from the origin than the
parallel lines through the points of the set (r+1). Continuing thus we can shew that for
all values of s less than 7+1, the line through R, is further from the origin than the
parallel lines through the points of the set (r+1). Supposing next that the second con-
dition is satisfied, namely that the line through R, parallel to o, is further from the
origin than the parallel lines through all the points of the set (r—1), and supposing
all these lines turned about their respective points to a less inclination with the negative
direction of the axis.of y, so as to become parallel to o,,,, the line through #, will remain
the furthest from the origin. But now the line parallel to o,,, is by hypothesis further
from the origin than the line through R, and is therefore also further from the origin
than the parallel lines through all the points of the set (r—1). Continuing thus we can
shew that the line through R, where s>r—1 is further from the origin than the parallel
lines through the set (r—1).
Thus we have only to consider how to satisfy conditions (1) and (2) for all values
of r, The first condition clearly is that all the points in the set (r+1) le on the same
side of the line through R, parallel to o” as does the origin. While the condition that
R,.,, corresponds to the highest term in # for y=a27™, of the set (r+1), requires that
all the terms of the set (r+1) lie on the same side of the line o,,, through R,., as does
the origin. We see in fact that the conditions only are that all the points must le within
the polygon, and further that this is perfectly obvious geometrically without the cumbrous
interposition of the conditions (1) and (2).
Considering now again our expression
= Oy, . Oy, ... OYn
it is clear that the first mm, factors give rise to the same power of # as their highest
power of z, namely the power [p,]+.0:. For 2 was the highest power of « in each of
Yi, Yor» Yn,u- We shall therefore have in the summation S@y, mp, terms each equal to
[p:]+p.c:. But it is convenient to write each of these my, terms in a different way,
thus
Let q.,.y* denote the general term corresponding to the first set on our chart, so that
a, is in turn equal to n—1, n—2,...n—m,. The degree of the term q,,.y" for y=” is
[a] +o, where [a,] means the degree of qg,,. We denote the difference
[pi] + pio — [a]—ao, by De
which gives [pi] + pres = [um] + 101+ D,,,
and this is the substitution which for the my, values of a we make for the ny, terms
of the form [p;]+ 0, arising in S0y. And the part contributed by these terms to the
summation S6y—g, since the values of g entering here have also been denoted by
{a,], 1s
oO; 3 a, + =D,,.
n— Mb,
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 409
The whole expression S0y—q can therefore be written
2 ape . Gt. == =Da,.
And %,o,. 2%, writing m,=o,[,,
1
F010 E - ed
aR o00006
+ 2M E = 4 fly — 0 — Np fra — tebe tS |
ta ce iereteia
Consider now =Da,.
We have Da, = [pr] — [er] — (pr — &r) or
My
=[p,]—[a,]+imtegral part of (p,—a,) Fi
(pr — Oy) MyNp
+ fractional part of
Mrly,
this fractional part being taken positive.
And Dg, may be constructed graphically by drawing a line through the point [a,], a,
parallel to c, to meet the line y=p,, say in A,. The line A,R, is Da, (and by the definition
of R, is necessarily positive and has a positive integral part). Since now it is our endeavour
to make 2@y—q as small as possible, and since the other part in the expression for this,
namely ¥,o,2.0, has a definite value prescribed by the curve y we shall make our sum-
mation 2@y — Zq as small as possible if we make the part =Dza, as small as possible. We
may agree then first of all that the imteger part of Da, shall vanish, and we may notice
here that this uniquely prescribes the chart-pomt (of @y) upon the line y=a, lying furthest
from the axis of y, namely thus,—imagine a line keeping always parallel to o, to move
from the position in which it passes through &, towards the origin, then the first unit
point it reaches upon the line y=a, is the point prescribed.
And > fractional parts of
(Pr = a,) MyNy _ ss r—1
at = on,
[rN 2
and this has a value dependent on the curve y only.
Thus on the whole
L6y —Sq—n+1
410 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
has for its least possible value
r=1
> n-m,NsMs + $En2mp — 4Xnm— bInw—}$Sn+4+1
8>r
and n= Xn,
and this is the number of ‘sequent’ intersections; and we may bear in mind that this
number was diminished by taking the curve @y such that the quantities D, were all
Tr
less.than 1.
But it should be noticed that this enumeration takes no count of possible infinite
intersections. It is in fact to be afterwards shewn that our conditions D,.<1 are
equivalent to prescribing a certain number of points at infinity on Oy. So that the
curve @y is not only specialised by the supposed prescribed values given to the 2g+n—1
coefticients left in it, but also by the prescription of these infinite points.
Returning now to the polygon formed by the lines o,, o2, ... its construction contains
necessarily a very large amount of arbitrariness. Writing for shortness
%%, Ts, --- for [\p:], [pa], ... respectively,
the points (7, pi), (7%, ps), ... are first to be taken arbitrarily, save only that p, is to
be one of the numbers n—1, n—2, ..., n—msy, p. one of the numbers
N— Mp, —1, ..., N— Np, — Noflg, etC.,
and 7, is to be sufficiently great for the line o, through A, or (7%, p,) to meet the axis
of y beyond the point (0, »—1)-—though the contrary only means that qj is identically
zero. Then (r,, p,) must be taken consistently with a certain condition that may be
thus expressed :
Denote the intersection of the o, line through R, and the o, line through R, by A,—and
suppose first that the line y=n—my,—1 meets these o,, o, lines at points further from
the axis of 2 than K, is, say in A,, A, respectively. Then A,A, must be less than 1.
This is required by the condition, which was necessary to make our quantities D, <1,
that the curve points of @ furthest from the axis of y and belonging to the second set
should all be at less than unit distance measured parallel to the axis of #, from the o, line,
combined with the condition that these points must be within the polygon. Or supposing
next that the line y=n—nyw,+1 meets the o,, o, lines in points not so distant from
the axis of # as the point K, is, say B,, B, respectively, then B,B, must be less than 1,
for a similar reason. This condition ensures that the pomts in which y=n—m, meets
the o,, o, lines shall not be beyond a certain limit of distance from the point K,. It
is of course easy to express this condition analytically—and a similar condition must
obviously be satisfied at each angular point of the polygon.
We should next remark that the conditions D, <1 together with the other conditions
for @y are really equivalent to prescribing that our curve @y shall behave as an ‘adjoint’
curve at the multiple points of yy that lie at infinity. This is really obvious from
Cayley’s proof that the number of ‘sequent’ points given above is the deficiency of the
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 411
curve yy. For we know that the number of sequent intersections of a non-adjoint
curve with a curve yy having 6 double points and & cusps is the deficiency +6+h.
And it is assumed in Cayley’s proof that the curve yy has no finite singularities. We
shall moreover give an independent proof of the fact that the expression found above
for the number of sequent points is in general the deficiency of yy (Part II. of present
paper).
We proceed now to shew that the above expression for the number of sequent points is
capable of a certain graphical interpretation.
Suppose as before a plane of rectangular axes to have its positive quadrant ruled
with lines at unit distance apart parallel to the coordinate axes. Let the intersections
of these lines be called unit points. Join now the points (0, n), (mm, n—mpy) by a
straight line. This will be parallel to the o, line before spoken of and will contain,
counting the end points, m,+1 unit points. We shall denote the coordinates of the ex-
tremities of this line by (a, y) and (#, y,) and call them P,, P;. Join P, to the point
(%, Y2) where 2,=1mm,+NM,, Y2=N— Mp —Nf,. Denote (x,, y,) by P,. P,P, will be
parallel to the o, line before mentioned, and contains, counting the end points, n,+1 unit
points. Proceeding thus we shall get a polygon whose sides are the two axes of co-
ordinates, and lines parallel to the o,, o,, o,, ... limes. We may call the number of these
latter lines k+1, so that the last of them is PyPyii, Pr being (teu, Ye) and Yer
being 0O—and Snm=a2y44,2n~=n. Then what we proceed to prove is that our number
previously found for the number of sequent points is the same as the number of unit
points within the polygon.
In proving this we shall not, except at first, need to assume that o,>o,>o,... or
that o,, o,,... are positive.
Pra
Neri Nr Ny Ny.2
Consider one side P,P, of our polygon.
Let P..N,,, P,N, be the ordinates from its ends to the axis of a, and let P,K,,
be drawn parallel to the axis of « to meet P,_.N, in K,,. Then the number of unit
points actually within the triangle P,K,.P,. together with the number of those (except
P,_, and P,) wpon the side P,P, is
Wa
5 (Mr? My fly — Ny My — Nyy + Ny),
Wor: XV. Parr TV. 54
412 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
as is easily seen by completing the rectangle of which P,P,, is the diagonal and re-
membering that the number of unit points on P,P,, is n,. [This gives a graphical
proof of the theorem
z=A-1 B 1
Sint t (eq) =3(4B-A-B+d
=, integer par x=) 3 ( +d)
where A, B are positive integers and d is the greatest factor common to both.]
Also the number of unit points within the rectangle P,K,.N,N, together with the
number of those (other than P,, K,_,) upon the line P,K,, is y,(a-—2,.—1)—and the
number of unit points on the line P,_,N,,, other than N,,, is y,,. Adding these three
numbers, and subtracting the number of unit points upon P,_,P, other than P,, namely n,,
and putting ¥4—-Y,=M-by, &—-@%,~=n,m,, and y,= =X nus, we obtain as the number of
s=7r+l1
unit points actually within the trapezium P,P,.N,,N, and upon the side P,_,N,_, (other
than P,_,) the result
Nyy + Ny~M, DY Ngitg + 2 [2,27 fy — Nyy — NyMy + Ny] — Ny,
s=r+1
which because n= = nw, is equal to
s=1
1 1 it
Ny poy + MyM, [NM — Noy — Noflg ... — Nyfly] + glee [Mby — 1] - 9 rbr — 5 Mr
Nyy +1 1 ‘
=N,M, [» = Myf -2. — Ny apy — = | ae 9 [ijt — LI] s sekosrences (NS
and if we assume for the present that all the quantities o,, o»,... are positive, it is obvious
that the whole number of points within the polygon is merely the arithmetic sum of such
expressions as these, except that we must subtract from this sum, in order to exclude the
unit points on the axis of y which occur for the trapezium P,P,N,N,, the number n—1.
If this arithmetic sum be formed it will be found to agree with our number. But with-
out this it is sufficient to notice that the expression (i) found above is identical with the
value before found for ¢,a,+ , fractional parts of Da,, and to recall that our number was
defined as the value of
= [o,2a,+ = fractional part of Da,]—n+1.
a a
The geometrical interpretation of the formula is then established in case oj, o2, ... be all
positive.
