If all the variables of a system with respect to which the equation is homogeneous are increased in the same ratio, the equation will still be true.
The general equations occurring in the application of mathematics to natural phenomena are equally true what ever units we employ for the measurement of the different quantities which enter into them, provided we employ the same units throughout the equation. Hence such equations must be homogeneous with respect to any system of vari ables which is referred to the same unit, and all quantities essentially numerical, such as exponents and exponentials, logarithms, angles, and circular and elliptic functions, must be of zero dimensions.
There are two methods of interpreting the equations relating to geometry and other concrete sciences.
We may regard the symbols which occur in the equation as of themselves denoting lines, masses, times, (fee. ; or we may consider each symbol as denoting only the numerical value of the corresponding quantity, the concrete unit to which it is referred being tacitly understood.
If we adopt the first method we shall often have difficulty in interpreting terms which make their appear ance during the calculations. We shall therefore consider all the written symbols as mere numerical quantities, and therefore subject to all the operations of arithmetic during the process of calculation. But in the original equations and the final equations, in which every term has to be interpreted in a physical sense, we must convert every numerical expression into a concrete quantity by multiply ing it by the unit of that kind of quantity.
Thus if we write [L] for the unit of length, that is to say, the actual concrete centimetre or foot, and if # denotes the numerical value of a certain line, then the complete expression for the line is x [L] ; and if y, z, &c., are the numerical values of other lines, then the complete expression for the quantity whose numerical value is xαyβzγ is
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and this quantity is said to be of α + β + γ dimensions with
respect to [L], the unit of length.
There must be as many different units as there are different kinds of quantities to be measured, but in all dynamical sciences it is possible to define these units in terms of the three fundamental units of length, time, and mass. We therefore suppose these three fundamental units to be given, and deduce all the others from these by the simplest attainable definitions.
The equations at which we arrive must be such that a person of any nation, by substituting for the different symbols the numerical values of the quantities as measured by his own national units, would obtain a true result.
This can only be the case if the equation is homogeneous with respect to each of the fundamental units. To ascertain if it is so we must count the dimensions of every term, and for this purpose we must know the dimensions of any derived units which enter into the equation. The theory of the dimensions of physical quantities were first stated by Fourier, Théorie de Chaleur, sec. 160.
By knowing the dimensions of any quantity we are able at once to deduce its numerical value as expressed in terms of one system of units from its numerical value as given in terms of another system.
Thus, magnetic measurements have been made according to the British system, in which the foot, the grain, and the second of mean time are the fundamental units. Other magnetic measurements have been made according to systems derived from the French metric system, using the metre, centimetre, or millimetre as unit of length, the kilogramme, gramme, or milligramme as unit of mass, and the second as unit of time. In recent times an effort has been made to procure the adoption for all scientific measurements of a system in which the centimetre, gramme, and second are the units. This is sometimes referred to as the C. G. S. system, and a copious list of examples of the measurement of physical quantities on this system, of its comparison with other systems, and of the dimensions of quantities occurring in all branches of physics, has been prepared by Dr Everett, and published by the Physical Society of London and by Taylor and Francis, under the title Illustrations of the C. G. S. System of Units.
The three fundamental units may be selected each independently of the others, in an entirely arbitrary manner. It is possible, however, by taking advantage of the permanence of the properties of natural substances, so to define the units that one or more of them may be re produced without reference to any material standard at present existing.
Thus, if the density of a standard substance in a standard state, such as water when at its maximum density under the pressure of its own vapour, is defined as the unit of density, then the unit of mass may be derived from the unit of length, or vice versa. In this system, therefore, the dimensions of mass in terms of length are L3 , or of length in terms of mass, M•1/3
We may define the three fundamental units without reference to any actual body, but by means of a natural substance such as water. For if the solid, liquid, and gaseous states of pure water are in equilibrium in a vessel containing no other fluid, the pressure and temperature of the system are determinate. We may therefore define the unit of density in terms of the density of the liquid water under these conditions, and the unit of pressure in terms of the pressure in the vessel. We may deduce the third unit from the law of gravitation, and define the unit of time in terms of the time of revolution of a satellite about a sphere having the unit density at a distance equal to the radius. This time must be calculated from the results of experiments on attraction. Having thus obtained a density, a pressure, and a time, the magnitudes of which are the same under all circumstances, we can derive from them standards of length and mass. For the dimensions of the unit of density [D] are [ML ], and those of the unit of pressure [F] are [ML T ], so that the dimensions of [L] are [P T D * TJ, and those of [M] are ???
This method of defining the three fundamental units is suggested, not as being at all comparable in point of accuracy with the usual methods, but as being an example of a method independent of the preservation of any material standards, whether artificial, as those kept by Government, or natural, as the earth, and its time of revolution.
Rajshahl Kuch-Behar division or commissionership, under the lieutenant-governor of Bengal, is situated between 24 43 o 40" and 26 22 50" N. lat. , and between 88 4 0" and 89 21 5" E. long. The district, which occupies an area of 4126 square miles, is a triangular tract of country with the acute angle towards the north, lying between the dis tricts of Jalpaiguri and Rangpur on the E., and Purniah on the W. ; on the S. it is bounded by the districts of Bogrd, Rajshahi, and Maldah. The country is generally flat, but towards the south becomes undulating, some of the eleva tions being about 100 feet in height. The district is traversed in every direction by a network of channels and water courses. Along the banks of the Kulik river, the undulating ridges and long lines of mango-trees give the
landscape an aspect of beauty which is nut found elsewhere,
