All gases diverge again from the law of uniform expansion by heat, but the divergence is less as the gas in question is less condensable, or examined at a temperature more removed from its liquefying point. Thus the perfect gas must have an infinitely high temperature. According to Dalton’s law each gas in a mixture retains its own properties unaffected by the presence of any other gas.[388] This law is probably true only by approximation, but it is obvious that it would be true of the perfect gas with infinitely distant particles.[389]
Mathematical Principles of Approximation.
The approximate character of physical science will be rendered more plain if we consider it from a mathematical point of view. Throughout quantitative investigations we deal with the relation of one quantity to other quantities, of which it is a function; but the subject is sufficiently complicated if we view one quantity as a function of one other. Now, as a general rule, a function can be developed or expressed as the sum of quantities, the values of which depend upon the successive powers of the variable quantity. If y be a function of x then we may say that
y = A + Bx + Cx2 + Dx3 + Ex4 . . .
In this equation, A, B, C, D, &c., are fixed quantities, of different values in different cases. The terms may be infinite in number or after a time may cease to have any value. Any of the coefficients A, B, C, &c., may be zero or negative; but whatever they be they are fixed. The quantity x on the other hand may be made what we like, being variable. Suppose, in the first place, that x and y are both lengths. Let us assume that 1/10,000 part of an inch is the least that we can take note of. Then when x is one hundredth of an inch, we have x2 = 1/10,000, and if C be less than unity, the term Cx2 will be inappreciable, being less than we can measure. Unless any of the quantities D, E, &c., should happen to be very great, it is evident that all the succeeding terms will also be inappreciable, because the powers of x become rapidly smaller in geometrical ratio. Thus when x is made small enough the quantity y seems to obey the equation
y = A + Bx.
If x should be still less, if it should become as small, for instance, as 1/1,000,000 of an inch, and B should not be very great, then y would appear to be the fixed quantity A, and would not seem to vary with x at all. On the other hand, were x to grow greater, say equal to 1/10 inch, and C not be very small, the term Cx2 would become appreciable, and the law would now be more complicated.
We can invert the mode of viewing this question, and suppose that while the quantity y undergoes variations depending on many powers of x, our power of detecting the changes of value is more or less acute. While our powers of observation remain very rude we may be unable to detect any change in the quantity at all, that is to say, Bx may always be too small to come within our notice, just as in former days the fixed stars were so called because they remained at apparently fixed distances from each other. With the use of telescopes and micrometers we become able to detect the existence of some motion, so that the distance of one star from another may be expressed by A + Bx, the term including x2 being still inappreciable. Under these circumstances the star will seem to move uniformly, or in simple proportion to the time x. With much improved means of measurement it will probably be found that this uniformity of motion is only apparent, and that there exists some acceleration or retardation. More careful investigation will show the law to be more and more complicated than was previously supposed.
There is yet another way of explaining the apparent results of a complicated law. If we take any curve and regard a portion of it free from any kind of discontinuity, we may represent the character of such portion by an equation of the form
y = A + Bx + Cx2 + Dx3 + . . .