In case however some of them be negative, e.g. o,4, in the figure [p. 411], it will be
found that the contribution corresponding to the trapezium P,P,,N,4,N, has the same form
as a function of the quantities n,, m,, p,, M., M, fly, -.. aS if o,4, were positive. In fact
having calculated the number of points as above for the trapezium P,,.P,N,N,., we must
subtract the number of points within the trapezium P,P,,,N,,,N, and also the number of
points upon the sides N,.,P,.;, P,i,P, (other than P,). If after this ¢,,. should also be
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 413
negative it will be needful to subtract an exactly similar quantity for the trapezium
PriPyi2Ny42Np41; while if o,,. be positive we shall have to add an expression for this
trapezium which is to be calculated in exactly the same way as was the contribution for the
trapezium P,.P,N,N,.. But the subtractive contribution above corresponding to the
trapezium P,.P,.,N,.,N,:is
1 5 S
~15 (= My papery + Up Mrya — Urgabrga — Ny) + (= MrpaMry —1) Z Maps + >> Nsbs | 5
s=rt2
8=7t2
which is exactly equal to
Ibe
Ny safer pi Np pyMy+1 = Mss + 5 [MW rtaMrsapertr — Mrsabertr — NrsiMyp1 + Nga] — Nr4r,
s=rt2 at
and this has exactly the same form as a function of 7+1 as had the expression found
above for the contribution of the trapezium P,.P,N,N,_, as a function of r. Thus our
geometrical interpretation is completely justified.
Parr II.
A priori proof of the significance of the number of points within Newton's polygon.
Taking once more our positive quadrant of rectangular axes ruled with lines at unit
distance apart and any arbitrary curve whatever, #\=0, mark on the chart, corresponding
to the term A,.,v’y* of the curve F, the poimt whose coordinates are v=r, y=s. This will
be called a curve point, the original points being called merely unit points. Then it is
possible to form a polygon each of whose sides shall begin and end in a curve point
and which shall be everywhere convex and have all the curve points (other than those on
its sides) in its interior. And in fact startmg from the curve poimt on the axis of y
which is furthest from the origin, say the point P, at distance n from the origin, let a
line passing through P, and coinciding with the positive axis of y turn about P, in a clock-
wise direction until it again contains a curve point. In this position it may contain several
curve points. Im any case let P,; denote the curve point on this line which is furthest
from P,. Let mm, be the abscissa of P, and n— my its ordinate, m, and y, being coprime
and y, possibly negative. Put o, for a and notice there are 7,+1 unit points upon P,P,.
In the same way let a line pivot in a clockwise direction about P, from coincidence with
the continuation of P,P, until it again contain curve points, P, beimg then the curve point
furthest from P;, the coordinates of P, being a,=1mm, +n, Yo=N—Nf,— Np, where
2
Z n : : :
Ms, fo are coprime; use o.= And so on until we ultimately come to a point P;,, on
2
the axis of w, this being the curve point on the axis of # which is furthest from the origin.
In a similar way let P’, be the curve point on the axis of y which is nearest to the origin,
at a distance nm’ say—and proceed from this to obtain in succession the straight sides
P/Py, PyPy, ... PueP ery
94—2
414 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
the successive rotations being now all in a counter clockwise direction. It is obvious that
all the fractions o,’, o;'... thus obtained are positive.
Then if in the equation of the curve we make a substitution
y =A’ + infinite descending series of powers of «
the highest power of « arising from any term A,,«"y® of our curve is the abscissa of the
point in which the axis of z is met by the line drawn from the curve point (7, s) in the
direction making with the negative axis of y the angle tano. If then o, o,... be all
positive, the terms in the curve corresponding to the unit points upon the side P,P;
become, for the substitution y= Az%+..., of the same order in a, this order being higher
than that arising for this substitution in any other terms of the equation of the curve.
Hence the curve has a series of infinite branches whose equations are of the form
Dele orae
the values of A being obtained by arranging the terms of the curve corresponding to
the curve points upon P;,P;, in the form
Cari-ryi [yi — kya) «2... [yt — kya]
(where a, Yin, %, yi are the coordinates of P;, and P;).
In what follows we assume that each of oj, o,... are positive. The method of proof
does not otherwise apply without considerably more detail in explanation. Various examples
are however given in which the main result obtaimed here holds when some of aj, op,...
are negative. But the consideration of this case is never necessary in practice, because
by the substitutions c=&+cn, y=n+c8&, it is always possible to reduce the equation
to one in which the highest powers of & and 7 that enter have, both, constant co-
efficients—in which case all of a, o»,... are positive.
In the same way as for the infinite branches, the diagram enables us to state the
first terms of the expansions
y = Aa® +infinite ascending series of higher powers of «,
of the curve near the origin, here supposed to be a multiple point.
Naturally we confine ourselves in the first instance to the most general curve
represented by the diagram—in that case its singularity at the origin and at infinity
is competently represented by the diagram. It is afterwards shewn how to represent
diagrammatically the corrections needful when the coefficients of the highest or lowest
terms in the equation are subject to certain particular relations, which are those of most
common occurrence.
Proposition. Consider all the unit points entirely within the polygon and write down
a curve with perfectly general coefficients whose curve points are just these unit points.
Since no one of these unit points has a zero abscissa, or a zero ordinate, the equation
of this curve will be divisible by «y.—Denote the curve then by azy¢. Then I say
that ¢ is of order N—3, where N is the order of the original curve F, and that it
is ‘adjoint’ to F at the origin and at each of the singularities at infinity. Limiting
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 415
ourselves to the case when all of o,... are positive the only exceptional case is when there
are only two of these, ¢4,.=2,o,=0. Then ¢ is of order N—4. This is the case in
Riemann’s canonical form for the equation of his surface. In this case ¢ is to be interpreted
as 2p where z=1, and z=0 is the equation of the line at infinity: then zH=0 is the
most general adjoint curve of order NV —3.
From this proposition it will follow that the number of unit points entirely within
the curve polygon is p+6+x, where p is the deficiency of F and 6+.« the number of
simple double points and cusps to which the finite singularities of F# other than the
origin are equivalent. This follows from the known number of linearly independent adjoint
curves of order N—3. And wf the curve have no finite singularities other than the origin
the number of interior points will be exactly equal to its deficiency.
To prove that the order of yp is N—1 we remark that if P,P, with coordinates
Ly, Yr and «x, ys, be the ends of the side of the polygon which represents the terms of Ff
which are of highest aggregate order, so that either s=r or else s=r+1 (in which case
P,P,.; is inclined at 45° to the negative axis of y), and if Q, be the unit point (#,—1, y;),
Qs; be the unit point (#,, y,s—1), then the side Q,Q, contains the points representing the
highest terms of the curve zyp and these terms are clearly of order N—1. The only
exceptional case is the Riemann curve just mentioned in which Q,Q, are not points for
the yp curve—being on the sides of the # polygon. But the modification and verification
of the result stated is obvious.
To prove that ¢ is ‘adjoint’ at the origi and infinity it is sufficient to prove that
the integral
ae a vy “y_&)
olan | On ( =)
dz
where z, =1, is introduced into the equation F to make it homogeneous, is finite on all
the branches at infinity and at the origin.
Consider the infinite branches and consider first the case where as above there is a
side P,P,,, of the polygon inclined at 45° to the negative axis of y. Then the curve has
branches at infinity, y= Aa + lower powers of «, along which (for «=rcos 6, y=rsin @)
dy fe dx _ dé
y « sin@cos@
; steel Ol
is zero of the same order as d@. The terms entering in a, can be represented in our
chart and will give rise to exactly the same curve points as F' with the exception only
of the points on the line P,P,,,. The points Q,Q,,, mentioned above, namely the points
whose coordinates are (#,—1, y,), (@1:,; Yri1—1), which represent the effectively highest
terms of the curve zyd for a substitution of the form y= Ae+..., will be outside poimts
of the polygon representing the terms of 2. Hence _ is finite on this branch and so
a
416 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
therefore the integral. With the exception of points on this line Q,Q,.,, all other points
oF
~
if P,.. be the angular point of the F polygon before P,, and P,,, the angular point
arising from the curve xy le within the polygon representing the terms of In fact
after P,.., P,»1Q-QriP+. are outside points of the = polygon. Hence for any substitu-
r
tion y=Ax® in which o >1, a will be zero like some positive power of : Ze say, and
a e
oz
xyd (dy dx i Th ; 1
[= te = =) =|C(e —1) wn +integral of higher powers of he
02
will be finite.
Exactly similar remarks apply to the case when there is no infinite branch for which
o=1, and to the case of the singularity at the origin, at which the a polygon entirely
encloses the zyd polygon.
Hence our proposition is completely proven.
We may give the following examples of the case when all the o,, o2, ... are not
positive—in both cases the curve ¢ obtained by the interior points of the polygon is ad-
joint at infinity and the origin.
(1) F=ya+y(a, 1);+ (x, 1,=0. :
Here the points inside the polygon give aes
zryp = avy (A + Bo)
and in fact, if »=ye+4(a,1);, the equation becomes
2 = (x, 1),
which is known to be of deficiency 2, the adjoint curve which gives rise to integrals of
first kind being A + Be—in fact |u + Bx) “ is always finite, and this is, for our original
form
| (A+ Ba) = -
2 oy
(2) F=yat+y? (a, 1)o+y(#, 1),+(e, 1),=90.
The diagram gives sea
yp = vy (A + By + Cay).
And in fact, by «= MY the curve becomes
g
PF=f=y+yE&(1, &.+yF (1, E+, &.=9,
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 417
shewing that, by the demonstration given,
y& (AE +B cao) (te)
a aa
0z
_ (AE+ ByE+ Cy
a nas
oy
_, (AtBy+ Cay)/x dx
“f | 1 oF e
: a dy
& A+ By + Cry
-— F da,
oy
?
is everywhere finite. So that A + By + Czy is ‘adjoint’ as desired.
The proof thus furnished that the curve ¢ is an adjoint curve of order NV —3, gives
then, in the case in which the origin is not a multiple point, another proof of the theorem
proved by Professor Cayley in the addition to Rowe’s memoir referred to.
But more; it gives an evaluation of the number, 6+, of simple double points and
cusps to which our complex singularity at the origin is to be reckoned as equivalent.
