# A-1 sentence example

a-1

- aggravated by the mistake of the new government in a 11
- His proposed tax of a 1 This phrase is always ascribed to Lowe, and has become history in association with him.
- Robertson Smith, on the other hand, a new era was reached, in which the recently recognized existence of Totemism was made the basis of an attempt to give a 1 Scipione de Ricci, bishop of Pistoia from 1780 to 1791, on the ex-Jesuits requesting him to consecrate a bell dedicated to this object, issued a pastoral letter (3rd June 1784) in which he pointed out that the spirit of true religion was "far removed from fetichism," and warned his flock against "cardiolatry."
- Possessing soil, climate and apparently all the requisite elements from nature for the production of cotton to an almost boundless extent, and of a 1 Approximately.
- Let there be given n 2 quantities all a,2 a13 ï¿½ï¿½ï¿½ aln a21 a22 a23 ï¿½ï¿½ï¿½ a2n a3, a32 a33 ï¿½ï¿½ï¿½ a3n and an, an 3 ï¿½ï¿½ï¿½ ann and form from them a product of n quantities ala a2 0 a37 ...Advertisement
- Similarly ali a21 a31 A =E a ik a2k a3k A li i > k > r, zï¿½k'r alr a2r air 23',!
- Consider then the system of ni equations a21xi+a22x2+ï¿½ï¿½ï¿½ + a2nxn = 0 a31x1+a32x2+ï¿½ï¿½ï¿½+a3nx,, =0 an1x1 + an2x2 + ï¿½ ï¿½ ï¿½ +annxn = 0, which becomes on writing xs = y 8, a21y1+ a 22y2 + ï¿½ ï¿½ ï¿½ + a 2,n-lyn-1 + a 2n = 0 a3lyl +a32y2+ï¿½ï¿½ï¿½ +a3,n-lyn-i+a3n =0 an1 y1 +an2y2 +ï¿½ï¿½ï¿½ +an, n-lyn-1 +ann = 0.
- Hence_li y ` A 1n where A li and A li, are minors of the complete determinant (a11a22...ann)ï¿½ an1 ant ï¿½ï¿½ï¿½an,n-1 or, in words, y i is the quotient of the determinant obtained by erasing the i th column by that obtained by erasing the n th column, multiplied by (-r)i+n.
- p 1 p 2 ...p n), it is necessary to bring under view other functions associated with the same series of numbers: such, for example, as P P3 P2 P4 Pn -2 /, /, /, F i a i 1 a 2 Fi a 1 a 2 ...
- +amaa.)sm, (ll, v; m P -"O a an + (l l + v) (ll +2 v) (m (11 +3v) +...], m - 2 2 !2 a 0 a 1 aan +2 !Advertisement
- If we restrict ourselves to this set of symbols we can uniquely pass from a product of real coefficients to the symbolic representations of such product, but we cannot, uniquely, from the symbols recover the real form, This is clear because we can write n-1 n-2 2 2n-3 3 a1a2 =a l a 2, a 1 a 2 = a 1 a2 while the same product of umbrae arises from n n-3 3 2n-3 3 aoa 3 = a l .a a 2 = a a 2 .
- We write;L 22 = a 1 a 2 .b 1 n-2 b2s 3 n - 3 3 n-3 3 n-3 3 a 3 = a 1 a 2 .b 1 b 2 .c 1 c2, and so on whenever we require to represent a product of real coefficients symbolically; we then have a one-to-one correspondence between the products of real coefficients and their symbolic forms. If we have a function of degree s in the coefficients, we may select any s sets of umbrae for use, and having made a selection we may when only one quantic is under consideration at any time permute the sets of umbrae in any manner without altering the real significance of the symbolism.
