487 



VARIATION OF THE COMPASS. 



VARIATION OF PARAMETERS. 



thus the height of the barometer (a length) varies as the pressure e 



the atmosphere on a given surface (a weight). But it is just as obviou 



that one magnitude cannot be equal to another, unless the two be o 



the same kind. When therefore a writer oil mechanics, with little o 



no previous explanation about the unite employed, states that tin 



miyht of a body is its man multiplied by the force of gravity, or thai 



the pressure on a mass is equal to the mass multiplied by its acccle 



ration, he writes effectively only for a reader who knows the subjec 1 



already. The weight of a body varies jointly as its mass and the 



acceleration which the force of gravity would create in one second. 



Alter either of these alone, and the weight is altered in the same pro 



portion. Hence, if v>, m, g be the numbers of units of their several 



kinds in the weight, the mass, and the acceleration caused by gravity 



the equation m = c m y must subsist, where c is a numerical constant 



depending on the units employed. If the weight which is called 1C 



(pounds, ounces, or whatever they may be) belong to the mass called 



5, when acted on by such gravity as produces an acceleration of 4 (feet, 



yards, or whatever the unit of length may be) in the time called 1 



(second, minute, or other unit of time), then 10=cx 5 x 4, or c = i. 



So long as the same units of length, time, mass, and weight are 



employed, the equation w= J my must subsist : change the unite, and 



the constant c must have another value, to be again determined from 



an instance. When the writer above mentioned says that v> = m g, he 



means, or ought to mean, that it is an agreement between him and his 



reader that whatever mass may be called 1, and whatever may be 



meant by 1 of length and 1 of time, the weight which is called 1 shall 



be that of the mass 1 acted on by the force of gravity 1. The older 



writers, who used variations, needed no specifications of this kind, 



since the actual concretes themselves were the subjects of reasoning, 



and the variation asserted was true both of the concrete magnitudes 



and of any system of unite which they might adopt. The introduction 



of their units was naturally and easily made ; and when variations 



became equations, the student could not help seeing the introduction 



of all conditions depending on the mode of measurement. In dropping 



the notation of variations, our writers passed into that want of distinct 



explanations of primary terms which was the characteristic of many 



f the French writers. 



The beginner must carefully bear in mind that one quantity does 

 not vary at another, because it varies in'fA that other. A square and 

 its root vary together, but the square does not vary as its root : if, for 

 instance, the root be doubled, the square is not doubled, but quad- 

 rupled. 



It is however most important to remember that when two quantitic * 

 change together, in any manner whatsoever, the increment of the one 

 varies as the increment of the other very nearly, if both the increments 

 be small, and the more nearly the smaller they are. Thus, if we know 

 that when x has a certain value, the addition of '01 lo x gives an addi- 

 tion of '001 to ite logarithm, we may be sure that the addition of 

 '01 x A to a; will give an addition of '001 x A to the logarithm, very nearly, 

 as long aa '01 x A is small. 



VARIATION OF THE COMPASS. [COMPASS COIIRECTIO.V ; 

 DECLINATION ; MAGNET ; TERRESTRIAL MAGNETISM.] 

 VARIATION OF THE MOON. [MooN.l 



VARIATION OF PARAMETERS. A parameter was a name origi- 

 nally given to a particular line connected with a conic section : being 

 the third proportional to a diameter and its conjugate. In time the 

 word was applied to any line which serves by ite value to distinguish, 

 or to help to distinguish, one individual of a family of curves from 

 another ; thus the radius of a circle, the axes of an ellipse, the co-or- 

 dinates of the centre of either, were called parameters. When a word 

 gets into the descriptive name of a method, it may happen, as part of 

 a jilir.-we, to outlive ite own separate use ; and such has been the case 

 with the word parameter. As this word is now generally abandoned, 

 element is the most frequent substitute for it, and it would be desirable 

 to speak of variation of dementi. 



Whatever phrase we may use, the thing occurs both in physics and 

 mathematics, in modes which are closely connected with each other. 

 A planet moves in a curve which is not an ellipse, but which would 

 change and become an ellipse if the disturbing attractions of the other 

 planets were removed, and that of the sun only continued. The easiest 

 way of calculating the planetary motions is to consider the planet as 

 moving in this ellipse, while during the motion the elements which 

 determine the ellipse are perpetually changing ; so that the form and 

 position of the ellipse both vary. This is done in such manner that 

 the ellipse of each moment is that which the planet would go on to 

 move in, if at that moment the disturbing attractions were all removed. 