For this equivalence is required only to be such as will give the proper value for the
deficiency of the curve: the value of « itself is independently determined by reference
to the number of cycles arising by all the branches at the origin—say by the number
of branch points at «=0 on the Riemann surface representing the equation F’ other
than those that arise by tangents of the curve parallel to the axis of y—which number
is clearly, in the notation explained, =n,’ (u,’-— 1), provided the expansions are of the form
1
y= (integral series in attr) and none of oa; a, ... are <1; and this is the number given
by Cayley (Quart. Jour. Vol. vii.). Considering then what are the additional points of
our polygon when the origin ceases to be a multiple point we have the
Proposition. The multiple point at the origin furnishes a contribution to the total
8+ of the curve F which is equal to the number of unit points between the axes and
the sides P,’P,', ... P’::; plus the number of those, other than P,’ and P’;.,, upon
these lines.
We proceed to verify that this is the number obtained by applying Cayley’s rules
(Quart. Jour., vol. vii.) to the expansions of the branches of the curve at the origin.
We have to consider the number of intersections. of all the branches corresponding
for instance to the side P’,,P’, among themselves, and the intersections of all the
branches corresponding to P’,,P’. with all the branches corresponding to iP Petoreall
values of s>r. For brevity we may be allowed for the present to drop the dashes, and
assume that each of o,, o,... is>1.
418 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
Then a branch y= A,x*" intersects a branch y = A,v* in o, points, in Cayley’s nomen-
elature. And the number of such pairs corresponding to ¢, is $n,u,(n,m,—1). So that
on the whole we get 4m,(n,u,—1) intersections. The number of intersections of y= Aa?"
and y=Bx%, where s>r and therefore o,<o,, is o,, and the number of such pairs is
Nyy. Mss. So that on the whole we obtain =n,nym,u, intersections. Thus Cayley’s
s>r
rules give the formula
33 1
8+ 5 © ==Tnnm,ps + 5 =N,M, (Myf, — 1),
— 8>r =
and hence, by « = =n,(u,—1)
1 1
S+e==UNNM ps + 5 Yn,M, (Np, — 1) — = Yn, (pw, — 1).
s>r
Using now the result before obtained for the number of unit points between the
axes and the sides P, ... P,,,, and remembering that the number of unit points on these
sides is =n,—1 (excluding P,, Py4,), the accuracy of our proposition above is verified.
The proof we have given of the Proposition makes it evident that it is not needful to
regard all of o;,’, o.’, ... as greater than unity. And it is easy to see that this result is
equally obtainable by Cayley’s rules: we divide, for this purpose, the sides into two sets
o,...0;,4 all <1, and o’,=1 and o7,,...0¢%4, all >1. The work is quite similar to
that given by Cayley in the addition to Rowe’s Memozr—but its expression is simplified
by the use of the diagram. The « of the point is in this case
t=r-1
= ni (m —1) + nm (m’ —1).
t=r t=r
We may notice that the contribution arising from a single branch y= Aa’ to 6++«,
being }n,p, (n,u,—1)o,—4n,;(u,—1) is capable of geometric representation. In fact if from
P,, P-K, be drawn perpendicular to the ordinate of P,_,, the contribution is equal to
the number of unit points inside the triangle P,A,P,. plus the number on P,P,_,
other than P, and P,_,. And the number of the intersections of this branch with all
following branches being n,m, =n,us, is equal to the whole number of unit points within
s>r
the rectangle P,N,, plus the number on the sides P,K,N,,., where P,,N, is the
ordinate of P,_,.
Part III.
Extension of foregoing to more particular forms of singular points.
In the previous cases we have assumed that the equation corresponding to any side
of the polygon for the origin has all its roots different. In particular we have assumed
that the branches which do not touch either the axis of «x or the axis of y have
separated tangents. This it is by no means necessary to assume. Moreover in counting
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 419
the number of cusps we have assumed that there are terms in the equation of the
curve corresponding to all the unit points within the polygon. This restriction also we
proceed to remove.
In fact, considering the branches that correspond to a side o of our polygon at the
origin, if a line coinciding with this o line move parallel to itself away from the origin
until it next contain unit points, and the point in which it intersects the axis of z in this
new position be called 7,, while its original position meets the axis of « in a point 7, then
1 : F : : Sic
TT,=~-. We have practically assumed that the unit points upon this new position of the
line are curve points. In what follows we assume that the first position of a line parallel
to the o line which contains curve points meets the axis of x in a point which is at a
: t : ;
distance from 7 equal to —. It will be found that the value of ¢ has an influence upon
the number of cusps corresponding to our singularity. (See for instance the examples, pp.
424, 425.)
It is necessary to consider the expansions with some particularity.
Consider the curve in the most general form possible
ahyk (yt — a,a™)™ ... (yt — aya) Ma + atayh (y#, am) + gloyh (y#, oat...
where h+ok+mn<h+ohk,+ 7m <ho+ok,+ryn<......
n=N,+N.+...+ Ny
and (y", #”)" means an integral polynomial homogeneously of degree r in the quantities
y*, «™; so that the terms are arranged to correspond to curve points on lines parallel to
the o-side.
1
Put £=2#, a definitely assigned value for each value of w, and y =v" = v2".
. 0 (vt—a,)™1...... (vu — ay) + vhES (ye, 1) 4 vbEL (wm, 1)2t......
(
where
f =h,-—h+o(k,-—k)+m(r,—n),
7
B= hy—h +o (k,—k)+m(r.—n)
(vt — a)": = Eng, (v) + E4h.(v) + «0.0. .
where
yh qd, yey
$:(v) = vE (yt — ady)N2 22. (ut — ay)NA
is a rational function of v which does not become infinite in the neighbourhood of
v= “/a,—and similarly for @.(v), etc. For the present I assume that ¢,(v) does not
become zero in the neighbourhood of v="/a,*. Then ¢, is the ¢ spoken of above as deter-
mined by 7;.
* Otherwise we proceed quite similarly with the first ¢ which does not vanish, and the corresponding t. See
an example in the Corollary to Part VI.
Won, OS IRA IIE 55
420 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
Then we may write
(oH — a, )¥i = E4g, (v) [2 i - © vase oS et aa |
= Ed (v) (1 + Ef, (v) + Mf, (0) + we ] say,
where f, (v), f:(v)... are rational in v and not infinite near v= fae
Then
1
a= RAYS) [1 + EA (0) + Befa(t) + eee
of which all the values for which & is small are given by
t ———
t= =o. YEO) [1+ By Alo) too]
where w, is in turn equal to all the N,th roots of unity,
t 1
say v =a, (1+ oya"™P (a, v)],
1 1
1
where w, a: have definite meanings and P(#, v) is a one-valued power series in
x, whose coefficients are rational functions of v, this power series not vanishing for z=0,
and the coefficients not becoming infinite for v=</a.
If now 6 be the greatest common divisor of N, and ¢, so that N,=A6, t= Bé, and we
1 8
put u = 744 = 9% then our equation becomes
vw =a, [1+ wy uFP (uA, v)].
Here A and B have no common factor.
It follows then that v can be expressed as an ascending series of positive integral
powers of uw, and cannot be expressed in integral powers of any root of wu. And all the
values of v near to u=0 are given by
1
v=0,0," ie += : wy uvBP (uA, v) + : =( - 1) 2B y,P (u4, v)+ =
and the continued substitution of this value of v in the right hand leads to the value
of vy as a power series in u,
1
v=0,0 + Ko,oyu* +...+higher ascending powers of wu.
To find the value of K we recall that
1
P (uA, v) =P (a*, v)
FA/ 600/14 By AW +],
is equal to
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNOTIONS. 421
ky
where $(v) = waned Ces
(uv — ay)... (UH — ay XA’
and ea ae h f
o(k,— esi Mire Vag Tea) te say ;
Int
so that Ope — co Ee Tem SAY,
N, i t Tp L t
= ++ — + (1, a)”
and a i) mN, mM, @@™, muN, | z
$ (Onc ) Oy a, (a, = An) Ne i (G— ay)Va
t Dp 1
lee less
=pCo,™™ ™™ a, “, say
where CO has a definite value.
So we obtain
LB ae oD
Y= 0,0" + Coo." ™iwy,u® + higher ascending powers of u,
1
= power series in 27,
where z2=Ap,
pies
and uB= oer;
eee on
1 y= Hoya +e" Co,oyou™™ ™™ + oe
But in this series the coefficients are in general functions of the w, and wy, chosen—and
certainly not always merely in multiplicative powers—see examples [on pp. 424 and 425}.
From this we are to obtain N, values of y.
These are in fact, arranging them in p» rows each of N, values,
tw res t+Le aia oui t+
= Y, V, ar OTN, i.
Th TPA as aa TO CaO), CIN aenies Hh, i= CoO, +e "Coo, wy,*;......
t t+L, 1 t
*EM Ow! [1+ am | Lip tea SRO) L [a+ mN,
Ou DN, ) sree Yj, i= U7 Wy!A" + & Co, d
pc er ce secccrsscesesvcecenseress
(where if we mean oy,’ as a i-th power we must assume wy, Was a primitive N,th root
of unity, etc.).
Suppose that underneath these w rows we write down the (A—1) w similar rows
belonging to the other roots dy,... a. It is easy to count the intersections of these wn
branches among themselves.
55—2
422 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
The intersections of any row are in number
Ie t
5M i-1) (+54),
iving in all Spin, (M—1)(e+—
giving in a =H ,(N,—1) orig ;
while any one of the branches belonging to the first ~ rows intersects each of the pn — Ny
branches, which are not in the same row with it, in o points, giving then
ee :
3 a 1 (un —N,)
since each branch is thus counted twice.
Thus on the whole we have
1 1 = 1 = 2 1 = ; :
5num=N, — gaa P+ 5m(2N2—=N,) + 3f= (N, — 1) intersections ;
: tn 1 1 ; :
that is gem —5mn + 5t (n —X) intersections.
The first ~ rows give either one branch point of order NV, or N, branch points of
i
order y, or possibly f, branch points of order * :
1
(hus f;=1 or N;),
and counting then f, (oe
1
-1) cusps, so that the first wN, branches give nu—Zf, cusps
we obtain 1 1 1 1 il
8+K = snmp —snm + gt (n —A)- ret 3 Zi
and this is greater than the normal value
A *m, as a pei
gv grm— nut 5m
1 1 us
r = = = =>
by at (n— 2X) 3 E 3],
which, when there is one branch point of order uN, is
i
and when there are for each N,, branch points of order y, is
1
oy (n—2).
The quantity f, above must in fact be equal to 8. For, if taking one of the Nye
series and thinking of the corresponding Riemann’s surface, we allow x to describe a
closed contour on one of the sheets round #=0, the new value of the series must clearly be
another of the N,p series. To see this we have only to notice that the original equation
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 423
remains completely unaltered, and we may imagine the N,w series calculated from it in its
new form. One of these newly calculated series will be the changed value of the series
first considered.