- A 1, A2 ï¿½ Ai, A 1 A 2, A2 and then Ao = al Ai+2a1a2AIA2+a2 A2 - (a1A1+a2A2) 2 = a?, A l = (a 1 A 1 +a2A2) (alï¿½l +a2ï¿½2) = aAaï¿½, A 2 = (alï¿½l +a2/-12) 2 = aM; so that A = aa l +2a A a u 152+aM5 2 = (aA6+a,e2)2; whence A1, A 2 become a A, a m, respectively and ?(S) = (a21+a,E2) 2; The practical result of the transformation is to change the umbrae a l, a 2 into the umbrae a s = a1A1 +a2A2, a ï¿½ = a1/ï¿½1 + a21=2 respectively.
- Again, if 0 is uneven =20+I, the condition is a 1 a 2 ...cr 241 II(a 1 +a 2)II(cr 1 +a 2 +(73)...II(a 1 +a 2 +...+ac) =0; and the degree, in the quantities a, is 20+1 + (42+1) +(21) ï¿½...-F(254)ï¿½1) =22°-1= 2e-1-1 Hence the lowest weight of a perpetuant is 2 0 - 1 -1, when 0 is >2.
- Solving the equation by the Ordinary Theory Of Linear Partial Differential Equations, We Obtain P Q 1 Independent Solutions, Of Which P Appertain To S2Au = 0, Q To 12 B U =0; The Remaining One Is Ab =Aobl A 1 Bo, The Leading Coefficient Of The Jacobian Of The Two Forms. This Constitutes An Algebraically Complete System, And, In Terms Of Its Members, All Seminvariants Can Be Rationally Expressed.Advertisement
- For the case 8=1, 0' =2, the condition is a i r 1 72 = A032=0; and the simplest perpetuant, derived directly from the product A 1 B 21 is (I)a(2)b-(21)b; the remainder of those enumerated by z3 I z.
- For the case 0=0' =2, the condition is a 1 a 2 T 1 T 2(a 1+ a2) (al +TO (al +T2) = -A2B 1 B 2 -A l A 2 B2 = 0.
- (2A2+ 41ï¿½i +2)a(3ï¿½3+12ï¿½2+ 1) b, due to the generating function 2 15 (1 -z)(1 -z 2) 2 (1 -z3) For the case 0=r, 0'=4., the condition is a 1 r 1 T 2 r 3 7 4 (a i + T 1)(a + T 2)(a 1 T (a i + T 4) II (T s T t) 0; the calculation gives Selecting the product A;B 4 B 3 B2, we find the simplest perpetuant A1B4(A 1 B2+A1B3+B4)(-B3-A1B2B3-ATB4) =0.
- The names in Greek are generally 'Iavvfis Kal 'laµ0piis (= a'1 on a'?') as in the Targ.-Jon.
- The theory may be extended to the cases of p= i and p = o; so that a 3 means a.a.a.1, a 2 means a.a.i, a 1 means a.i, and a° means I (there being then none of the multipliers a).Advertisement
- To multiply A 1 by a scalar, we apply the rule A = A1E = E (Eat) ea, and similarly for division by a scalar.
- pp. 503-509); "Chronique parisienne anonyme de 1316 a 1 339
- The agent chosen to preside at the nomination ceremony was Mr (afterwards Sir) Theophilus Shepstone, who was in charge of native affairs in Natal and had won in a 1 Bishop Schreuder, a Norwegian missionary long resident in Zululand, gave Sir Bartle Frere the following estimate of the three brothers who successively reigned over the Zulu: " Chaka was a really great man, cruel and unscrupulous, but with many great qualities.
- St a e Fo e i Q1ne w se E; ott cc't O` appetiama au1 on Stratf Brae unvesjon oaklo r 'jv't11„, I?
- Line a 1 liter (2 pint) pudding basin or 4 individual basins with suet paste.Advertisement
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- A computer with a 1 Terabyte (TB) hard drive is overkill for most, but neither should you settle for an 80 Gigabyte (GB) drive.
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- You should get at least one movie on a 1 GB Memory Stick.
- The Action Bundle comes with a 1 GB memory card, an extra remote and Nunchuk, and the games The Legend of Zelda: Twilight Princess, Super Paper Mario, Super Monkey Ball: Banana Blitz, and WarioWare: Smooth Moves.
