VARIATION OF PARAMETERS. A parameter was a name origi nally given to a particular lino 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 its value to distinguish, or to help to distinguish, one individual of a family of curves Irons another : thug 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 phrase, to outlive its own separate use ; and such has been the ease with the word parameter. As this word is now generally abandoned, element is tho most frequent substitute for it, and it would be desirable to speak of variation of elements.
Whatever phrase wo 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 tho disturbing attractions were all removed. The advantage Is that in this ease the elements will vary very slowly, or A will be long before the disturbing attractions produce much effect. In theory, any curve might be taken. A planet for instance might be supposed to move in a parabola, which varies its dimensions and position in a manner to be determined. In TROCHOIDAL CORVEA, 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. lotions of different magnitudes, the beet 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 is 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 manlier. 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 instetui of each constant, to be determined by the conditions of the question. Two species of cases have frequently arisen.
I. 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, laving 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 eases 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 an 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 ease above noted, be so taken as to provide for the effect of the altered circumstances. But in the second ease they must destroy ono another's effects altogether. We shall take a few instances in which the variation of elements is successful or unsuccessfuL I. TLo equation y' + r being n function of x, is solved by y C being a constant. Now alter tho equation into y' + Py =Q, and to meet the alteration, let c become a function of x. Ou this supposition V+ IV becomes - + + ere/ Ns But this ought to be Q : therefore we must have = Q, or o (gess' ra') + E E being another constant. Hero y' + ay = Q is solved by y + ry = 0 and subsequent variation of an element.
Now try y' + and y' + y==la in the same manner. The first Li solved by and if c be made variable, and y thus altered be introduced into the second, it is found, making =x + c, to require the solution of as difficult an equation as the original. In this case then we are unsuccessful.
du du 1 2. Let - dx dy + - =x. One solution of this is u= 2 a (x-y) + a and b being constants. To find a more general solution of this same equation let b be a function of a, a being a function of x and y. We have then du db)da da - a + (x- des (Ty g • da /Ty and the equation will obviously still be satisfied if b and a be so related that d x-y + b (I-4=0 db Now as b is what function of a we please, so also is xi: hence it follows that if b =Oa , and x-y= -tiVa, we may make a what function of x-y we please. Let a 4.(x - y) and let xv=f el/adv. We have then 1 u = + (x + of which the last two terms merely amount to an arbitrary function of x-y, so that the complete solution is 1 u =- + (x - y) .p meaning any function whatever.
This subject has many developments. We have-Introduced it here under the idea that some studeuts of the differential calculus may be led to consider it at an earlier period of their reading than books will give it to them.
It is to ho remarked that this method does not merely search for some solutions of a question : if the number of constants be sufficient, it goes direct to tho most general solution. In our first example tiler° is no function of x but what is capable of being represented by ; in our third there is no funetiou of x and y but what is 1 1 capable of being represented even by + a y) or 5, and also with a relation between a and b. Whatever function of x, or of x and y, will solve these equations, is sure to be found, if the method be successful. This point would need a little more development than we have here space to give.