152 SCIENCE PROGRESS 



i.e. when its dimensions are ignored ; in other cases it is on the same category 

 as the other factors concerned and does not require any distinctive name. 



We do not think that, as stated on p. 391, the choice of mass, length, and 

 time as fundamental magnitudes is made because they can be measured 

 more accurately than others ; but because in the historical development of 

 physics it was, more or less explicitly, decided to base the definitions of newly- 

 discovered magnitudes (electric charge, current, magnetic pole, etc.) on their 

 mechanical actions. 



There is a well-known difi&culty in this part of the subject when the funda- 

 mental nature of temperature is inquired into. TaMng the case of an 

 ideal gas as an illustration for which the theoretical equation can be written : 

 pvM = RT where v is the specific volume and M is a characteristic for each 

 gas (in reahty the mass of a molecule), the coefi&cient R is the same for all 

 gases. The product RT has the same dimensions as energy. We may treat 

 R as a no-dimensional constant and T as of the same nature as energy. 



We think that this is the course which Campbell considers necessary 

 " if we want to convey by a statement about the dimensions of temperature 

 the most significant assertion and not one merely about an arbitrary and 

 artificial method of measuring it . . . ." We may express this in rather a 

 different but equivalent way by pointing out that in the gas equation R 

 is introduced only to make the equation conform numerically with the 

 Centigrade or other arbitrary scale, and that the product RT might itself be 

 defined as the temperature mccisured according to the (say) universal scale. 



One of the chief uses to which dimensions are put is " the argument 

 from dimensions." After stating this weU-known argument, he gives a simple 

 illustration and at once falls foul of it. He assumes that the period of a 

 simple pendulum depends only on its length, L, and its acceleration due to 

 gravity, g ; and finds that the period is a a\/L./g, where a is presumably the 

 non-dimensional factor. We think that the way he presents the method 

 in this illustration is very imperfect. Why is the mass of the pendulum 

 bob ignored ? But Campbell's criticism of the usual reasoning, which he 

 regards as false, seems to depend upon whether the acceleration is uniform 

 or not. If b in the formula s == ^bfl can be regarded as an acceleration by 

 a change of the " formal constant " J, why should we not also change the 

 exponent 2 in ^ ? " But if we allow that b is an acceleration when it occurs 

 in a law of the form s = J bi", where n is different from 2, the argument fails." 

 Who does aUow this ? "Again, an acceleration does not mean a constant b 

 occurring in any numerical relation of the form s = i bfl, but only one occur- 

 ring in a numerical law stating, besides a numerical relation, a physical 

 relation " and so on. We are not ashamed to say that we do not see what 

 he is driving at. So far as we can attach significance to it, it appears to mean 

 that we must have defined g before making use of it ; but he may mean more 

 than this. We may point out that there is no obvious reason why g should 

 be used at all. In choosing the magnitudes upon which the time-period 

 may possibly depend, its length L, mass m, and weight W, all of which have 

 definite values during the vibration, would seem to be the proper choice ; 

 not forgetting the amphtude also in a complete appUcation of the method. 

 Putting the last aside for the moment, we obtain the equation T — aVLm/W. 

 But as Newton showed, the time is independent of the mass of the bob, and 

 therefore W/m must be independent of the mass and is denoted by g. Of 

 course this last part of the argument may be made first and g employed 

 thenceforward instead of W/m. Thus T is determined except for a no- 

 dimensional factor a. When the amplitude A is also considered, we have 

 two independent lengths to take into account between which any ratio 

 whatever may exist. But the dimensional equation remains undisturbed 

 if a is taken as any function whatever of this ratio, for it will then remain 

 of no dimensions as required. 