- The test can be performed at 10 to 14 weeks; test performance is associated with a 1 percent risk of miscarriage.
- For example, with one exposure of unprotected sexual intercourse, a woman has a 1 percent chance of acquiring HIV, a 30 percent chance of acquiring herpes, and 50 percent chance of contracting gonorrhea if her partner is infected.Advertisement
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- However, it's a very popular summer shoe, well-liked for its comfort, and with a 1 1/2 inch heel, you won't feel too much like a hippie.Advertisement
- Then, cut a 1 inch by 1 inch square out of its center.
- Three is considered normal so a -2 or a 1 are the same font size stated differently when your current font size is 3.
- The impression we get of the man is that, whether or not he actually enjoyed the full rights of Roman citizenship, he was a 1 "If it were permitted that immortals should weep for mortals, the divine Camenae would weep for Naevius the poet; for since he hath passed into the treasure-house of death men have forgotten at Rome how to speak in the Latin tongue."
- We consider the quantic to have any n number of equivalent representations a- b n -c n So that a 1 -k a 2 = b l -k b 2 - c 1 -k c 2 = ...
- When the cylinder r =a is moved with velocity U and r =b with velocity U 1 along Ox, = U b e - a,1 r +0 cos 0 - U ib2 - 2 a, (r +Q 2 ') cos 0, = - U be a2 a2 (b 2 - r) sin 0 - Uib2 b1)a, (r - ¢2 sin 0; b and similarly, with velocity components V and V 1 along Oy a 2 b2 ?= Vb,_a,(r+r) sin g -Vi b, b2 a, (r+ 2) sin 0, (17) = V b, a2 a, (b2 r) cos 0+Vi b, b, a, (r- ¢ 2) cos h; (18) and then for the resultant motion z 2zz w= (U 2 + V2)b2a a2U+Vi +b a b a2 U z Vi -(U12+V12) b2 z a2b2 Ui +VIi b 2 - a 2 U1 +Vii b 2 - a 2 z The resultant impulse of the liquid on the cylinder is given by the component, over r=a (§ 36), X =f p4 cos 0.ad0 =7rpa 2 (U b z 2 + a 2 Uib.2bz a2); (20) and over r =b Xi= fp?Advertisement
- Helmholtz, of the efflux of a jet between two edges A and A 1 in an infinite wall, is obtained by the symmetrical duplication of the above, with n = 1, b = o, a' = - oo, as in fig.
- ', In Bobyleff's problem of the wedge of finite breadth, ch nS2 = ?a b' s h n S 2 = V b a a 1 u u b, (6)?
- The system of forced loans was abolished and a 1 o% tax on real property introduced in its stead, and a law of amnesty for political offenders enacted.
- On the election in 1846 of Pius IX., who appeared to be a Liberal and an Italian patriot, the eyes of a]1 Italy were turned on him as the heaven-born leader who was to rescue the country from the foreigner.
- 3(A-1-,/AB+B).Advertisement
- (ii) Similarly, if m is a multiple of 3, the repeated application of Simpson's second formula gives Simpson's second rule A 1?
- Thus we have x f x udx dx = (x + 2 4 5 - ri o 6 3)2h2u = a2 62, A 1 80 64 rs2 RI-vs 66 + ...)h2u = a2(h2u - 40 6 4 h 2 u).
- If a wave front is in a given position, as a 1 (fig.
- Hasselquist, who had travelled in that country as a 1 Some derive the name sal ammoniac from Jupiter Ammon, near whose temple it is alleged to have been found; others, from a district of Cyrenaica called Ammonia.
- i'C) C: 4.1 A,1 A 1.A O Y 1 t 1 x 1 t)-e1(y FIG.
- above Bordeaux, and the highest point to which ordinary spring-tides ascend, the river is accompanied at a distance of from a 1 to 3 m.
- He took a conspicuous part in arranging the Union of Utrecht, and in 1583 was chosen burgomaster of Antwerp. In 1585 he surrendered the city, after a 1 3 months' siege, to the Spaniards.
