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.
The unit to which they are ordinarily referred is I electrostatic unit of electricity per cubic metre of air.
For Tartaglia's discovery of the solution of cubic equations, and his contests with Antonio Marie Floridas, see Algebra (History).
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.
Its weight varies from 48 to about 55lb the cubic foot, but in very hard slowly-grown trunks sometimes approaches 60 lb.
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.
The ternary cubic has been investigated by Cayley, Aronhold, Hermite, Brioschi and Gordan.
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.
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°.
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.
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.
We will write the cubic covariant (f, i) 2 =j, and then remark that the result, (f,j) 3 = o, can be readily established.
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 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.
On account of its cubic form the mineral was early known as "cube ore" (Ger., Witir felerz); the name pharmacosiderite, given by J.
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.
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.
This method of solution fails when the discriminant R vanishes, for then the Hessian has equal roots, as also the cubic f.
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.
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.
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.
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.
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 cubics, and then, by supposing the two cubics to merge into identity, find the expression of the quadratic covariant, of the single cubic, commonly known as the Hessian.
For the quadratic it is the discriminant (ab) 2 and for ax2 the cubic the quadratic covariant (ab) 2 axbx.
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?
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.
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.
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.
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.