Thus the «J, series consist of one or more cycles.
1
But in fact, since they are all of them rational in the quantity 244, revolutions of «
round «=0 can only change any given one of the series into ~A—1 other series. There
will therefore be eae 6 cycles.
pA
Substituting then 6 for f, in the previous formula the excess there found is equal to
Lena) 1] n-'Fach
star) —5[n- = ( »].
Putting t= B,d;, N;=Axzd;, where A;, B, are coprime, this excess is
t
{t= (M,-—1) —= [NV — 6 (N;)]}
> [Bi 8; (Azoe— 1) = Ady + 8x]
Nli—= Nl Nl dle
re [A,B é.— B,.—Axt+ 1] j
> [07A;, By, = Boy - Axo; + ox] = = [Nit —t-— N; + 8; ]-
And the quantity within the square bracket here is easily susceptible of a graphical
representation—thus, take in a plane, whose positive quadrant is ruled with unit lines as
before, a point on the axis of x at distance =t, from the origin, and a point on the aais
of y at distance N;, from the origin, and join these points.
The number within the square brackets is equal to the number of unit points within the
right-angled triangle so formed, plus the number on the hypotenuse, less two.
As an example of the previous, consider the curve
y (y? — ax) (y* — ba) + yar (y?, z+ a? (y?, x) = 0.
It can be shewn that the branches of this corresponding to (y?—az)* are of the form
y=eE a. 2% + ewrt at ew'x8 + ewx' [ey +o%S]+...,
where € is a square root of unity, and is a fourth root of unity, and where a, 8, y, 6
are perfectly definite.
Giving then to e and @ all their possible values we obtain the eight expansions:
y= Saat aka HERA 2 (GD) cccccceccnceseccssenscevecenee (1),
j= Jaat— atat+aB- CATON ee re eS (2),
424 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
y= Saad + tad a — 08 + tat (y—S) cecccccsvsesesveccrsveseceees (3),
y= Saad — iat a — 2B — tat (y —8) ooccceccccccccccccceveeceenees (4),
y =— Saad — ab a — aR — wt (—y $8) ccccsssccseceesessccseeeees (5),
y=—Saat+ aba abt at (—y 48) oocccccccccccdeccccceccrens (6),
y =— Ja ad — tak 2 + 0B + tak (7 + 8) ccesesccescccccscoensessernes (7),
y = — Saad + tat a + 0B — tat (ry 48) oe cccccsecccseeersssrcereeene (8).
And if we allow z to make a cireuit on the Riemann’s surface round «=0, which
changes 2 into ix, these series break up into the two cycles
Gi, 298, 1),
(SG) 405.19):
In fact here fee eos
6,=2, and the excess in the value of 5+x« due to the facts that
J, is not equal to 1, and ¢ is not equal to 1, is
lp. 1
52(4-1)—5(4—2)
=2,
which is the number of unit points within the triangle ABC and upon the hypotenuse
other than the points A, B.
The diagram for the curve is as follows :—
Here the circles round the unit points indicate that
they are not curve points. In fact t=2. From this diagram,
taking count of the correction, we infer that for the origin
6+%*=27: and that the deficiency is 8.
We may remark that if in
y (y? — ax) (y? — bx) + yo (y*, a+ar(y, a) =0,
: at: c (y? =. aw)?
we put eT n= ay ;
leading to
__(l—aky DiGi ery
£ we pa a F
we obtain
WE(1—bE)+nE(1, E+ —a€p(1, &=0,
which, writing y for 7 and # for 1—a€ is of the form
PUY, + YUU, + LU, = 0,
where 4, %, Us, U; are polynomials in « of the degree indicated by the suffixes.
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 425
The figure for this form is seuacen which gives 8 for the deficiency—and in-
dicates that the general finite integral is, in these new coordinates (see Part II.)
daz
2 3 le Sag Ea
[[4+ Be+ Cy + De Te ee ia ain aCe ZA argmraseacy
Another example of the theory is the curve represented by
the diagram—
The equation of this curve is
ry (y? —ax)? (y—ba)+y(y, a +a(y, xP + ay (y’, x)
+ ya (y?, x) +y%a = 0.
Here m=1, #1,$=4% m=1
m=1, jo=2, m=3|&=2, &=2.
m,=2, w=1, n=
The values of y corresponding to the factor (y*— az) are
given by
y=e Van + etaa+t x! ef [eB + fy]+.-.
where e, £ are square roots of unity, and a, f, ¥ are definite
functions of the original coefficients.
Thus the four values of y are
y= Vawh+ wat ah bay) A... ck cdescve see sve wotedeseaces (1),
p= Neat ca [Bey ace an enteneteveenssnust- saccsemes (2),
p= — Nat = wa iat [BS yb oe sel sacegsoanetersevonsenersdee: (3),
y= — Vo oh a BAB + gy] ice, nsrsrosesencerenonnnoevest aces (4).
And if we make 2 describe a contour round 2=0, so that 2? changes into —z*, then
the series (1) changes into the series (4), and the series (4) changes into (1), while also
the series (2) changes into the series (3), and the series (3) into the series (2). So
that there are two cycles, as there should be according to our theory.
Various other examples of the rules of this Part are given below.
426 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON'S POLYGON
Part IV.
Some examples of the foregoing theory. Consideration of the normal form of any curve
given by Weverstrass.
1. In the paper by Rowe referred to in Part I, the deficiency of the
curve &
y+y (a, 1),+y(a, 1),+(a, 1),=0 es
is determined. (=3.)
The result is immediately obvious on inspection of the figure.
2. In the Math. Annal. ix. p. 174, Noether gives as example of his method of re-
duction the curve
YY (®, Yst+(@, Ys=9,
and obtains that the multiple point at the origin is equivalent to a quadruple point
and two double points, that is in all that +«=8 (beside that «= 2).
This result is obvious from the figure.
Sae
We shall have further occasion for this Example in Part V. SK
Our diagram gives moreover the deficiency = 2. Hence the curve can oN
be transformed to 7?=(1, &),. Put in fact Be
x
E= 5
qi 2
Fe
3. The hyperelliptic curve can always be put in the form
y? (@, 1)p+2+ = (@, ee
wherein 7 is arbitrary.
The number of unit points within its polygon is p.
The figure is drawn for p=7, r=3. HEARS
The figure gives, according to the theory here developed, the adjoint curves of order
n—8, viz 1, @,..-
einer.
I believe that in all cases in which the deficiency of a hyperelliptic curve is
accurately given by the number of unit points within its polygon, these unit points will
be collinear, whatever be the form of the curve polygon.
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 427
4. An example is quoted by Forsyth (Zheory of Functions, page 355) from Burnside
(London Math. Society, May 14, 1891).
The curve Opel 1))s=|ke, UIE
has deficiency two. EH
This is obvious from the figure.
We see further, from previous work, that the finite integrals are
d. dx
[rem lgeue
5. In a paper (in the Journal de I’Ecole Polytechnique) Raffy has given three
examples of a method there developed by him for determining the deficiency of a curve.
Two of these are
yi-e(e+e+1)=0, y+ a°—5a*y =0,
having respectively deficiencies 1 and 2.
These results are obvious from the figures.
The other of these examples is
y — 5y* (a2 + @+1)+5y(a@+ae4+1)P—2c(e@+e417=0, LINN
for which Raffy obtains p=0. The equation can indeed, by an obvious transformation,
be made to take the form of a conic. But the equation is hyperelliptic and this trans-
formation not reversible.
But by putting
pe __ o(@+2)(+2+1~P
az y 3 2y° 2
tra? 2 fal
ot [n+ +5 0-56 458)| -0F
leading to C= =, , Where w= 1,
Qn tn” (1 5E+ 5) + of
Ye = ,
(1-5 +58) + of
27? + 9
ye
“a
we can transform to
1-—«a
ae ( ; ya — BE +. 5E4)* + wif’.
Thus the curve has p=2.
This curve forms a good example of the failure of our rule owing to the very ex-
ceptional forms of the coefficients. (It is treated by these rules in Corollary to Part VI.)
Wor, XV. Parr EV. 56
428 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
6. The following example is given by Cayley (Quarterly Journal, Vol. vii., p. 217):
(yY— ap — Bary (y + 2a*) + a” (8y? — 2) = 0.
The value of the singularity at the origin is obtain-
able from the figure with the help of the rule developed
in Part IIL.
Here m=4, p=3, t=7, N=2, and in addition to
the 20 given by the first diagram there is to be counted 1,
given by the second diagram, where ee
AB=t=7, AC=N=2,
giving on the whole 8+" =21.
SSSR RAE ees
[|
L TNE
(See the expansions given by Cayley.)
7. The followig example is quoted from Miss Scott by Harkness and Morley
(Theory of Functions, p. 147), and furnishes another example of
Part III.
y + 2x*) (y — 2° — 2° (y + 2x”) + 9x"y =0.
y y y J)
Here m=2, w=1, t=2, N,=1, N.=2, and we have a correc-
tion=1, given by the second diagram, where
A — ee Ve SW
. 84+«=7.
Also the curve has p=2 and can be transformed to 7’= (&, 1).
8. In case the curve be
(y + 22x") (y — 2°) + 9a7y = 0,
the figures are slightly modified. But as in (7) there is a correc-
tion =1. The difference is that in this latter case there is a branch
point.
Here Nia
and 6+x=7, as_ before. (See the expansions in Harkness and —
Morley.)
9. Of Weierstrass’ normal curve.
If g, be the algebraic function of lowest order which is only infinite at one point A
of a plane curve, and g, be the function of next order prime to a, the equation of the
curve can be transformed to
F= 9," + 97° (ga, 1)a,+---+(Ga, 1),=9.
Every algebraic function can be rationally expressed by gq and g,. Every expression
which is integral in g, and g, becomes infinite only at A. But conversely there exist
in general algebraic functions only becoming infinite at A where g, and g, are infinite,
which are nevertheless not expressible integrally by gq and g,. We can indeed prove the
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 429
Proposition. Of algebraic functions which become infinite only at the point A where gq
and g, become infinite, there exist functions of as many different orders (of infinity at A),
which are not integrally expressible by gq, and g,, as there exist simple double points and
cusps of the curve F above; in other words, the part of the +x of the curve F above
other than that furnished by the place g,=%, g,=%, is equal to this number of different
orders of existent functions.
In order to prove this we notice that a function of order z cannot be expressed
integrally in gq, and g, unless we can find positive integers x and y such as to make
an+ry = Z,
and thence put G2 = Cat gd +....