- In consequence Winnipeg is already a 1 Incorporated in this year as a city.
- On the 21st of August, while the Indians were demoralized by a sortie from the garrison, the town was evacuated, and the inhabitants made a 1 The exact date of the founding of Santa Fe is not known, but the best opinion has fixed the date between 1604 and 1608, and favours the year 1605.
- During the centuries preceding the birth of Christ there grew up an apocalyptic literature which regarded as a primary truth the conception of a 1 Lactantius, Inst.
- 4 a 1 4,747 sq.
- This alliance between the two monarchs led to a 1 In Judges x.
- But a younger member of the household, Willie Douglas, aged eighteen, whose devotion was afterwards remembered and his safety cared for by Mary at a time of utmost risk and perplexity to herself, succeeded on the 2nd of May in assisting her to escape by a 1 It is to be observed that the above conclusion as to the authenticity of the Casket Letters is the same as that arrived at upon different grounds by the most recent research on the subject.
- Its chief buildings are the citadel and many mosques, one of which is an ancient Byzantine basilica, originally a 1 Prince von Billow was credited with suggesting in his correspondence on the question of the Bundesrath that a tribunal of arbitration should be instituted to deal with all questions of capture.
- There is a 1 Clarendon, State Papers, iii.
- The Federal government completed in October 1907 the construction of a 1 According to the report of the State Geological Survey, the value of the total mineral product in the state for 1907 was $152,122,648, the values of the different minerals being as follows: coal, $54,687,382; pig iron, about $52,228,000; petroleum, $ 16, 43 2, 947; clay and clay products, $13,351,362; zinc, $6,614,608; limestone, $4,333,651; Portland cement, $2,632,576; sand and gravel, $1,367,653; natural slag, $174,282; fluorspar, $141,971; mineral waters, $91,700; lead ore, $45,760; sandstone, $14,996; and pyrite, $5700.
- His unresolved antithesis' is interpreted either diversely or with a 1 Or antitheses.
- from Corinth, an ancient inscription written 1 30vvrpoc 66v has recently been discovered, which shows that though Cleonae for B wrote E {, like the Corinthian ?j, and, as at Corinth, wrote for a vowel sound, the vowel thus represented was not short and long e and n) as at Corinth, but Il only, as in Xp g A, (X p i i a 1 Here 'a represents and the spurious diphthong is represented by a, as in (dycv, Doric infinitive -= a form which shows that c has at Cleonae the more modern form I as distinguished from the Corinthian Regarding three other questions controversy still rages.
- where = {(i,o p cos a)1 +2(fm p cos a)(u cos a v) cos a +
- For a long time Faraday's observation on the rotation of the plane of polarized light by heavy glass in a 1 H.
- In the case of the fraction a 1 - a2 _ a3 _a4 _ we have the relations p n = = ang n-1 - bngn-2 Taking the quantities a l..
- Resolving vertically we find that the weight of the liquid raised above the level must be equal to T(sin 0 2 - sin 01), and this is therefore equal to the area P 1 P 2 A 2 A 1 multiplied by gp. The form of the capillary surface is identical with that of the " elastic curve," or the curve formed by a uniform spring originally straight, when its ends are acted on by equal and T 2 opposite forces applied either to the ends themselves or to solid pieces attached to them.
- Since the plates are very near one another we may use the following equation of the surface as an approximation: y= h1+Ax+Bx 2, h2=h1+Aa+Ba2, whence cot a 1 = - A, cot a 2 =A+2Ba T(cos a i +cos a 2) = +3Ba2), whence we obtain h, = g (cos a, +cos a 2) +6 (2 cot a, - cot a2) h 2 = - T (cos a,+cos a 2) +6 cot a2 - cot al) Let X be the force which must be applied in a horizontal direction to either plate to keep it from approaching the other, then the forces acting on the first plate are T+X in the negative direction, and T sin a t + 2gph 1 2 in the positive direction.
