TRANSFORMATION OF CO-ORDINATES. We intend this article purely for reference ; that is, supposing the subject already known, we mean only to put together the formula) in such a manner that any one can be used at once.
Rectilinear co-ordinates are the only ones which are usually trans. formed ; such a thing rarely, if ever, happens with polar co-ordinates, except in investigations each of which has its peculiar method. And, first, we shall consider rectilinear co-ordinates in one plane, and after wards in space. What is usually wanted is to express the co-ordinates. of a first system in terms of those of a second, and subsequently given, system.
And, first, as to co-ordinates in one given plane.
1. Both systems oblique. Let x and y be the old co-ordinates of a point, x' and y' the new ones. Let a and v be the old co-ordinates of the new origin ; 0 the angle made by the old co-ordinates ; 1) the angle made by the axis of x' with x; tp the angle made by y' with x. Angles are to be measured as explained in the article SIGN ; thus the angle made by a' with x means the amount of revolution which would bring the positive part of x into the direction of the positive part of x', the revolution being made in the positive direction.
In any of the preceding cases, if the new and old origin coincide, we have only to make A = 0, r = 0, and use the formula) accordingly.
Next, when the co-ordinates are those of points of space. The only two cases which are particularly useful are when both systems are rec tangular, and when the new one only is oblique. Let x, y, z be the old co-ordinates, and y„ ; the new ones. Let A, A, v be the old co ordinates of the new origin, and let the angle made by x, and y, be C, that of y, and z, be t, and ;hat of z, and x, be 77, which we may thus denote :— This brings us to the mention of a defect of reasoning which has fre quently vitiated mathematical works, namely, the assumption of the species of a transformation, and the supposition that only the character of the details remains to be settled, or the individual of the species to be picked out. In the preceding case, for example, it is often stated as follows : " Required the expansion of a. in a series of powers of x."
The form of the series is then assumed, say p + , and by the use of the property above alluded to, it is found that the series must be of the form I + ax + lAke+ .... But all that is hero proved is, that if a. be capable of expansion in integer powers of x, the expau.
Mon must be of the form 1 + . . It is true that, looking at what we see in algebra, that science might be strongly suspected to have a peculiar power of rejecting false suppositions, or of indicating their falsehood by refusing to furnish rational results : thus it certainly does generally happen that when we attempt to select from among series of integer powers the one belonging to an expression which really has no such series, we find infinite coefficients, or some other warning. But it is too much to ask of a beginner that he should take it for granted that algebra has so peculiar a property ; nor, in fact, is it true that such a property is quite universal. It is necessary, therefore, to watch all transformations narrowly, both in their general as well as their specific form : first, because there can be no sound reasoning without such caution ; next, because, though it be true that This cane is not much required. The following, in which both sys tems are rectangular, is of the highest importance. When we speak of the angle made by two axes, we mean, as Lefore, the angle made by the positive side of one with that of the other ; but, since only cosines are used, the direction of revolution is immaterial. If both systems be rectangular, and if they have the same origin, we have two seta of equations, each of which follows from the other, one set being in each column ; the meanings of a, a', &c being as before, For the mode in which these nine quantities are made to depend upon three, we must refer to works on mechanics, in which such reduc tion is particularly useful We avoid giving it here, because trifling differences exist in the manner of taking the quantities to functions of which all the rest are to be reduced, so that no set of equations can be given which can be called universal. So far as we have gone, the expressions of all writers are the same, though the letters used are not always alike.