And this equation being =? =e ;
wherein we may suppose y <a, requires, for any value of y,
Z=ry, ry+a, ry+2a,...,
and therefore cannot be satisfied by those values of z=ry (mod. a) which are <ry—
that is, cannot be satisfied by
z=ry—a, ry—2a, ry—3a,....
The number of these values is H (=), the greatest integer in 2.
The number of values of z thus excluded is
y=1 &
which is equal to Sr) (a- 1), as we see by noticing that it is equal to the number of
unit points inside a right-angled triangle having one side =r and the other equal to a.
Any value of z other than these of the form ry—a, can be expressed in the form ax +ry
—so that for such values of z a function g,=Cg,"g," certainly exists, and the most general
function of this order, infinite only at A, is of the form Cg,*g,¥+gz, where 7 is <z and
gz is, possibly, not expressible integrally by gq. and g,.
Of the not integrally expressible orders, in number 5-1) (@-D), there are, as we
know, (see note at end of this paper), just p which correspond to actually non-existent
functions.
Hence there remain just
1
3(@—-1)(r-1)—p
orders, of functions which exist, are infinite only at A, and are not expressible integrally by
Ja and g,.
56—2
430 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
‘ yw o8R : ; :
Consider now the function oa It is a function of degree a—1 in g, and therefore of
r
order +(a—1); and vanishes therefore at »(a@—1) points of the original curve. These
points consist of (1) those at which dg, is zero of the second order, namely those which
become the branch points of the Riemann surface which represents g, as a function of
Qa and are therefore in number =2a+2p—2, of which a—1 fall at A where the a
values of g, are all infinite, and (2) of those which become multiple points of the curve F
or of the Riemann surface, the number of these for any multiple point other than those
already counted among the branch points being 28+ 2« (6, « being Cayley’s equivalent
numbers of double points and cusps for the multiple point).
Hence +x for the whole curve F is
1 See
5 i” (@—1) — [2a + 2p —2— a —1)}
=3(r-1)(a—-1)—p.
The comparison of this number with that previously obtained for the not integrally
expressible functions, proves our proposition.
Hence also
p+8+e=5(r-1)(a-1)
= whole number of unit points with the curve polygon of F, this curve polygon being
a right-angled triangle of sides r, a, if we do not take count of finite multiple points.
This verifies the general proposition of Part II.
Before considering how these exceptional functions are to be expressed we may consider
as examples the cases p= 3, p=4.
For p=3, we may have
(1) a=2, r=7. The orders of non-existent functions being 1, 3, 5. This is the hyper-
elliptic case, the number of moduli being 5: the equation is SS
97 +9: (G2 1)s + (G2, 1), =9.
(2) a=3, r=4. The orders of non-existent functions are 1, 2, 5. This is the case of
a point of undulation on a plane quartic. The number of moduli is 5. The equation is
92 +92 (Gs, 1), +96(9s; 1). + (9s; 1),=0,
reducible to he +h (gs, o+(9s, 1)s=9,
or, say, WE +E (n, E)o+(n, &),=9,
which for 7=0 gives &=0.
(3) a=8,r=5. The orders of non-existent functions are 1, 2, 4. There is a function
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 431
9, not integrally expressible by g, and g,. The (g;, gs) curve has therefore a double point.
Its equation is (cf. Schottky. Crelle, 83)
F= 93+ 98 (9s, 1). + 959s (9s; 1), +92 (9s; 1), =0
and depends on six moduli. The double point is at
9s = 95 = 9.
In fact by taking for triangle of reference of a plane
quartic
z=0 any inflexional tangent,
y=0 the tangent at the remaining poimt B where the
inflexional tangent meets the curve,
a=0 any line through 4,
we may put
Y vy
zg? os A?
Is =
these being infinite at A in the orders indicated, and so reduce the quartic, which
takes the form
f= ty + H2(y, Zr+#2(Y, Z2+2(Y, 2)=0
immediately to the form above, with
The diagram for F is RES
Notice 7 =(a—1)r—a.
There is no need to consider cases in which a>3. On every curve for which p=3
there exist points for which g, exists.
Considering next p= 4, there are five possibilities.
(1) a=2, r=9, The non-existent orders are 1, 3, 5, 7. The equation is
9s +9s(92, I)st+ (G2, 1) = 0.
(2) a=8, r=5. The non-existent orders are 1, 2, 4, 7. Equation is
Is + 95° (Js; 1) +95 (9s; 1); + (gs, 1); =0.
Figure is
a
(3) a=3, r=7. Non-existent orders are 1, 2, 4, 5. There exist functions gs, gu,
which are not expressible integrally by g, and g,, so that the (g:, g,) curve has two double
points.
432 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
The equation is
92 + 97Bot Grays + HAs = 0,
where a, 82, ys, % represent integral expressions in g, of the order given by their suffixes
For this form the figure is
and the polygon contains p+é6+x«=4+4+2=6 points, as it should.
Gz
But by putting = Ne obtain
NO + 4°B.+ NY2+ a= 0.
For this form the polygon contains only p= 4 points. EN
We notice 8=(a—1)r—2a, 1l=(a—-1)r—a.
(4) a=4, r=5. Non-existent orders are 1, 2, 3, 6. There exist functions g;, gy
which are not integrally expressible by g, and g;, so that the (g,, g;) curve has two
double points. Its equation is
gs + gs + 9sBs i IsBrY2 + ay? = 0.
|
|
For this the figure is
and polygon has p+8+«=4+2=6 interior points.
But if we put y,=&, g,;=1, the equation becomes
(€& n 1 +(& ot (& neot(& a En] +(& a) En? =0
for which the figure is Ly
and now the polygon contains only p=4 points.
We notice that 11=(a—1)r—a, 7=(a—1)r-— 2a.
(5) a=4, r=7. Here non-existent orders are 1, 2, 3,5. There exist
Is» Yor Gro» Gis» Jr
which are not integrally expressible by g, and g,. Thus there are five double points on
the (g9,, g;) curve. We notice that
17=(a-—1)r—a, 13=(a—1)r—2a, 9=(a—1)r—3a,
10=(a—2)r—a, 6=(a—2)r—2a.
Passing from these particular cases to the consideration of the forms of these not
integrally expressible functions, we see first that we can always build such a function
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 433
corresponding to a double point. For if O denote the point of the original curve at
which g, and g, are infinite, namely the point which becomes the infinite point on the
(9a; gr) curve and d, denote the double point supposed to be reached from one branch
of the double point, the other point being denoted by d,, and Poa, be the integral of
the third kind which is once logarithmically infinite at O and at the double point on this
first branch, which is therefore finite on the other branch at this double point, then
Vem
fg a
where f(a, gr) is the (ga, g-) equation and
; aye
St (9) =ao Se Ir),
a is once algebraically infinite there and f (g,) is once zero,
and is infinite at O to an order r(a—1)—(a+1)+1=r(a—1)—a.
is not infinite at d,, for
From this remark, recalling the ordinary method of expressing P.a,, we have a rule for
forming this function as a rational expression in g, and g,. Viz. it is
10)
Lea
where Ig represents a linear function in g, and g, which vanishes at O and for the
values which g,, g, have at the places which become the double point, and is for the
equation /(ga, g-) an adjoint curve which touches the branch d, at the double point and
passes through the a—2 finite points other than O and d, at which Z,g meets the curve ihe
We know that such a curve can be expressed as 0,+2,¢¢6, where , is a special curve
of the kind and ¢ an integral function in g, and g, such that
dda
765
is an everywhere finite integral: and one form for Q, is immediately obvious—viz. let t,
be the tangent to the branch d, at the double point of the curve f and y be such an
Ya
F (Gr)
than the one under consideration, and such that wy, while not vanishing at this double
point, vanishes at the a—2 points other than d and O at which Z,, meets the curve ie
The multiplicity of such a curve y after passing through all the other double points,
is known to be p+1, and to prescribe that it passes through a—2 points of the line
L,q leaves it with a multiplicity p+1—(a—2), which is certainly not negative. Hence,
integral expression in gq, and g, that Iys is finite at all the double points of f other
noticing that since O is at ; "=, gp= 2, we may take D.q=g,—D, we may write
a
our function
434 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
In the same way we obtain another such function
and, attaching proper numerical multiplers to them we may write
t,— OP aa,
RaDYa¥=f Ona
This representation is in accord with the previous results. If the most general integral
expression in g, and g, formed by such powers as are represented, in accordance with
Part II. of the present paper, by the points within the polygon of the (ga, g,) curve, be
represented by gag,P, we know (see for instance Clebsch and Gordan, Abelian Functions,
page 16), since ® is of order N—3 (see Part II.), that
G@,-G! =
ieend
J? ray = CPi, +. + O5irPee, + AY +... + App +
where e,, @ refer to the (6+ .«)th double point, and »,...v, are the everywhere finite
integrals, namely
D = 0, (G,— GY’) + 0 + Ohne (Foie — Moin) + O+ M
where ¢ is the general adjoint curve of order V—3, or
D =O t+... Os eeret P+ be
Of course on the other hand, the form of G, can be variously altered. For instance,
in the example previously considered where p=3, a=3, r=5,
a a
Is = J A oR S a )
Zz
the double point of the (95, g;) curve arises from the points D,, D,, where the quartic is
cut by the tangent at B. And we may write
= Shey
I= 2? (x— nz)’
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 435
where #—z is the line OD,, and U is an arbitrary conic through A and Z: then, easily,
dg, _ U_ zdy—ydz
eG) eae
Ox
where
Pa=a'y + a2 (y, zh+az[y, 2+ 2(y, 2),
and this is in agreement with the remark on page 433.
The expression = D above can be put into the form
=
(Ge = 4) (ge — oe (gr PD 0) +++ (Gr= Ca-2) + integral expression in g,, ga,
whence as ai" and the integral expression in gq, g, only become infinite when g, and
gr are infinite, we see that g,=D, g,=E is the double point and g,=c,,... are the
values of g, at the points other than the double point in which g,—D=0 meets the
(Ja, Jr) curve. We may thence put
— (Gr = E) (Gr = G1) «+ (Gr = Ca-2)
Ja—D
J (a) r-a
’
and this is obviously only infinite when g, and g, are infinite.
We might expect to be able to form thence functions of order (a—1)r— 2a, etc. for,
since g°(q—1);-a has an order which is =2a(mod. 7) we might expect to put
Gr (Ja, 1), + Gr * (Ja » 1), Hass
(9a — D)? ;
G (a) r-a = mtegral expression in gq and g,+
and thence, putting (g., 1,=%(ga—D)+-p, to obtain
HG + Gr (gas, Dit <=
(9a — DY :
Ga) r—a — AY (aa) r—a = integral expression in g, and g, +
and thence be able to infer the existence of a function
HG One (a Lit...