- Accordingly Ga Cg T Z I M A 1 Vat Since X Is Undetermined, All That We Can Conclude Is That M Is Of The Form M= Tax(Q 2?, (I) Where F Denotes An Arbitrary Function.
- The Tom and its numerous tributaries rise on the northern slopes of the Kuznetsk Ala-tau, and their fertile valleys are occupied by a 1 Mr S.
- It formed the rallying-ground for the new generation which chafed under the tyranny of a 1?
- On warming with caustic soda it is converted into the A 1 ' 4 acid.
- There is no manual of Jainism as yet published, but there is a 1 The Hatthi Gumpha and three other inscriptions at Cuttack (Leyden, 1885); Sravana Belgola inscriptions (Bangalore, 1889); Vienna Oriental Journal, vols.
- When the wise primate Lanfranc, his A 1
- In the case where X, Y, Z are of an order exceeding 2 the required number n 2 - r of common intersections can only occur by reason of common multiple points on the three curves; and assuming that the curves X=o, Y=o, Z = o have a 1 +a 2 +a 3 ...
- (n-I)2an-1 =n2-I; but to this must be joined the condition a 1 -F3a 2 +6a 3 ...
- In analytical geometry, r the equation axe+2hxy+bye+2gx+2fy+ c = o represents an ellipse when ab > h 2; if the centre of the curve be the origin, the equation is a 1 x 2 +2h 1 xy+b i y 2 =C 1, and if in addition a pair of conjugate diameters are the axes, the equation is further simplified to Ax e +By 2 = C. The simplest form is x 2 /a 2 +y 2 /b 2 = 1, in which the centre is the origin and the major and minor axes the axes of co-ordinates.
- A A 1 n outer surface.
- The two currents are therefore added in one wire and subtracted in the other, and produce a differential heating effect which causes unequal expansion, and this in turn is made to tilt a 1 H.
- The foul incurs a 1 minute penalty.
- Using the frozen glycerol stock a 1 liter Terrific Broth + 100 µg/ml ampicillin culture was inoculated.
- Paddy is a 1 1/2 year old cocker spaniel.
- The main buildings on the site are a 1 hectare greenhouse for salad cress production and a 5 hectare greenhouse growing pot herbs.
- About two hours later our path was blocked by a 1 1/2 meter wide deep crevasse.
- Pour into a 1 gallon demijohn, add the citric acid and tea.
- To left of central section a 1 light dormer with stack with fixed 1 light.
- Is there anyone out there who could give us a 1 or 2 ton dumper with four-wheel drive and a roll bar?
- melt the ghee and use to grease a 1 liter ovenproof dish.
- The problem was overcome by using a 1 ton chain hoist for the raising.
- inoculated with virus in a 1 square kilometer pen behind two sets of double fences.
- Lightly cook 1 sliced leek in a 1 tbsp olive oil to soften.
- marsh orchid at Brooklands In 1996 the Trust decided to create a 1 ha.
- The device will carry a 1 megaton nuclear warhead.
- metreave a friend who uses a 1 meter diameter dish on this band!
- A References A.1 normative References IETF RFC 2119 RFC 2119: Key words for use in RFCs to Indicate Requirement Levels.
- The foliage effects he has made using a 1 inch paintbrush are truly stunning.
- People who carry this condition have a 1 in 10 chance of developing a sarcoma.
- spindle pulleys was 6.5 inches in diameter and took a 1 3/8 " -wide belt.
- The diving boards included a 1 meter springboard and a symmetrically stepped stage up to 8 feet high.
- The DPF80 series (Dual Rate totalizer) is a 5 digit Ratemeter, 6 digit Totalizer in a 1 /8 DIN package.
- walkers Paradise with numerous munros within a 1 hour drive.
- A supplementation of up to a 1 mg of elemental zinc / kg body weight / day is therefore recommended.
- We do not know what faculty the master-thinker may have had for captivating this ardent spirit; at any rate Alexander carried with him through life a 1 The use of the surname,is proved as far back as the 1st century B.C. (Nepos, De Reg.
