# Cubic Sentence Examples

- Xy 2 -4z 3 +g2x 2 y+g3x 3, and also the special form axz 2 -4by 3 of the cuspidal
**cubic**. An investigation, by non-symbolic methods, is due to F. - Now geometry deals with points, lines, planes and
**cubic**contents. - Its weight varies from 48 to about 55lb the
**cubic**foot, but in very hard slowly-grown trunks sometimes approaches 60 lb. - For the
**cubic**(ab) 2 axbx is a covariant because each symbol a, b occurs three times; we can first of all find its real expression as a simultaneous covariant of two, and then, by supposing the two**cubics**to merge into identity, find the expression of the quadratic covariant, of the single**cubics****cubic**, commonly known as the Hessian. - 2 - 9 m 2 (1 3 m 2)) 2 we have a
**cubic**equation for determining m 2 as a function of the absolute invariant. - We will write the
**cubic**covariant (f, i) 2 =j, and then remark that the result, (f,j) 3 = o, can be readily established. - For the quadratic it is the discriminant (ab) 2 and for ax2 the
**cubic**the quadratic covariant (ab) 2 axbx. - The ternary
**cubic**has been investigated by Cayley, Aronhold, Hermite, Brioschi and Gordan. - From the invariant a2 -2a 1 a 3 -2aoa4 of the quartic the diminishing process yields ai-2a 0 a 21 the leading coefficient of the Hessian of the
**cubic**, and the increasing process leads to a3 -2a 2 a 4 +2a i a 5 which only requires the additional term-2aoa 6 to become a seminvariant of the sextic. A more important advantage, springing from the new form of S2, arises from the fact that if x"-aix n- +a2x n-2. - Again, for the
**cubic**, we can find A3(z) - -a6z6 1 -az 3.1 -a 2 z 2.1 -a 3 z 3.1 -a4 where the ground forms are indicated by the denominator factors, viz.: these are the**cubic**itself of degree order I, 3; the Hessian of degree order 2, 2; the cubi-covariant G of degree order 3, 3, and the quartic invariant of degree order 4, o. - The closed figure a c d e a is variously called a hysteresis curve or diagram or loop. The area f HdB enclosed by it represents the work done in carrying a
**cubic**centimetre of the iron through the corresponding magnetic cycle; expressed in ergs this work is I HdB. - Fleming, it 47r requires about 18 foot-pounds of work to make a complete mag netic cycle in a
**cubic**foot of wrought iron, strongly magnetized first one way and then the other, the work so expended taking the form of heat in the mass. - When the magnetizing current is twice reversed, so as to complete a cycle, the sum of the two deflections, multiplied by a factor depending upon the sectional area of the specimen and upon the constants of the apparatus, gives the hysteresis for a complete cycle in ergs per
**cubic**centimetre. - Denoting by W the work in ergs done upon a
**cubic**centimetre of the metal (=_fHdB or f HdI), he finds W =nips approximately, where n 47r is a number, called the hysteretic constant, depending upon the metal, and B is the maximum induction. - Working with two different specimens, he found that the hysteresis loss in ergs per
**cubic**centimetre (W) was fairly represented by o 00125B 1 6 and o o0101B 1 ' 6 respectively, the maximum induction ranging from about 300 to 3000. - In many experiments, however, different inductions and frequencies are employed, and the hysteresis-loss is often expressed as ergs per
**cubic**centimetre per cycle and sometimes as horse-power per ton. - In one case the hysteresis loss per
**cubic**centimetre per cycle was 16,100 ergs for B =1 5,900, and only 1200 ergs for B = 20,200, the highest induction obtained in the experiment; possibly it would have vanished before B had reached 21,000.2 These experiments prove that actual friction must be almost entirely absent, and, as Baily remarks, the agreement of the results with the previously suggested deduction affords a strong verification of Ewing's form of the molecular theory. - But some phenomena are difficult to reconcile with pressed into less than one five-hundredth of a
**cubic**foot, or, if allowed to expand, the air originally occupying the**cubic**foot can be made to fill, apparently uniformly, a space of a million**cubic**feet or more. - Owing to these natural "locks," the Senegal never discharges less than 1700 or 1800
**cubic**ft. - The unit to which they are ordinarily referred is I electrostatic unit of electricity per
**cubic**metre of air. - The wood is very heavy and hard, weighing 70 lb the
**cubic**foot; the colour is dark brown; it is used in Spain and Italy for furniture, and in the former country for firewood and charcoal. - Thar equal to 148930 foot pounds 4 2.37
**cubic**feet. - West Virginia, estimates that in fairly good producing sand a
**cubic**foot of rock contains from 6 to 12 pints of oil. - For Tartaglia's discovery of the solution of
**cubic**equations, and his contests with Antonio Marie Floridas, see Algebra (History). - Problems in artillery occupy two out of nine books; the sixth treats of fortification; the ninth gives several examples of the solution of
**cubic**equations. - A standard sodium hydrate solution can be prepared by dissolving 42 grammes of sodium hydrate, making up to a litre, and diluting until one
**cubic**centimetre is exactly equivalent to one**cubic**centimetre of the sulphuric acid. - Hexachlorethane is trimorphous, forming rhombic, triclinic and
**cubic**crystals; the successive changes occur at about 44° and 71°, and are attended by a decrease in density. - Telluric acid forms
**cubic**and monoclinic crystals from a hot nitric acid solution, and ammonium fluosilicate gives**cubic**and hexagonal forms