ERRATA. p. 26, line 9 from top, for "+007" read "- 021." 199, line 7 from bottom, for "199" read "201." 206, last line, for "207" read "209." 222, line 6 from top, for "light" read "homogeneous light." OUR senses and judgment may be trusted up to certain limits, beyond which they begin to be subject to errors. Thus if we wish to measure a length of say 5 centimetres by means of a millimetre scale, no one will feel any difficulty in obtaining a result accurate to a millimetre or to half that quantity. But as soon as we wish to push the accuracy much further, even the most experienced observer will find the estimation difficult, and his measurement may be wrong by a quantity which is called an "error of observation." If he repeats the observations a great many times he will obtain a number of different results, which will group themselves round their average or mean value in a manner which will always shew a certain regularity, if the number of observations is sufficiently large. The study of the law of distribution of errors is of importance because it allows us to form an estimate of the accuracy with which under given conditions the measurements can be made. If there is no bias, which will cause the observation to err more often in one direction than in the other, common sense is sufficient to tell us that the arithmetical mean of a number of observations will give us the most probable result. And common sense will S. P. 1 רי. also allow us to form a rough estimate as to the limits within which the result may be trusted to be right. Suppose for instance a certain observation three times repeated has given the numbers 3·1, 3·3, 34, and in another case the three observations have been 1·1, 1.5, 7·2. In both instances the most probable value, being the arithmetical mean, is the same, viz. 3.27, but in the first case the observer may conclude with some confidence that his result is right to within ten per cent., i.e. the actual value will lie between 3 and 3'5, while in the second case he will attach little value to the mean obtained from such discordant measurements. Common sense like the sense of sight or of hearing may be trusted up to certain limits, and just as we can increase the efficiency of our ordinary senses by suitable instruments, so may we increase the efficiency of this common sense by an instrument which in this case is the theory of probability. To apply that theory we must in the first instance study the laws of grouping of errors, and this is best done by a graphical method. Let the curved line in Fig. 1 have the property that if N represents the number of observations supposed to be very large, the area EFHK will be a measure of the ratio n/N, where n is the number of observations which shew an error greater than OH and smaller than OK. It follows of course that the unit of length chosen is such that the total area included between the curve and the line PQ is unity. It is found that in all cases which it is necessary to consider here, the curve has the same shape and may be represented analytically by the equation where x is the "error," i.e. the deviation of an observation from the arithmetical mean. It is seen that different cases can only differ owing to a difference in the value of h, and Fig. 2 gives Fig. 2. the curves for three different values of h. Inspection of these curves shews that the greater the value of h, the steeper the curve in the neighbourhood of the central ordinate OA, and this means that the observations are grouped more closely round their average value. We might therefore take h to be the measure of the precision of our observations, but it is usual to choose for this purpose another quantity which we proceed to define. In Fig. 1 draw two lines LM and L'M' at equal distances from OA such that the area included between these lines the curve and the horizontal axis is equal to half unit area. By the definition of the curve this means that half the total observations shew errors numerically smaller than OL. The quantity OL is called the "probable" error, meaning that errors larger and smaller than that quantity are equally probable. The probable error (r) may be calculated from the equation to the curves in terms of h and is found to be given by The probable error varies therefore inversely as h, and the smaller the probable error the more confidence may we have in our result. The quantity which interests us most, however, is not the probable error of an observation but the probable error of the result which is obtained by taking the mean of all the observations. Assuming the curve which has been given to represent |