- The lower vibration number is justified by due consideration of the three divisions of the male voice, bass, tenor and alto, as given by Praetorius, whose Cammerton very closely corresponds with Bernhardt Schmidt's Durham organ, 1663-1668, the original pitch of which has been proved by Professor Armes to have been a 1 474.1.
- Such an arrangement would distribute a 1 For a history of the discovery of the earth return, see Fahie, History of Electric Telegraphy to the Year 18 37, pp. 343-348.
- ductor A 1 and creating electrical oscillations (see Electrokinetics) in the wire.
- The various theories with regard to the origin of the name are all somewhat unconvincing; it is conceivable that, with the other homonymous Greek towns, such as Athenae Diades in Euboea, A01 vac may be connected etymologically with iivOos, a flower (cf.
- Let there be given n 2 quantities all a,2 a13 Ã¯¿½Ã¯¿½Ã¯¿½ aln a21 a22 a23 Ã¯¿½Ã¯¿½Ã¯¿½ a2n a3, a32 a33 Ã¯¿½Ã¯¿½Ã¯¿½ a3n and an, an 3 Ã¯¿½Ã¯¿½Ã¯¿½ ann and form from them a product of n quantities ala a2 0 a37 ...
- Similarly ali a21 a31 A =E a ik a2k a3k A li i > k > r, zÃ¯¿½k'r alr a2r air 23',!
- If the jth column be identical with the i ll ' the determinant A vanishes identically; hence if j be not equal to i, k, or r, a 11 a 21 a31 0 =I alk a2k a3k A11.
- Similarly, for the order 3, we take 1 v Ab= -v 1 A =1 +x2 + 1, 2 + Ã¯¿½ - A 1 and the adjoint is 1+A v +AÃ¯¿½ -Ã¯¿½ +Av -v +AÃ¯¿½ 1+11 2 A +/-tv pt+AvA +Ã¯¿½v 1 +1,2 leading to the orthogonal substitution Abx1 = (1 +A 2 - / 22 - v 2) X l +2(v+AÃ¯¿½)X2 +2(/1 +Av)X3 1bx2 = 2(AÃ¯¿½ - v)Xl+(1 +Ã¯¿½2 - A2 - v2)X2 / +2(Fiv+A)X3 Abx3 = 2(Av +Ã¯¿½)X1 +2(/lv-A)X2+(1+v2-A2- (12)X3.
- Consider then the system of ni equations a21xi+a22x2+Ã¯¿½Ã¯¿½Ã¯¿½ + a2nxn = 0 a31x1+a32x2+Ã¯¿½Ã¯¿½Ã¯¿½+a3nx,, =0 an1x1 + an2x2 + Ã¯¿½ Ã¯¿½ Ã¯¿½ +annxn = 0, which becomes on writing xs = y 8, a21y1+ a 22y2 + Ã¯¿½ Ã¯¿½ Ã¯¿½ + a 2,n-lyn-1 + a 2n = 0 a3lyl +a32y2+Ã¯¿½Ã¯¿½Ã¯¿½ +a3,n-lyn-i+a3n =0 an1 y1 +an2y2 +Ã¯¿½Ã¯¿½Ã¯¿½ +an, n-lyn-1 +ann = 0.
- Hence_li y ` A 1n where A li and A li, are minors of the complete determinant (a11a22...ann)Ã¯¿½ an1 ant Ã¯¿½Ã¯¿½Ã¯¿½an,n-1 or, in words, y i is the quotient of the determinant obtained by erasing the i th column by that obtained by erasing the n th column, multiplied by (-r)i+n.
- Thus to obtain the resultant of aox 3 +a i x 2 +a 2 x+a 3, 4, =box2+bix+b2 we assume the identity (Box+Bi)(aox 3 +aix 2 +a2x+a3) = (Aox 2 +Aix+ A 2) (box2+bix+b2), and derive the linear equations Boa Ã‚° - Ac b o = 0, Boa t +B i ao - A 0 b 1 - A 1 bo =0, Boa t +B 1 a 1 - A0b2 - A1b1-A2bÃ‚° = 0, Boa3+Bla2 - A l b 2 -A 2 b 1 =0, B 1 a 3 - A 2 b 2 =0, = = (y l, y2,...ynl `x1, x2,...xnl for brevity.
