282 



NATURE 



{July 19, 1888 



The first part of the multiplication is the representation of a 

 complex quantity which is proportional to the several factors in 

 the numerator, and inversely proportional to those in the deno- 

 minator ; the second part is the comparison between the particu- 

 lar complex quantity and a standard of the same kind. The 

 representation may be temporary, i.e. adopted for the solution 

 of a particular problem ; or it may be permanent, i.e. adopted 

 throughout a whole subject. 



Thus, if a, b are two lengths, the product ab is always used to 

 represent a rectangle whose sides are a, b respectively ; though 

 we might have agreed to use it as a representation of a parallelo- 

 gram with sides a, b containing an angle of (say) 6o° ; and of 

 course we might find a number of things which in some par- 

 ticular problem might be represented by ab, but all such quan- 

 tities must agree in this property, viz. that in the problem in 

 question they shall vary jointly as a and b. 



Our right to cancel among concretes may be established once 

 f jr all in some such way as the following : — 



Let a = aa', b = 0b', and therefore ab = a$ . a'b', as before. 

 Now, if we proceed to deduce a formally from the equation 



ab = a& . a'b', we shall get a = a & \ a b , which reduces down to 



b 

 its known value aa' if we allow b in the denominator to cancel 

 against its equivalent $b' in the numerator. (This cancelling is 

 really an application of the law of association to the quotients.) 



By such methods as this we can establish once for all our right 

 to apply the formal laws of multiplication and division to con- 

 crete products and quotients, when such concrete products and 

 quotients represent quantities varying directly as the concrete 

 numerator and inversely as the concrete denominator ; though, 

 indeed, for that matter a very little practice in the use of such 

 concrete representations renders one's perception of that right 

 almost intuitive. In fact, in all cases a student would very soon 

 perceive that the standards involved in the various equations 

 might be treated exactly like numbers, and he would also learn 



from the resulting expressions (e.g. ^— , , foot . &c.Vo appre- 



V sec. (sec.) 2 / 



ciate the meaning of the dimensions of quantities with a 

 thoroughness unattainable in any other way. 



All questions dealing with mixed standards, or change of 

 standards, present no difficulty when this method is adopted. 



Here is a good example of the concrete method. Two ton- 

 masses p'aced a yard apart attract each other with a force equal 

 to the weight of one-eighth of a grain. Calculate the mass of 

 the earth in tons. 



Solu 'ion. 



earth x | grain _ I ton x 1 ton 

 (I yard)'- 



(4000 miles)- 



mass of earth = **?£_ x f4°°° miIes Y tc 

 I grain \ ! yard 

 = &c. 



It is most important that the student should be taught to 

 notice that physical equations can only be among quantities of 

 the same kind, or that, if there are quantities of different kinds 

 in the equation, then the equation is really made up of two or 

 more independent equations which must be separately satisfied, 

 each of these being only among auantities of the same kind. 

 So we may consider generally that, in any equation, all the terms 

 must represent quantities of the same kind. 



But I want to call attention to the fact that merely the dimen- 

 sions of a quantity do not always fix the kind of quan'ity. For 

 example, the moment of a force is of the dimensions of work, 

 and yet it is not work, and cannot exist as a term in an equation 

 involving work terms. Again, the circular measure of an angle 

 is not a pure number, though it is of zero dimensions as a pure 

 number is ; and that it is not a pure number is evident physically, 

 for a moment of a force x an angle = work. 



Now these are special cases of certain general laws as to 

 direction which hold among the terms of an equation involving 

 directed quantities, but in ivkich the symbols themselves do not 

 include the idea of direction (for I wish to confine myself strictly 

 to ordinary algebraical equations). 



The laws are : firstly, if any term is independent of direction, 

 every term must be also independent of direction, or involve 

 ratios between parallel vectors, and so by cancelling direction 

 become independent of it. 



E.g. if a body is projected with velocity V at angle o with 

 the horizon, it reaches its greatest height in the time Y sin a . 



g 



Here both numerator and denominator are vertical vectors, 

 and therefore the directions cancel as they ought. 



Secondly, if any term involve only one & vector, the other 

 terms must also, after such simplification of directions as possible, 

 involve the same vector only. 



E.g. Horizontal range of projectile = 2Y " sin a c g!_g , where 



V sin a and g are vertical vectors, and V cos o is horizontal, so 

 that the whole expression is a horizontal vector, as it should be. 



Again, if any term involve a product (or ratio) between two 

 vectors including any angle, every term must, after such can- 

 celling and simplification of directions as possible, also involve 

 a product (or ratio) between two vectors including the same 

 angle. 



The most frequent cases are those whe-e a term consists of a 

 product of parallel, or mutually perpendicular directed quanti- 

 ties, in which case every term must do the same. 



It is not easy to see what law holds in cases where a greater 

 number of directed quantities occur in each term, except in the 

 simple case where one term consists of a product of a number of 

 parallel vectors, in which case every term must do the same. 

 _ The general law is, I believe, that if any term consists in its 

 simplest form of a product or quotient of certain vectors, which 

 will form a kind of solid angle, then every term must also 

 involve an exactly similar solid angle of vectors. However, I 

 have not followed this out, as it dors not seem likely to be a 

 useful test in its general form. 



The following are simple examples of some of the above laws : 



b = a cos C + c cos A ) . . . , 

 a"- = //-' + c" - 2bc cos A } ,n a tnan S le I 



y = mx + c ; 

 sin (A + B) — sin A cos B + cos A sin B. 



This last example should be considered in connection with the 

 ordinary geometrical proof, where it will be seen that each term 

 on the right is a ratio between lines inclined to each other at the 

 angle 90 - (A + B), just as the left-hand side is. 



An angle is the ratio between the arc and radius of a circle, 

 and if it multiplies a radius, changes it into an arc. Thus, if by 

 applying a force P at the end of an arm a, a body is turned 

 through a small angle 0, the work done is Tad ; i.e. the product 

 of P into the arc through which it has been acting, which is a 

 product of parallel vectors, as it must be besides having to be of 

 right dimensions if it is to represent work. This expression is 

 also the product of the moment of the force into the small angle 

 turned through, so that, if we wish to connect the moment of a 

 force with work, we must say : — 



The moment = the worker radian which can be done, 



or simply, moment = W0 - rk done . 



angle turned through 



Now I do not wish to insist that in dealing practically with 

 mechanical problems it is necessary always to include the 

 standards as well as the numerical multipliers in the equations, 

 for it. would be an intolerable nuisance to have to do so. In com- 

 plicated cases, however, I think the student should test the dimen- 

 sions of each term in his equation, so as to avoid gross mistakes. 

 But it is in trying to understand the fundamental equations in 

 any subject that it appears to me important to express particular 

 examples of them as fully as possible. 



For practical purposes any numerical equations we may 

 desire may be deduced from the fundamental equations. 



For example, the connection between the height (h) of an 

 observer above the sea with the distance (d) of his horizon, is 

 d' 2 = 2RI1, where R is the radius of the earth ; and we can 

 deduce from this the numerical relation between the height in 

 feet, and the distance of vision in miles. For if/ be the number 

 of feet in h, and m the number of miles in d, so that /z = /Teet, 

 and d s= m miles, the equation becomes 



• "• / = » 



(in miles)- = 2R x/feet, 



= 8000 miles x /"feet ; 

 (miles) 2 = 5280 m „ 



8000 miles x 1 foot 8000 



= I i/r approximately ; 



i.e. the observer's height in feet = § of the square of the distance 

 of his view in miles. 



This is a strictly numerical equation, deduced for practical 

 purposes from the concrete equation d' 1 = 2R//. 