from aqueous solutions between 6° and 13°. - Ammonium iodide assumes
**cubic**forms with perfect**cubic**cleavage; tetramethyl ammonium iodide is tetragonal with perfect cleavages parallel to {100} and {o01} - a difference due to the lengthening of the a axes; tetraethyl ammonium iodide also assumes tetragonal forms, but does not exhibit the cleavage of the tetramethyl compound; while tetrapropyl ammonium iodide crystallizes in rhombic form. - We may therefore regard the nitrogen atoms as occupying the centres of a
**cubic**space lattice composed of iodine atoms, between which the hydrogen atoms are distributed on the tetrahedron face normals. - The following are the " lowest level " discharges of the principal affluents of the Brahmaputra in Upper Assam, estimated in
**cubic**feet per second: Lohit river, 9 m. - D x the first evectant; and thence 4cxdi the second evectant; in fact the two evectants are to numerical factors pres, the
**cubic**covariant Q, and the square of the original**cubic**. - The Binary
**Cubic**.-The complete system consists of f=aa,(f,f')'=(ab)2a b =0 2, (f 0)= (ab) 2 (ca)b c=Q3, x x x x x x and (0,0')2 (ab) 2 (cd) 2 (ad) (bc) = R. - The transformation to the normal form, by the solution of a
**cubic**and a quadratic, therefore, supplies a solution of the quartic. If (Xï¿½) is the modulus of the transformation by which a2 is reduced to 3 the normal form, i becomes (X /2) 4 i, and j, (Ap) 3 j; hence? - There are four invariants (i, i')2; (13, H)6; (f2, 151c.; (f t, 17)14 four linear forms (f, i 2) 4; (f, i 3) 5; (i 4, T) 8; (2 5, T)9 three quadratic forms i; (H, i 2)4; (H, 23)5 three
**cubic**forms (f, i)2; (f, i 2) 3; (13, T)6 two quartic forms (H, i) 2; (H, 12)3. - Hence, solving the
**cubic**, R 2 j = (S -m i a) (S - m 2 a) (S - m3a) wherein m 1 m2, m 3 are invariants. - Now, evidently, the third transvectant of f, expressed in this form, with the
**cubic**pxgxrx is zero, and hence from a property of the covariant j we must have j = pxgxrx; showing that the linear forms involved are the linear factors of j. - The complete covariant and contravariant system includes no fewer than 34 forms; from its complexity it is desirable to consider the
**cubic**in a simple canonical form; that chosen by Cayley was ax 3 +by 3 + cz 3 + 6dxyz (Amer. - Hesse showed independently that the general ternary
**cubic**can be reduced, by linear transformation, to the form x3+y3+z3+ 6mxyz, a form which involves 9 independent constants, as should be the case; it must, however, be remarked that the counting of constants is not a sure guide to the existence of a conjectured canonical form. - By the x process of Aronhold we can form the invariant S for the
**cubic**ay+XH:, and then the coefficient of X is the second invariant T. - Of the Memoir he discusses bi-ternary quantics, and in particular those which are lineo-linear, quadrato-linear, cubo-linear, quadrato-quadratic, cubo-
**cubic**, and the system of two lineo-linear quantics. - The wood is variable in quality and, though hard in texture, is less durable than the best oak of British growth; the heart-wood is of a light reddish brown varying to an olive tint; a Canadian specimen weighs 524 lb the
**cubic**foot. - He assumes that in what is considered a good producing district the amount of petroleum which can be obtained from a
**cubic**foot of rock would not be more than a gallon, and that the average thickness of the oil-bearing rock would not exceed 5 ft. - The mere retention of the same crystal form by homologous substances is not a sufficient reason for denying a morphotropic effect to the substituent group; for, in the case of certain substances crystallizing in the
**cubic**system, although the crystal form remains unaltered, yet the structures vary. - Of these all except points are quantities: lines involve lengths, planes involve areas, and
**cubic**contents involve volumes. - The Flos of Leonardo turns on the second question set by John of Palermo, which required the solution of the
**cubic**equation x 3 -{-2x'-}-lox = 20. - It became clear that only very rough estimates of the numbers of planktonic organisms in a volume of sea-water as large as (say) 10
**cubic**metres could be made, but that these estimates could nevertheless be trusted to show very marked regional and seasonal differences. - Let x be the number of molecules which dissociate per second when the number of undissociated molecules in unit volume is unity, then in a dilute solution where the molecules do not interfere with each other, xp is the number when the concentration is p. Recombination can only occur when two ions meet, and since the frequency with which this will happen is, in dilute solution, proportional to the square of the ionic concentration, we shall get for the number of molecules re-formed in one second ye where q is the number of dissociated molecules in one
**cubic**centimetre. - This method of solution fails when the discriminant R vanishes, for then the Hessian has equal roots, as also the
**cubic**f. - We find that Di must be equal to p x g x for then t x (p x) 3 +, u (g x) 3, Hence, if px, qx be the linear factors of the Hessian 64, the
**cubic**can be put into the form A(p x) 3 +ï¿½(g x) 3 and immediately solved. - The discriminant, whose vanishing is the condition that f may possess two equal roots, has the expression j 2 - 6 i 3; it is nine times the discriminant of the
**cubic**resolvent k 3 - 2 ik- 3j, and has also the expression 4(1, t') 6 . - For, since -2t 2 =0 3 -21f 2, 6,-3j(-f) 3, he compares the right-hand side with
**cubic**resolvent k 3 -21X 2 k - j 2.