- Taking the same example as before the process leads to the system of equations acx 4 +alx 3 +a2x 2 +a3x =0, aox 3 +a1x 2 +a2x+a 3 = 0, box +bix -1-b2x =0, box' +b i x 2 -{-h 2 x = 0, box + b i x + b:: = 0, whence by elimination the resultant a 0 a 1 a 2 a 3 0 0 a 0 a 1 a 2 a3 bo b 1 b 2 0 0 0 bo b 1 b 2000 bo b 1 b2 which reads by columns as the former determinant reads by rows, and is therefore identical with the former.
- It can be shown that the number 0 enumerates distributions of a certain nature defined by the partitions (,Ã¯¿½i,Ã¯¿½2...), (sT1sÃ‚°2...), 1212 an = a 1 a 2 a 3 ...
- For the substitution rr xl =A 11 +1 2 12, 52=A21+Ã¯¿½2E2, of modulus A1 Ã¯¿½i = (AlÃ¯¿½.2-A2Ã¯¿½1) = (AM), A 2 Ã¯¿½2 the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A lÃ¯¿½l +ai(A1/.22+A2Ã¯¿½1) +7,2X2/22, A2 = aoÃ¯¿½l +2a1Ã¯¿½1/Ã¯¿½2 +a 2Ã¯¿½2 Ã¯¿½ We pass to the symbolic forms a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/ by writing for ao, al, a2 the symbols ai, a 1 a 2, a?
- A 1, A2 Ã¯¿½ Ai, A 1 A 2, A2 and then Ao = al Ai+2a1a2AIA2+a2 A2 - (a1A1+a2A2) 2 = a?, A l = (a 1 A 1 +a2A2) (alÃ¯¿½l +a2Ã¯¿½2) = aAaÃ¯¿½, A 2 = (alÃ¯¿½l +a2/-12) 2 = aM; so that A = aa l +2a A a u 152+aM5 2 = (aA6+a,e2)2; whence A1, A 2 become a A, a m, respectively and ?(S) = (a21+a,E2) 2; The practical result of the transformation is to change the umbrae a l, a 2 into the umbrae a s = a1A1 +a2A2, a Ã¯¿½ = a1/Ã¯¿½1 + a21=2 respectively.
- Put M 1 For M, N I For N, And Multiply Through By (Ab); Then { (F, C6) } = (Ab) A X 2A Y B X 1 M N I 2 (Xy), ?) 2, = (A B)Ax 1B X 2B Y L I Multiply By Cp 1 And For Y L, Y2 Write C 2, C1; Then The Right Hand Side Becomes (Ab)(Bc)Am Lbn 2Cp 1 M I C P (F?) 2 M { N2 X, Of Which The First Term, Writing C P =, ,T, Is Mn 2 A B (Ab)(Bc)Axcx 1 M 2 N 2 P 2 2222 2 2 _2 A X B X C (Bc) A C Bx M N 2 2 2 M2Ã‚°N 2 N 2 M 2 2 A X (Bc) B C P C P (Ab) A B B(Ac) Ax Cp 2 = 2 (04) 2 1 (F,0) 2.4 (F,Y') 2 Ã¯¿½?; And, If (F,4)) 1 = Km " 2, (F??) 1 1 M N S X X X Af A _Af A Ax, Ax Ax Ax1 Observing That And This, On Writing C 2, C 1 For Y 11 Y 21 Becomes (Kc) K X 'T 3C X 1= (F,0 1 ', G 1; Ã¯¿½'Ã¯¿½1(F,O) 1 M 1=1 M 2 0`,4)) 2 0, T (Fm 2.4 (0,0 2 .F ' And Thence It Appears That The First Transvectant Of (F, (P) 1 Over 4) Is Always Expressible By Means Of Forms Of Lower Degree In The Coefficients Wherever Each Of The Forms F, 0, 4, Is Of Higher Degree Than The First In X 1, X2.
