In general his object is to reduce the final equation to a simple one by making such an assumption for the side of the square or cube to which the expression in x is to be equal as will make the necessary number of coefficients vanish.
The mass-equation then becomes an energy-equation.
Per second, the total resistance R, which the engine can overcome at this s p eed, is by equation (10) R=(1190X550)/88=7.400 lb.
Hence if W is the weight of the vehicles in tons, and the weight of the engine and tender be taken at too tons, the value of W can be found from the equation 1 4 W +33 00= 744 0, from which W =296 tons.
Like Berthelot, he writes the chemical equation of the reaction, but in addition he considers the chemical formula of each substance to express not only its material composition, but also the (unknown) value of its intrinsic energy.
The distribution of weight in chemical change is readily expressed in the form of equations by the aid of these symbols; the equation 2HC1+Zn =ZnCl2+H2, for example, is to be read as meaning that from 73 parts of hydrochloric acid and 65 parts of zinc, 136 parts of zinc chloride and 2 parts of hydrogen are produced.
Nagel (Ber., 1898, 31, p. 2009), this oxide does not exist, the reaction leading to the formation of an hydroxide according to the equation: Mo 3 C1 4 (OH) 2 + 4KHO 3H 2 O = 3Mo(OH) 3 -l-4KBr+3H.
Riccati's name is best known in connexion with his problem called Riccati's equation, published in the Ada Eruditorum, September 1724.
In equation (4) there is a fixed relation between w, V and D given by the expression.
Equation (3), § I expresses the fundamental condition which must be satisfied when a locomotive is starting a train.
If p is the mean pressure at any speed the total tractive force which the engine is exerting is given by equation (25) above.
Zero for carbon and oxygen, and x for carbon dioxide, we obtain the equation o+o=x+94300 cal.
With knowledge then of the heats of formation of the substances involved in any chemical action, we can at once calculate the thermal effect of the action, by placing for each compound in the energy-equation its heat of formation with the sign reversed, i.e.
A few problems lead to indeterminate equations of the third and fourth degrees, an easy indeterminate equation of the sixth degree being also found.
Its cartesian equation, when the line joining the two fixed points is the axis of x and the middle point of this line is the origin, is (x 2 + y 2)2 = 2a 2 (x 2 - y 2) and the polar equation is r 2 = 2a 2 cos 20.
Such curves are given by the equation x 2 - y 2 = ax 4 -1bx 2 y 2 +cy 4 .
The solution may be worked out directly or through the determination of the equation of the centre which, being added to the mean anomaly, gives the true anomaly.
Thus, the equation 2112+02 =2H20 not only represents that certain definite weights of hydrogen and oxygen furnish a certain definite weight of the compound which we term water, but that if the water in the state of gas, the hydrogen and the oxygen are all measured at the same temperature and pressure, the volume occupied by the oxygen is only half that occupied by the hydrogen, whilst the resulting water-gas will only occupy the same volume as the hydrogen.
One other instance may be given; the equation 2NH3=N2+3H2 represents the decomposition of ammonia gas into nitrogen and hydrogen gases by the electric spark, and it not only conveys the information that a certain relative weight of ammonia, consisting of certain relative weights of hydrogen and nitrogen, is broken up into certain relative weights of hydrogen and nitrogen, but also that the nitrogen will be contained in half the space which contained the ammonia, and that the volume of the hydrogen will be one and a half times as great as that of the original ammonia, so that in the decomposition of ammonia the volume becomes doubled.
If a very large quantity of water be added, the chloride is entirely decomposed in the manner represented by the equation BiC1 3 -fOH, = BiOCI -F2HC1, Bismuth chloride.
In pure algebra Descartes expounded and illustrated the general methods of solving equations up to those of the fourth degree (and believed that his method could go beyond), stated the law which connects the positive and negative roots of an equation with the changes of sign in the consecutive terms, and introduced the method of indeterminate coefficients for the solution of equations.'
Soc., 1902, 81, p. I) showed that this can be almost entirely avoided by replacing the manganese oxide by hydrated ferric oxide, the reaction proceeding according to the equation: 2Fe(OH) 3 3S0 2 = FeS 2 0 6 FeS0 3 3H 2 0.
The coefficient of friction is a variable quantity depending upon the state of the rails, but is usually taken to be This is the fundamental equation between the forces acting, however the torque may be applied.
Multiplying through by w we obtain Tw = 2FwD = 2µWwD = RV (4) This is a fundamental energy equation for any form of locomotive in which there is only one driving-axle.
The relation between the b.h.p. and the torque on the driving-axle is 55 o B.H.P. =Tu., (9) It is usual with steam locomotives to regard the resistance R as including the frictional resistances between the cylinders and the driving-axle, so that the rate at which energy is expended in moving the train is expressed either by the product RV, or by the value of the indicated horse-power, the relation between them being 55 0 I.H.P. =RV (Io) or in terms of the torque 55 0 I.H.P.X€=RVe=TW (II) The individual factors of the product RV may have any value consistent with equation (to) and with certain practical conditions, so that for a given value of the I.H.P. R must decrease if V increases.
Thus by transposition we may write the last equation as follows 2HI =H2+12+12200 cal., and thus express that hydriodic acid when decomposed into its elements evolves 12200 cal.
Thus the equation Cl 2 -1-2KI, Aq=2KC1, Aq+12+52400 cal., or (C12) +2KI, Aq =2KC1, Aq+-I-52400 cal., would express that when gaseous chlorine acts on a solution of potassium iodide, with separation of solid iodine, 52400 calories are evolved.
The elliptic lemniscate has for its equation (x 2 +31 2) 2 =a 2 x 2 +b 2 y 2 or r 2 = a 2 cos 2 9 +b 2 sin 20 (a> b).
The hyperbolic lemniscate has for its equation (x2 +y2)2 = a2x2 - b 2 y 2 or r 2 =a 2 cos 2 0 - b 2 sin 2 B.
The hydrocarbon C20H42, for example, might be resolved into C5H12+C15H30, or CEH14+C14H28, or C7H16 +C13H26, &c., the general equation of the decomposition being C„1-1 27, ± 2 (paraffin) =G_rH2(, - P)+2 (paraffin)+C P H 2 n (olefine).
A simple equation like this, therefore, when properly interpreted, affords a large amount of information.