(9a — Dy?
only becoming infinite for g,, g, infinite, obviously of order (a—1)r—2a, which is not
integrally expressible by gq and g,. But in fact this function will sometimes be integrally
expressible by gq and g,. For instance, when p=3, a=3, r=5, the curve being
9s + 9s (9s—) + 959s (9s, Lo +9" (9s, 1)s=9,
= 9(9s—¢)
Ii Qs
is not integrally expressible, yet we can easily verify that
9° + 9: (Gs, Y2=9s +95 (Gs, L2— (Gs, Ds] +(gst+e¢) Gs, 1)s,
though
or again, when pet =, PST,
the curve being IF +97 Bo + Greer + a.°a; = 0,
and %=C (Ys = ky) (Ys = ks), B= (9s —k,) fi +b,= C(9s — i) h,+b,,
Vou. XV. Parr IV. 57
436 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
where 7,, h, are of the first order in g, and ¢, b,, b, are constants, though
_ IG +h)
I, —f,
is not integrally expressible, we can easily verify that
Iu + Inl%e (9s — k,) = aS; +9; lef, (9, — k,) — ca, (9s— k,)"] Te Ca, (Js — k,)* (8, — 2b,),
and similarly that
OF (ope =; 2 Ah, —
€ (gs — ky) (9s — (Grt¥2 + G25) (Gr — Bo + b, — b.).
That such expression as given by these examples should be possible in case of a
curve having only one double point, is obvious from our proposition that the number
of orders of existent not-integrally-expressible “integral” functions is the same as of
double points—for we have shewn how to form a function gq+))~«=
sponding to that double point.
But we can form functions of order (a—1)r— 2a ete. in another way.
In the case of a curve having two double points and known to have a not-
integrally-expressible function giq—);-.1, We may form the difference
Jia r-a — J (a1) ra
of the two such functions formed as above for the two double points. This will be at
most of order (a—1)r—a—1 or r(a—1—A)+a(R-1), where A, R are integers less
respectively than a and r such that Ar—Ra=1. Subtracting from this difference a
proper multiple of ga*~%g,"+ we shall obtain a function of lower order. Proceeding
thus we may expect to arrive at an equation
9a») r—a — J (a—1)r—a = integral expression in Ya, Jr +Ja—1) r—aa-
For instance, in the example just cited, p=4, a=3, r=7,
1 _ 9G: +b) - 2) k yee
In — ke, Is — ee EG: —k)g,- ky)?
so that we may take, unless k, =k,,
Iu -IJu
of which other forms are, in this case,
Pg Gag et A, + oF,
beg, * p, J ‘+ b, :
oy fle 4 pee —b,=b, , a4 (Bib; — bs)
2 b, Ir .
or
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS, 437
In the same way for a curve with any number of double points we can, from any X
of these double points, form a function of order 7(a—1)—2a, namely
(1) @) (A)
Yr (a1) —a 5 (a—1) —a Yr (a—1) —a
(hh)... mh) (a= Tee) + =) .. «(ky = hy)
Dr a—1
= + 9, a60 ]|Er oso 5
(9a—'h) .. » (Ja — kt) a ‘l ]
()
where Oana = Go, etc.,
as before explained, and the double pomts sare jat Gq@— i, Ma -.., these hi, Kas one being
supposed different. The function thus obtained is necessarily only infinite when Ja and
Jr are so, and it is not expressible integrally, since such integral expression must be of
the form Py,""+..., where P is integral in gq.
Thus in the case of a curve with no higher multiple points than double points of
which no two have the same value of ga, we can always express in this way as many
not itegrally expressible functions of orders of the form r(a—1)-2a, as there are
double points. Since however every number r(a—1)—2a is prime to a, we see that we
must have r(a—1)—Aa>r, namely X$r—1—E (=) . Hence if (6+), be the number
of the double points
(+e) pr-1-B (7),
and this is verified in all the examples considered (pages 430 and 432). For instance when
p—A =A — ie ial -E(*)=3,
and we found that there were functions gy, 913, gi. The other two g, Qi are of orders
(a—2)r—a, (a—2)r— 2a.
In the case of a curve having double points for which the values of Ja are not all
different, we may suppose the previous expression applied only to those double points for
which the values of gq are different. We obtain thus as many not integrally expressible
functions as the number of these. If then there be a value g,=k, for which there are
m separated double points at g,= F,, Hiei ...9,=4,, there exists a function
(Gr =H) +++ (Gr = Eu) (Grs Vayu
Ja—k
of order r(a—m)—a, which is ‘integral’ and not integrally expressible, the function
(Gr, 1)a—y. being determined to vanish at all the points for which g,=k other than the
double points. The consideration of how we should proceed to obtain functions of other
#&—1 orders may be omitted. Especially as the orders of the existent functions do not
57—2
438 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
necessarily determine the nature of the curve. For instance the function g»= Ja—» ra
(9r — E,) (Gr — 2)
gr —k
curve at g,=E, and g,=£,. In accordance with Kronecker’s theory (Crelle, 91) there is
no need in general to consider the normal curve to have higher singularities than double
points. The examples here given should be compared with his theory.
above might arise as where g,=k is a double tangent touching the
Part V.
On the Graphical Meaning of Noether’s (Cramer's) Resolution of the Multiple Singularity
at the origin, by means of the Quadratic Transformation.
We use the same notation as in Part I. save that for o,, m,, pm, we write
c,, m,, wy; b being the actual degree (= degree in «+degree in y) of the lowest terms im
the equation of the curve. So that if the side of the polygon for which «=1 be present,
l=distance from the origin of the point in which this side meets the axis of y. And
if this side be not present, /=distance of P, from the origin. Then according to Noether
the singularity is resoluble into a simple multiple point of order /+an additional number
of multiple points which happen to be coincident with the multiple pomt of order /—
and these latter in their tum are similarly resoluble. This result is arrived at by a
particular case of a reversible quadratic transformation, as follows—
Substitute in the equation of the curve «=£&, y=&m, where », m are connected
by a linear relation py+qm=1. Then in the transformed curve we may either substitute
for m, in terms of » and regard & » as the new coordinates, or substitute for in
terms of , and regard &, », as the new coordinates, The inverse substitution is
as y
=prr+ Y, =— =. = =
a aa aero ep eso
so that to a point near the origin and on a branch y & # corresponds a point near the
axis £€=0 for which
h ame 1
when o<l n aera nm Re }
—1
when o>l yee n= ae =;
p+ quar pt que
For «<1 we shall regard £ 7 as the new variables, and for o>1 we shall regard & m
as the new variables. Then the part of our singularity for which «<1 becomes a
singularity at £=0, 7=0, and the part of our singularity for which o>1 becomes a
singularity at €=0, » =0. The part for which o=1, say yx ka, becomes a singularity
at £E=0, n,=kn. If then there be ¢ branches for which o=1, we obtain ¢+2 singularities
corresponding to our original singularity. And since the transformation is reversible
every point on these new singular branches corresponds to a point at the original singu-
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 439
larity. Noether uses the substitution in one of the forms in which either p or q is zero,
but when this is chosen to be effective for a branch for which «<1, it is imeffective for
a branch for which o>1. In the form here no finite point of the original curve (except
the pomts other than the origm upon the line px+qy=1) becomes represented by an
infinite point of the new curve, Also there is no multiple point on the new curve
arising by transformation from a simple point of the original curve. For if
Se, y) =f (En, Em) = EF (E, 0)
the equations
0a (hen)
y
give j T 0,
x= 0.
We imagine now the polygon constructed for the new curves and each of the ¢+2 new
singular points obtained. We proceed first to enquire what the values of the o’s will
be at these new points. And, defining provisionally the word ‘multiplicity, applied to
our original singularity, as the number of unit pomts within and upon the origin-polygon,
save those upon the axes of coordinates, we shew that this is equal to
1(J—1)+ the sum of the multiplicities arising from the +2 new points.
bo]
The reapplication of this theorem to the new singularities obtained, and so on continually,
enables us to give a geometrical meaning to the number which we call the multiplicity.
Consider then the effect of «= £&n, y=£&n, where », is regarded as a linear function
of »(=a+ bn), upon the branches at the original singularity for which «<1. The lowest
terms in the new equation will be of the same dimensions as if we put w=&y, y=€.
From a term #/y! there arises a term &*%y/n,7, so that the whole equation divides by
£’. and this term becomes effectively &/*’'y’. For instance corresponding to the point
P, in the diagram of the original curve, for which f#=0, we obtain in the new curve
the term £&-~, which gives on the representative chart a point lying on the axis of z.
And corresponding to the points (f, g), (/’, g’) in the old diagram, wherein f</’, and
g>g’, we obtain in the new diagram the pomts (f+g—J, f) and (f'+g' —l, f’), wherein
fi+g—l<f+g—l and f>f And the o’ of the corresponding side in the new figure
reckoned away from the axis of & is
oe f-f __m
“GED =F 2) eae
m f—f
where — = ; is the o of the original figure.
(Pat)
440 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
If then in the new diagram all the points are marked corresponding to the points
Po; Eayssw ben (where 07, Ga, ...10,-7 ake each) <1, oy — ly aye woyensm are (alles),
the point corresponding to P,., for which the sum of the coordinates =/, that is to
say in the notation above f+g=/, will be on the axis of ». We shall not mark in this
diagram the points corresponding to P,, P,.,,.... We desire only to obtain the number
of points within and upon the polygon Q,... Q,. which corresponds to the part P,... P,_;
of the old. Call this number A and notice that the greatest common measure, say n,
of the quantities f/—f, g—g'—-(/’—f), is equal to the Gaom. of f—/f and g’-g. A is
formed from the quantities m, m:, #e—7™ in the same way as was our original number
from the quantities m, mm, #:—and in the new polygon ¢ varies from 1 to r—1. Con-
sidering next the points of the transformed curve corresponding to the x, branches for
which ¢=1 on the original curve, the effect of our hypothesis, that in the corresponding
n, expansions of the form y=dAw+... all the coefficients A... are different, namely that
the n, branches have separated tangents, is that on the transformed curve we have n,
simple points lying on the axis €=0, and the multiplicity of these is zero. With
reference finally to the branches for which o>1 we imagine 7 expressed as a linear
function of 7, and regard & 7, as our new coordinates. So that so far as regards the
lowest terms of the new equation, our substitution is equivalent to =& y=£&n,. The
effect of this upon a term ay? is to transform it to &*%%J/, which after division of the
equation by & becomes &*9—y’. So that for instance to the term a*-y% where z,+y,=1
corresponds the term £%,%r. And to the terms ay’, «#/’y" correspond in the representative
diagram of the new curve, the points (f+g—-Jl, g), (f/+g'-1 9’), giving
eee ihe odes) ie a= (g— id ee m— pb
io 9-9 BB?
where “=o - Ls . We have to determine the multiplicity B given by the new polygon
b =
which is formed from the quantities m, m:— pz, #, aS was our number from the original
polygon with the quantities n,, m,, w,, t having here the values
Pelee ce ,k+1.