- Again, if 0 is uneven =20+I, the condition is a 1 a 2 ...cr 241 II(a 1 +a 2)II(cr 1 +a 2 +(73)...II(a 1 +a 2 +...+ac) =0; and the degree, in the quantities a, is 20+1 + (42+1) +(21) Ã¯¿½...-F(254)Ã¯¿½1) =22Ã‚°-1= 2e-1-1 Hence the lowest weight of a perpetuant is 2 0 - 1 -1, when 0 is >2.
- (2A2+ 41Ã¯¿½i +2)a(3Ã¯¿½3+12Ã¯¿½2+ 1) b, due to the generating function 2 15 (1 -z)(1 -z 2) 2 (1 -z3) For the case 0=r, 0'=4., the condition is a 1 r 1 T 2 r 3 7 4 (a i + T 1)(a + T 2)(a 1 T (a i + T 4) II (T s T t) 0; the calculation gives Selecting the product A;B 4 B 3 B2, we find the simplest perpetuant A1B4(A 1 B2+A1B3+B4)(-B3-A1B2B3-ATB4) =0.
- The theory may be extended to the cases of p= i and p = o; so that a 3 means a.a.a.1, a 2 means a.a.i, a 1 means a.i, and aÃ‚° means I (there being then none of the multipliers a).
- Every ordinary algebraic quantity may be regarded as of the form a+0 - I, where a, 6 are " real "; that is to say, every algebraic equivalence remains valid when its symbols of quantity are interpreted as complex numbers of the type a-1-(-V - I (cf.
- The inverse operation is free from ambiguity, and, in fact, A 1 - B 1 = E (a i - 130ei.
- 2A 0 - 2AI 2 a 1 3 - a3 When the spheres are not concentric, an expression for the effective inertia can be found by the method of images (W.
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- Similarly, for the order 3, we take 1 v Ab= -v 1 A =1 +x2 + 1, 2 + ï¿½ - A 1 and the adjoint is 1+A v +Aï¿½ -ï¿½ +Av -v +Aï¿½ 1+11 2 A +/-tv pt+AvA +ï¿½v 1 +1,2 leading to the orthogonal substitution Abx1 = (1 +A 2 - / 22 - v 2) X l +2(v+Aï¿½)X2 +2(/1 +Av)X3 1bx2 = 2(Aï¿½ - v)Xl+(1 +ï¿½2 - A2 - v2)X2 / +2(Fiv+A)X3 Abx3 = 2(Av +ï¿½)X1 +2(/lv-A)X2+(1+v2-A2- (12)X3.
- For the substitution rr xl =A 11 +1 2 12, 52=A21+ï¿½2E2, of modulus A1 ï¿½i = (Alï¿½.2-A2ï¿½1) = (AM), A 2 ï¿½2 the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A lï¿½l +ai(A1/.22+A2ï¿½1) +7,2X2/22, A2 = aoï¿½l +2a1ï¿½1/ï¿½2 +a 2ï¿½2 ï¿½ We pass to the symbolic forms a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/ by writing for ao, al, a2 the symbols ai, a 1 a 2, a?
- =0, a 1 a ?
- But, under the guidance of a 1 The full title of the palatine (Mag.
- If A 1 = X a i e i, B i = /if i e i, the distributive law of multiplication is preserved by assuming A1B1=E(a0 i 3)eiej; it follows that A 1 B 1 = - B 1 A 1, and that A l 2 = o.
- A2B 1 B 2 gives (22) a (21) b - (221) a (2) b, and A i AgB 21 (2 2 I) a (2) b -(22)a(21) b; these two merely differ in sign; and similarly A 2 B 1 B2 yields (2)a(2 2 I) b -(21)a(22)b, and that due to A 1 A 2 B2 merely differs from it in sign.
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