But or F= = [M] [L] [T]-2. •. [μ]2 = [M] [L]3 [T]-, [μ] = [M]} [L]} [T] ̃1. When the dimensional equations for the different units have been obtained, the calculation of the factor for conversion is a very simple matter, following the law given on p. 26. We may recapitulate the law here. To find the Factor by which to multiply the Numerical Measure of a Quantity to convert it from the old System of Units to the new, substitute for [L] [M] and [T] in the Dimensional Equation the old Units of Length, Mass, and Time respectively, expressed in terms of the new. We may shew this by an example. To find the Factor for converting the Strength of a Magnetic Pole from C.G.S. to Foot-grain-second Units— I C.G.S. unit of magnetic pole -1 = 1 × [M] [L] [T]-1 = 1 × [gm.] [cm.] [sec.]-1 = 1 × [154 gr.] [0.0328 ft.] [sec.]-1 = 1 × (15 ̊4)1 (0·0328) [gr.] [ft.) [sec.]-1 0233 foot-grain-second unit. That is, a pole whose strength is 5 in C.G.S. units has a strength of 1165 foot-grain-second units. Conversion of Quantities expressed in Arbitrary Units. We have shewn above how to change from one system of units to another when both systems are absolute and based on the same laws. If a quantity is expressed in arbitrary units, it must first be expressed in a unit belonging to some absolute system, and then the conversion factor can be calculated as above. For example: To express 15 Foot-pounds in Ergs. The foot-pound is not an absolute unit. We must first obtain the amount of work expressed in absolute units. Now, since g= 32'2 in British absolute units, I foot-pound 322 foot-poundals (British absolute units). .. 15 foot-pounds =15×322 foot poundals. We can now convert from foot-poundals to ergs. . The dimensional equation is 15 foot-pounds = 15×32*2 × 454 × (305)2 ergs. Sometimes neither of the units belongs strictly to an absolute system, although a change of the fundamental units alters the unit in question. For example: To find the Mechanical Equivalent of Heat in C.G.S. Centigrade Units, knowing that its Value for a Pound Fahrenheit Unit of Heat is 772 Foot-pounds. The mechanical equivalent of heat is the amount of work equivalent to one unit of heat. For the C.G.S. Centigrade unit of heat, it is, therefore, or the mechanical equivalent of one C.G.S. Centigrade unit of heat 4'14 X 10' ergs. If the agreement between scientific men as to the selection of fundamental units had been universal, a great deal of arithmetical calculation which is now necessary would have been avoided. There is some hope that in future one uniform system may be adopted, but even then it will be necessary for the student to be familiar with the methods of changing from one system to another in order to be able to avail himself of the results already published. To form a basis of calculation, tables showing the equivalents of the different fundamental units for the measurement of the same quantity are necessary. Want of space prevents our giving them here; we refer instead to Nos. 9-12 of the tables by Mr. S. Lupton (Macmillan & Co.). We take this opportunity of mentioning that we shall refer to the same work whenever we have occasion to notice the necessity for a table of constants for use in the experiments described. CHAPTER III. PHYSICAL ARITHMETIC Approximate Measurements. ONE of the first lessons which is learned by an experimenter making measurements on scientific methods is that the number obtained as a result is not a perfectly exact expression of the quantity measured, but represents it only within 1 Numerical Tables and Constants in Elementary Science, by S. Lupton certain limits of error. be given as fifteen miles, we do not understand that the distance has been measured and found to be exactly fifteen miles, without any yards, feet, inches, or fractions of an inch, but that the distance is nearer to fifteen miles than it is to sixteen or fourteen. If we wished to state the distance more accurately we should have to begin by defining two points, one in each town-marks, for instance, on the doorsteps of the respective parish churches-between which the distance had been taken, and we should also have to specify the route taken, and so on. To determine the distance with the greatest possible accuracy would be to go through the laborious process of measuring a base line, a rough idea of which is given in § 5. We might then, perhaps, obtain the distance to the nearest inch and still be uncertain whether there should not be a fraction of an inch more or less, and if so, what fraction it should be. If the number is expressed in the decimal notation, the increase in the accuracy of measurement is shewn by filling up more decimal places. Thus, if we set down the mechanical equivalent of heat at 4'2 x 107 ergs, it is not because the figures in the decimal places beyond the 2 are all zero, but because we do not know what their values really are, or it may be, for the purpose for which we are using the value, it is immaterial what they are. It is known, as a matter of fact, that a more accurate value is 4'214 × 107, but at present no one has been able to determine what figure should be put in the decimal place after the second 4. If the distance between two towns Errors and Corrections. The determination of an additional figure in a number representing the magnitude of a physical quantity generally involves a very great increase in the care and labour which must be bestowed on the determination. To obtain some idea of the reason for this, let us take, as an example, the case of determining the mass of a body of about 100 mass. grammes. By an ordinary commercial balance the mass of a body can be easily and rapidly determined to 1 gramme, say 103 grammes. With a better arranged balance we may shew that 103.25 is a more accurate representation of the We may then use a very sensitive chemical balance which shews a difference of mass of o'r mgm., but which requires a good deal of time and care in its use, and get a value 103 2537 grammes as the mass. But, if now we make another similar determination with another balance, or even with the same balance, at a different time, we may find the result is not the same, but, say, 103 2546 grammes. We have thus, by the sensitive balance, carried the measurement two decimal places further, but have got from two observations two different results, and have, therefore, to decide whether either of these represents the mass of the body, and, if so, which. Experience has shewn that some, at any rate, of the difference may be due to the balance not being in adjustment, and another part to the fact that the body is weighed in air and not in vacuo. The observed weighings may contain errors due to these causes. The effects of these causes on the weighings can be calculated when the ratio of the lengths of the arms and other facts about the balance have been determined, and when the state of the air as to pressure, temperature, and moisture is known (see §§ 13, 14). We may thus, by a series of auxiliary observations, determine a correction to the observed weighing corresponding to each known possible error. When the observations are thus corrected they will probably be very much closer. Suppose them to be 103.2543 and 103 2542. Mean of Observations. When all precautions have been taken, and all known errors corrected, there may still be some difference between different observations which can only arise from causes beyond the knowledge and control of the observer. We |