results of a number of observations be added together and the sum be divided by the number of the observations, the average or arithmetical mean obtained is probably more near the correct result than any one result taken hap-hazárd. And it is possible to determine by calculation how far this mean probably is from the true result.

Further, if, when the condition of a substance is altered in one respect, it is found to alter simultaneously in another, it is often required given certain changes in the first condition to find the most probable values of changes in the second condition which have been determined by experiment, and also to calculate values for the second condition which have not been determined experimentally.

There are two methods of effecting this, the one geometrical (the Graphical Method), the other analytical (the Method of Interpolation).

(36) THE METHOD OF LEAST SQUARES.

In any series of equally precise observations upon a single quantity, the most probable value is the arithmetical mean obtained by adding all the results together and dividing the sum by the number of observations.

To ascertain the trustworthiness of this mean result:(i) Find the residuals, these are the excess or defect of each result from the mean.

(ii) Square the residuals, and add the squares together. (iii) Divide by the number of observations multiplied by one less than the number of observations, and take the square root of the quotient. This gives the mean error of the mean result.

(iv) Multiply the mean error of the mean result by 0.6745, and the probable error of the mean result is obtained.

The probable error of the mean result is a test of its trustworthiness, since it is a probability of 1⁄2, or the odds are even, that the true result lies between the mean result + or the probable error.

Thus 100 grams of tin, when oxidized by hydrogen nitrate, give according to

Berzelius 127.2 grams of stannic oxide.

Mulder 127.56 99

Dumas 127.11 99

Hence, assuming that the result obtained by each chemist is of equal value, 100 grams of tin most probably give

381.87

3

= 127·29 grams of stannic oxide.

=

The probable error of this mean result, n the number of experiments being 3, is:

This result is written 127.2909, which means that it

is an equal chance that the true value lies between 127.2

The precision of the mean increases as the square root of the number of observations.

If the values of the results are not all equal, each result must be multiplied by a number expressing its worth relatively to the others called its "weight." Or the calculation is effected as though each experiment had been repeated a number of times corresponding to its assumed accuracy.

The mean of the measured quantity is found by multiplying each observation by its weight, and dividing the sum of the products by the sum of the weights.

Thus the quantity of hydrogen sulphate in 1 c.c. of a solution was determined by three different methods :

(a) By barium chloride as 0847 gram, weight 3.
(B) By sodium
(y) By sodium carbonate as '084

as 085

gram, weight 2.

gram, weight 1.

And the result is 08470002, or it is an equal chance

that the true result lies between 0845 and 0849.

(37) THE GRAPHICAL METHOD.

The graphical method enables an experimenter to compare a number of results in which two quantities vary simultaneously, to find the most probable value of each result, and to trace out a general law which serves to determine intermediate values of the results.

A sheet of copper or paper is ruled into a large number of equal squares, and commencing with the left hand bottom corner which is called the origin, the lines are numbered consecutively according to the two quantities which vary simultaneously.

Along the bottom line which is spoken of as the x-axis, the variable is measured, and portions cut off from it are called abscissæ. The variant is measured along the vertical line or y-axis, lengths parallel to which are called ordinates.

х

To express a result the value of x is measured parallel to or along the x-axis, and the corresponding value of y is measured parallel to or along the y-axis. Lines are drawn through these points parallel to the y-axis and to the x-axis respectively, and their point of intersection is the required point.

Thus if it be required to express a value of x= 11.6 and y=104 or as it is usually written to find the point (11.6, 10-4), the point must evidently lie in the small square contained by the lines drawn through the abscissæ 11 and 12, and through the ordinates of length 10 and 11. By means of a dividing machine or pair of compasses and ruler, ad is cut off equal to of ac, and ae equal to of ab.

Then ex and dx are drawn parallel to ac and ab respectively, the point of intersection (x, y) is the required point (11·6, 10.4).

In exactly the same way all the other experimental results (x,, y1) (x,, Y2) (x ̧, Y ̧) &c. are expressed on the paper, and a permanent record of all the experiments is thus obtained.

it

But further, if the results are sufficiently close together may in most cases be assumed that no sudden change of value occurs between them, and therefore that the true values would lie on a continuous line, passing among the points expressing the experimental results. This line in most cases is either a straight line or a continuous curve, passes through some of the points which have been experimentally determined, and leaves about an equal number of these points above and below it.

Such a continuous line then drawn by the free hand shows the most probable value of the experimental results, since its position at any point is determined not by one but by a considerable number of experiments; it also enables the value of the result for any intermediate value of the variable, which has not been determined by experiment, to be obtained by direct measurement, for it in fact expresses the probable