 The advantage is that in this case the elements will vary very slowly, 

 <>r it will be long before the disturbing attractions produce much 

 Affect. In theory, any curve might be taken. A planet for instance 

 ' be supposed to move in a parabola, which varies ite dimensions 

 :,ii.l position in a manner to be determined. In TROCHOIDAL CURVES, 

 all the curves given are produced by a point moving in a circle with 

 variable elements; that is, of variable centre, though given radius. -If 

 it were required to investigate trochoidal curves with loops and undu- 

 lations of different magnitudes, the best way would be to consider 

 them as made in the same manner, with a circle of variable radius also : 

 or else to make both circles variable, 



In the differential calculus the variation of elements ia introduced 



__ _ 



thus : If an algebraical expression containing some variables and some 

 constant elements be proper to answer a certain purpose, it is not, impos- 

 sible that it may answer the same purpose when the constants are 

 made variable, provided they be made to vary in a proper manner. 

 Now, if the purpose which is to be answered involve differentiation, 

 the infinity of the number of suppositions which may be made as to 

 the variation of the (former) constants is equivalent to introducing an 

 arbitrary function instead of each constant, to be determined by 

 the conditions of the question. Two species of cases have frequently 

 arisen. 



1. When under certain circumstances a problem is solved by an 

 expression containing certain constants, and the circumstances are then 

 altered ; it is often convenient to inquire whether the altered problem 

 might not be solved by the same expression, on the supposition that 

 the constants become variable. And the question then is, how the 

 (former) constants are to be made to vary. 



'2. Without any alteration of the circumstances, having a solution 

 which contains constants, it may be asked how to substitute variables 

 in place of constants, so that the altered expression may still be a 

 solution. 



In both cases it is obvious that so soon as the constants are made 

 variable the differential co-efficients of all expressions into which they 

 enter will receive au accession of terms above what they had before. 

 These new terms, which we may describe as functions of the variations 

 of the elements, must, in the first case above noted, be so taken as to 

 provide for the effect of the altered circumstances. Cut in the second 

 case they must destroy one another's effects altogether. We shall 

 take a few instances in which the variation of elements is successful or 

 unsuccessful. 



1. Tie equation y + py=0, p being a function of x, is solved by 



y = e-/ 



c being a constant. Now alter the equation into y" + vy = Q, and to meet 

 the alteration, let c become a function of x. Ou this supposition 

 l/ + r</ becomes 



But this ought to be q : therefore we must have 



E being another constant. Here y 4- vy = Q is solved by y + TO = and 

 subsequent variation of an element. 



Now try y + yJ = and y' -t-y'-' = Q in the same manner. The first u 

 solved by y = (.c + c)- ' and if o be made variable, and y thus altered bo 

 introduced into the second, it is found, making ; = x + o, to require the 

 solution of 



as difficult an equation as the original. In this case then wo are 



unsuccessful. 



da, du 1 



ite + d~ii ~ x ' One solution of this is u= -z o? + a(x y) +b, 



a and ft being constants. To find a more general solution of this same 



equation let 6 be a function of a, a being a function of x and y. Wo 



lave then 



du / db\da 



dl>\da 



.7 x f ~"g 



and the equation will obviously still be satisfied if i and a be so 

 related that 



_ ^_ 



f ow as 6 is what function of a we please, so also is y : hence it 



ollows that if & = $xj,and xy=<t>'a, we may make a what function 

 >f x y we please. Let a=<Kx-y) and let xv=fc<^vdv. We have 

 hen 



1 



.f wliich the last two terms merely amount to an arbitrary function of 

 ty, so that the complete solution is 



meaning any function whatever. 



This subject has many developments. We have introduced it hero 

 mder the idea that some students of the differential calculus may bo 

 ed to consider it at an earlier period of their reading than books will 

 five it to them. 



It is to be remarked that this method does not merely search for 

 omc solutions of a question : if the number of constants be sufficient, 

 t goes direct to tho most general solution. In our first example there 

 no function of x but wliat is capable of being represented by 



