position of all the loci, which would be obtained were the experiments made infinite in number by taking infinitesimal changes of the variable.

Thus OA expresses the increase in volume of 273 volumes of gas at 0° C. when gradually heated to 16° C.

OB shows the number of grams of nitre dissolved by 100 grams of water from 0° C. to 106° C. (20 grams of nitre are expressed by 1 of ordinate, and 10° C. by 1 of abscissa).

And OC shows the expansion of 100 volumes of mercury at 0° C. when heated to 160° C. (1 of ordinate expresses 1 volume of mercury, and 1 of abscissa expresses 10o C.).

(38) INTERPOLATION.

When two values vary simultaneously it frequently happens that, having for a number of values of one variable the corresponding values of the other variable, it is required to determine the value of the second variable corresponding to some intermediate value of the first variable. In simple cases this is effected with sufficient accuracy by assuming that the change in value of one variable is proportional to the change in value of the other variable.

Thus, if 4100 volumes of gas at 100° C. become 5200 volumes at 200° C. to determine the volume of the gas at 150° C.

total change in temp.: partial change in temp.
:: total change in vol. : partial change in vol.

Hence the volume at 150° C. will be 4100 + 550 = 4650

volumes.

But in complicated cases, especially when the intervals are large, the method by simple proportion becomes so inaccurate that it has to be replaced by another.

The most simple form of the problem is-given that a series of consecutive equidistant values of one variable Xo X1 &c. make the second variable have the values u。, u1, u2, uz, &c. to find the value of u, when x lies between

х

and

Write the values of the first variable in a column, and near them the corresponding values of the second variable. Subtract each value of the second variable from the succeeding one, and write these differences also in a column. Subtract each difference from the succeeding one to obtain a column of second differences, and in like manner if necessary obtain columns of third and fourth differences.

These differences are signified by Au, A3u, ▲3u, A*u, &c.
The value of u can then be found from the equation:—

For the greatest accuracy the differences must be taken until they become constant for all values of u, that is, until each number in the column is identical.

Thus, to determine the tension of sulphur vapour at 440° C. from the tensions at the following temperatures:

The difference (x) between 400° C. and

the difference between 400° C. and 450° C.

mula becomes;

440° C. is of Hence the for

u = 329 + 1 × 451 + 1 (−1) 404 + 1 (− ) ( − 2) 193.

=

10

329+360.8- 32·32 ÷ 6·176.

u ̧ = 663·65 m.m,

The tension at 440° C. found by experiment is 663.11 m.m. Had the ordinary mean only been taken, the result would have been found to be 329 + 360·8 689.8 m.m.

=

Again, to find the tension of aqueous vapour at 11° C. having given:

The difference (x) for 1° C. is of the difference for 5o C.,

hence,

u = 9·165 + 1 × 3.524 + } (− 2) × 1·158 + ( − ) ( − 3) × 309. =9·165+705·093+ ·015.

u = 9.792 m.m.

A simple interpolation formula is of great use in reducing observations with the spectroscope from angular or scalereadings to wave-lengths.

If lines in a spectrum, which are tolerably close together, are found to have the scale-readings n1, n, n ̧, and the wavelengths λ, λ, λ, these values are connected by the equation,'

ין

from which, if n1, n,, n ̧ are determined by observation and the wave-lengths of two of the lines are known, the wavelength of the third line can be calculated.

Thus in the case of the three brightest lines of magnesium,

The problem becomes much more complicated, when it is required to find a formula to express the results of all experiments, which have been or might be made on any given change.

No general rules including all cases can be given. If nothing suggests any one form of equation as more probable than another, it is usually most convenient to assume an equation of the form

y= a + bx + cx2 + dx3 + &c.,

and to determine the constants a, b, c and d from the experimental results, since equations of this form are found to include most of the more commonly occurring cases in Chemistry and Physics.

To determine the constants widely separated values x,, x, x, of x are taken and y1, y,, y, are determined in each case; another experiment in which if possible x is made equal to 0 serves to determine a.

Now writing y1-α=α1, y2-α = а2, Y2-α = α ̧·