It may be noticed that the total number of sides other than the axes in the two polygons
corresponding to the summations A and B is either equal to, or less by one than the
number of sides other than the axes in the original polygon. With these explanations,
and putting
Ve
C= a> nm (np — 1) 45 = nyNs (Myfls ~ Msfly) — 5 5 Sn (u—1)
2s>r
which, as is easily seen, is another way of writing the number previously obtained of
the unit points within our original polygon and upon the sides other than upon the axes,
and writing this in the abbreviated form
53a Eb 5 Sa 5 3b+5 3 (arb, — ayb,) +5 Sn,
74
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 441
where a, =N;M,,
, b= yb,
the theorem is
=S1@ a) eaeee
In the same’ way, making the assumption that in the multiple point at €=0=7, the
branches which do not touch either €=0 or »=0 have all simple contact with their
tangents, we can write
A=5m(m—1)+ A+B,
and similarly at €=0 =,
pa m’ (m’ —1)+ A” + B”,
g
and therefore
Ld 1) +5 m(m—1)+5 m (m'—1)+ A’+ B’+ A” +B"
a Ld
and so on continually—and it is perfectly obvious geometrically that the polygons corre-
sponding to A’B’A”B” diminish indefinitely as their number increases, and eventually
correspond to only simple points, in which case the corresponding multiplicities are zero.
We thus resolve our compound singularity into a coincidence of simple singularities so
far as the “multiplicity” is concerned, and are thus able to shew that this multiplicity
is really to be interpreted as the contribution to 6+« which is due to the singularity.
It is immediately obvious that the « of the singularity = ,n(m—1)+2=m(u—1) is the
sum of the values of the « due to the simple singularities into which it is so resolved.
Thus we again prove Cayley’s rules.
The proof of the equation stated is as follows—the work is quite similar to that
of Cayley in the addition to Mr Rowe’s memoir. Putting a,=n,m,, b,=n,u,, denoting
the number of points on the side for which e=1 by v +1, and the corresponding values
BR
of a,, b, by ay, b, (each of these being in fact =v), putting also =, to denote a
L é -
summation extending from r=p to r=A—1, and =, to denote a summation extending
from r=p to r=k+1, it being understood that when the p is absent the summation =,
begins with r=1 and the summation =, begins with r=2+ 1, we have
Sa=S,a4+.a+v=,a4+ 2. (a—b)+d.b+v
Sb =3,)04+ L.b+v=%,(b—a)+>2.b+ a+
Sn= n+ ln+v
20 = Ya>db — Ta — Tb+ n+ = (a,b, — agb;)
8>r
2A = Sad, (b — a) —S,a—-— 3, (b— a) + Sn + & (4b; — ugb,)
s>r
2B=%,(a—b)>,.b - 2a (a — b) — >.) + Lan + &, (a,b, — agb,).
s>r
442 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
For the values of o corresponding in these two cases are
nm m—p
m—m’ pb
the former being reckoned in a particular way.
And Lath — Ya — 2b + En —[E,ad, (b-—a) — Sa —- 3, (b-a) + En]
—[=.(a—b) ¥.b — ¥, (a — b) — 5.6 + En]
=r+y (S,a4+ 5.) + d.a + ¥.b) + S,a3,b + ad.) + Dad.) + E,bE.a —Ta— =
+ v— Z,a>,b + (2,4)? + 2b — 2.ab.b + (2b + Sa
(i) =+v(3,a+ 2,04 .a+ 2.b-1)4+(S,a)P + (2.0) + ad.) + 2,b2.a — Sa — >,
while
2 3
= a,b, = a, (2b +v + Ub) +a, (2b + v4 Eb) +... +4 (vy + Eb) + aad.b + E, a,b,
s>r s>r
= Y, a,b, +v (Sa + Sb) + E,ad.b + E, a,b;
8>r s>r
(ii) oe = 2 (Gr b, —b,a)— > 1 (ar b, — bas) — 2 (a,b; — bas)
s>r
=p (2a + >.) — 5b — 3.0) + T,a>.b — =, bd.a.
Adding this to the expression above we obtain
v?+ Qv(Z,a + Sb) —v + (2,2)? + (2b)? + 2B,aF.b—-Ta— Tb .
and l= d,a+ 3.b+ 7;
. this is P—l;
~ C=51L-1) + A+B.
It would, I imagine, be easy to give a similar interpretation of Noether’s work for the
case in which the vy roots of the equation corresponding to the line for which o=1 are
not all different—for instance, to investigate the branches that correspond to a repeated
factor y—kae we must put y—kw=£y, and «=&m where is a linear function of 7.
As an example of this method we proceed to determine the 6+« of the singularity
at the origin for the curve
PP (YY, +H (Y B+ PY e+ Py +H¥(Y, SPY, P+(y, xP =O.
If the polygon be drawn, the angular points nearest the origin are
(0, 15), (5, 11), (10, 7), (14, 4), (17, 2), (22, 0)
of which the first three are upon one straight line. The number of points between the
sides given by these points and the axes, with those upon the sides that are not upon
the axes, is 130—so that
6+«=130,
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 443
5 4 3 5
Also =7) Or T=) t= ls os= 5) Te o.=5> 1,= 1
so that «= Yn(w—1)=10.
We proceed to prove that this is in accordance with the results given by Noether’s
method.
I. All the o’s being greater than unity, we put as explained
= a
YAY
and obtain after division by 2,”
y+ ay” (y, 1 + e2yk Yrs 1) + eryt A, 1)
+aty? (yi, D+ ery? (Ys D8 + arn YH 1)" +4,’ (%, 1)” =0,
wherein a=
II. Putting now «= &n, y= &(E+ 7), we shall have three branches at &=7=0,
and one branch at £=0=H+~y. At this latter point will be s viz. E+n x &, that
is, we have an ordinary contact with &=0, and the “multiplicity” as defined will be 0.
Considering then only £=0=7” and putting » for E+, we obtain, after division by &,
Bui + £0 (1, Evy + Ernie’ (L, Evy! + Entot (1, Evy + tv? (1, Ev)"
+ Eniv? (1, Ev) + En'v (1, Ev" + En’ (1, Ev)” =0,
and the values of o are o,=3, o.=2, o,;=1 (reckoned from the axis of &).
Ill. Putting now €=&m, n=&m,
we shall have a simple point corresponding to E=0=H+yn and # +n
two branches at &=0=7)
one branch at £,=0=%,’—kn corresponding to the terms
no? [B® (1, Eo) (1, £0)"
T assume that this is a simple tangent to nm —kn and put in consequence, simply
£=£,(L,.+m), 1=fm.
Then at these two branches at &=0=m (reckoning o from the axis of &, as in IL.)
oon —
and putting for EZ,+ we obtain, after division by m/,
EPusv® + EPmvv" (1, P+ E,n,70;' ( ye + nr, ( ye
ae ni ( ye ee Erm, ( ys or Em? ( ye ae Etn/ ( ye =0,
Vou. XV. Part IV. 58
444 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
IV. Putting now
&,=&(£.+m), m=&m
we obtain after division by &° a curve having a double point at &=0=m.
V. And thence putting & =&(H;+ m3), m= &m; we obtain after division by & two
simple points on &=0.
Reckoning now the 6+ as indicated in the general theory given, by the indices
of the factors that have divided out, we obtain
Ste=5[15(15—1)+6.54+4.3843.242.1]
= 130, as before.
The transformations are
T=2, , %=Ey , E=£,(m, + 4), &, = &, (#.+ m), &, = & (EH; + 3)
Y=X, W=E(L+n), n=Em > m= Em > Mo = Ens.
Part VI.
On a particular monomial transformation.
We give now an identity which is useful in a particular kind of transformation—It
will be seen that it leads to a resolution of the same kind as Noether’s.
ee Roe S of ea bi iepet
+" Ky + Kyat... +R" + KB’ 4+K +... + Kom + Kona
be any continued fraction, and let the convergents corresponding to the elements
ry ek Be
ve
qo
Then if A, B be any quantities
(qA + pB-1)(7'A+p'B-1)-(q¢A+p'B-1)(q’A+p"B-1)=(¢A+pB-1)(K7A+ Kp)
= K[(q4 + p'BY— (7A +p'B))
or, if k=qA+pB, ete.
(1) (k —1)(k' —1)-—(k -1)(k" -1)=K (k?-— kh’) = Kk (kh - 1).
Take now A, B, so that A=Pa+ Pb, B=Qb — Qa,
where 7 5 are the two actual last convergents of our continued fraction, so that
a=QA+PB, b=QA+PB,
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 445
so that a is the last, b is the last but one of the quantities kh’, k’, k... and notice
that if our fraction begin with
a
so that k,=A+K,B, k,=K,A+(K,K,+1)B
and we put ones
then (k, — 1)(k, — 1) — (4-1) (B-1)=(h,- 1) K,(A + KB)
= Kk, (k, —1)
and (k, - 1)(B-1)—(B-1)(A —-1)=(B-1) KB
= K,B(B-1)
= Kk, (ky — 1).
Therefore adding all the equations of the form (I.) and using these initial forms of
that equation we have
(a—1)(b—1)—(A—1)(B-1) = Kk, (&) — 1) + Kuk, (hy —1) 4+... + Kemsabburn (Kam — 1),
where in fact b=
If now a= >a,=2n,m,,
b = 3b, = 2n,p,
and we put C—O frat Ora O10) tbe
leading to a,' = Pa,’— Pb,, b,” = Qb, — Qa,’
and Az=%Xa,/, B=b,,
and > (a,b; — asb,) = & (a,'be — a'b,’),
ee Sy
we obtain the identity in question (wherein n,=n,’, since clearly any divisor common to
a,, b, is common to a,’, b,’; and conversely)
nm (Zn —1)+ E ns (mM,ps— Msp) — In (wu —1)
8s>r
—[2n’m’ (Sn'p! —1) + & n,'n6' (mpg! — mg'p,’) — Sn (w’ = 1)]
8>r
= Kk, (kK —1)+ Kk, (k, -1)+...... + Kom+ikom (Kiem — 1)
where, as may be recalled,
IP 1 1
Or ume re
ie’ 1 1
qe A at eeeeee ie
Dre i Be
oe ie ae “Mira
58—2
446 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
ky= 0’, ky = rte’ +p,20' > kom = Yb
= q,(P’Sa— PXb) + p, (— Qa + QS)
= (q.P’ — p,Q) Sa ~ (qpP — p,Q) %b.
If now we make the substitution
aera’, y= en?
equivalent to
f= at %y@ baer =p ae
since OE O ale
this being the result of a combination of such substitutions as
«= En, y= oe y= n€,
the terms a/y!, a y% of our original equation become
pvr +9P Se aoe. pe ee ee +9Q
and corresponding to
ni} =f 9 — 9 —,
we have m =f'P’+ 9 P—(fP’ + 9P)=mP —-pP,
w =f +9Q— (FO +9Q) = wQ — mQ,
and thus m=Qm'+Pw, p=QUm +P,
which are in accordance with the equations of the previous page,
and one =, is positive if b> o> 5
and is negative if 2 >o a
A 0, —- 0.
while a — oy Sa
es (Q- aiQ’)(Q a ox’) ‘
so that, if o,<o,, then o,’<o,' if o,, o, are (both greater or) both less than Q
ae
P 1 1 1
oe qe is + Fe WE) ae
Qs: rae | 1
aoe ay 76
Noticing now that &=x%y-% give when you’, Ex re-@
n = a~PyP no er -P
we see that the points of our branch y xa? that are near the origin will not
projected to infinity provided
Q
J
g++ Pp”
be
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 447
If now for instance there be only one o (- _) Jor the singularity at the origin and
we put a into a continued fraction
i
= Koni. sts es
ais
7 Pp
then the transformed value m’ is equal to 0, and corresponding to the nw branches
y=A,e, y= A,oz", ...... Y= Avo a
DSA, SEA seone y = Aw" 27
TSG, (SAAT, cacsor y= A,o" 27
where o=er,
we have points where
Gea Ab Ss osc, eZ nas5 agoone n= A,o""+...
PO emer eee ete e wwe teweseeee sees
that is, mw branches cutting the axis €=0 at, in general, different points. (When these
points are not different the transformation can be reapplied.) And the transformed value
of our expression
Tnmnpe t+...
is 0, and the original singularity consists of
K, k-ple points
with K, k-ple points
etc,
For instance Noether’s example (Math, Annal. 1x. p. 174)
ys + y? (x, YP +(e, YY) + vee
eo eee lee ae
Here aa er eee
Orasene lee
dg hae ee
Om su ienie
Deel eee
Gen agggok ngs
448 Mr BAKER, ON EXAMPLES OF THE APPLICATION OF NEWTON’S POLYGON
Sa’ = P'Sa— Pb, Yb’ =—Q'Sa+ Q3b
=6P.=4P =— 6Q'+4Q
earn (Sey (Sach
namely, our singularity is resolvable into two double points and one quadruple point—
(which gives 8 as the contribution to 6+; as is obvious from the figure).
Corollary. An Application of the preceding transformation.
If CHS <S ceebee < Op
be the values of c for a multiple point at the origin, and we make the transformation
w= £Pn®, y=bPn®
‘ IE
taking care only to choose P SOE:
and therefore a > On,
the branch y x 27, leading to Ea a®-%, » 2a” ~” becomes always represented by a point at
infinity on the axis of , for all the values c=a,, oy, ... oi; Namely on the new curve
the singularity corresponding to the singularity at the origin on the old curve is entirely at
&=0, n=. If the old curve be
F(a, W=F(En®, En) = PF (E, 0) say,
where F'(€, 7) is the new curve, the conditions for a singularity on the new curve, viz.
oF
Mane er ge E - EP-1n% 4 P ver 4°
05, Ere [QL erat OE erat]
give (PQ-PQ) Ley? Er H=0
(PQ POE Bae. EM =0,
and can only be satisfied, unless of =0 and of 0, and excluding infinite values of & and
ox
n for the present, by £=0 or »=O0 or both, namely at points arising from «=0=y—at
which both - and ze are by hypothesis zero. So that the new curve has no finite singularity
that does not arise from a singularity on the old curve. The infinite values of € and
that are possible, can, since
eC — EPn®, 5 Pn,
only have arisen from points c=%#, y=«. Now suppose that after the transformation
TO THE THEORY OF SINGULAR POINTS OF ALGEBRAIC FUNCTIONS. 449
above has been applied to the singularity of the old curve at the origin, we transform the
axes of £& » by writing F=£+4A, n=7,+B, to a point (A, B) which is a singular
point of the curve F(&, 7). By a similar transformation to that just applied, viz.
writing
£= Xu yr rep Gad) ais
X=b%a" Yak hyY
we can transform this singularity to be at X=0, Y=o. The singularity of
FE, )=0
which is at -— oor
that is, also, at &£=0, m=, changes to X=0, Y= —-viz. our new curve in X and
Y has the singularities corresponding to the two already considered, both at Y=0, Y=o.
Let this process of changing axes and subsequent transformation be continued.—Hence* we
at length obtain a curve whose only singular points are on the line infinity—there being
a very complex singularity at the infinite end of the axis of zero abscissae and, beside,
possible singularities at other points of the line infinity which have persisted throughout.
For instance, Raffy’s example previously discussed, .
w= 5a (ye ty +1)+ de (ye +y + IP 2y (ye ty + DP =
becomes by z=é&, y=o+&n!
1 — 584° (Ent + c) + 58m! (Ent + c)° — 2En* (w + &n') (c+ En*? = 0
where ¢=o— oa’.
All the singularity of this curve is on the line infinity of the & 7 plane.
Note. We may put further 1s ae
and hence obtain
(i) a — Baty (y* + ca) + dary? (y® + cx?) — Qy (wa? + y*) (cx? + y*)° = 0
which we may treat by the rules of Part III. Putting
Fadi, y=oF
we obtain v(e+v*y [2 (@ + v*) — 5vE] + 5vE (e+ 0?) — & =0
and here, for the branch v=/—c+..., we are to count t=5 (see page 419, note)
while N=2; the correction is therefore 2; the diagram for the curve (1)
above gives 102 as the number 6+« for the singularity at the origin, with
4 interior points. Hence, admitting the correction, we see that, for the origin,
§+«=104 and the deficiency is 2, as previously obtained. The value 104
for 6+.« can be verified by expansions. The curve (i) gives six expansions of the form
re eed =
=—-S¢fz! $= WC LB pee ssisnsls
y J2
* If, in such a curve, y be an integral function of <, all integral functions are expressible integrally.
450 Mr BAKER, ON EXAMPLES OF APPLICATION OF NEWTON’S POLYGON, ETC.
where #=1,
beside three expansions of y in powers of «* with different initial coefficients, each series
beginning with the term a, and one series for y in integral powers of x, beginning with
x. Hence by Cayley’s rules the total number of intersections is
b+ 3e=T+743(5 +2) +3 (542)
6)
=1044+ 541
The first six expansions give «=5, and the second three expansions give « = 2.
LS akg
Notwithstanding the crucial nature of this example and that at the end of Part V. as
tests of the method of this paper, the change of the origin of coordinates used in this
Corollary may quite well render the coefficients in the resulting equation so mutually
dependent that the method of counting the deficiency by the number of interior points
of the curve polygon becomes inoperative, For instance the deficiency of
(y — a) (y — b) + cary? + dary’ + fry + gary’ + hary’ + katy = 0
is quite properly given by the diagram as 1. But by putting y-a=7 we obtain a curve
having eighteen terms, among the coefficients of which there are nine quadratic relations ;
and the polygon of this latter contains seven unit points.
Re p. 427. Cf. Noether, Crelle, 97, p. 224. Also a paper by Hensel, Crelle, 109—which I had not seen when
this paper was written. His results are not universally true. But they enable us to write down the integral
functions when, by some such method as here, we can write down the finite integrals. Or conversely.
IN DEX TOs yO XV.
Baker, H. F. Concomitants of three ternary quadrics,
62
—— On application of Newton’s polygon to the singu-
larities of algebraic functions, 403
Newton’s polygon and plane curves, 403
Curves, method of determining deficiency of, 403
Cayley, on singular points of plane curves, 403
Weierstrass, Normal form for plane curve, 427
Cremona’s transformation of plane curves, 438
Abel, on Deficiency of plane curves, 406
Cayzey, A. On non-Euclidian Geometry, 37
On the kinematics of a plane, 391
On three-bar motion, 391
CurEE, C. Changes in dimensions of Elastic Solids,
313
—— Equilibrium of elastic solid having an axis of
material symmetry, 1
— On some compound vibrating systems, 139
Isotropic Elastic Sphere and Spherical Shell, 339
Crystallization, 119
Elastic Solids, changes in dimensions under stress, 313
Application to the Earth, 327, to ellipsoids, 334,
Betti’s Theorem, 316
Vor. XV. Parr Lv.
Elastic Spheroid, equilibrium of, 1
Elastic Spherical Shell, 339
Geometry, non-Euclidian, 37
Kinematics of a Plane, 391
Liveinc, G. D. On Solution and Crystallization,
119
Lover, A. E. H. On the Rigidity of the Earth, 107
Macponatp, H. M. The self-induction of ‘two parallel
Conductors, 303
Pascal’s hexagram, 267
Ricumonp, H. W. On Pascal’s hexagram, 267
Rigidity of the Earth, 307
Self-induction of two parallel conductors, 303
Solution and Crystallization, 119
Ternary quadrics, concomitants of, 62
Three-bar motion, 391
Vibrating Systems, on some Compound, see Index on
p- 139
’
Cambridge:
PRINTED BY C. J. CLAY, M.A. & SONS,
AT THE UNIVERSITY PRESS.
Camb. Phil Soc. Trans Vol. AV Partly Plate VI
Prof Cayley on Three-bar Motion .
A B Db 3;
~ ~~ 4---=--- -----~--} —____J--------}----
Fig 2-3 Fig? c=atbh-d.
hig 4. c-arb+d.
Fig 3-4
Extertor Series.
7
eo nanae 4%
—
Camb. Phil. Soc Trans Vol XV PartlV
Plate Vi
[ Prot. Cayley on Three -bar Motion
Interior Series.
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i at [ad te, - “|
cs 7 nl witha Bly:
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;
7
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BINDING SECT. JUL 19 1968
Q Cambridge Philosophical
AL Society, Cambridge, Eng.
C19 Transactions
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