APPENDIX B.

Let the case be supposed of a school, in which the number of scholars remains constantly the same, one always coming to it when another leaves; let, moreover, each child be supposed to remain in the school the same time. Take a line a p, and divide it into as many equal parts as there are children, draw ba and p c at right angles to it, and take ab to represent the time during which each child remains in the school, and make a c equal to it; complete the parallelogram a bp q, and join q c, and through the points of division of a p draw lines parallel to cab; then if a represent the time when the scholar who has been in the school a period, represented by b a, is supposed to be leaving it, ek will represent that during which the next to this senior scholar has been in the school, and i k the time at school of the third in order, and so of the lines which fill the triangle a bp, whose sum will therefore represent the sum of the times in the school of the scholars at present attending it.

x y z C

Also the lines which compose the parallelogram a bpq, represent the sum of the times that the scholars at present in the school will have been in when they leave it. But the triangle is half of the parallelogram; therefore the sum of the times at school of the children at present attending it is equal to one-half the sum of the times they will have been at school when they leave it; and, therefore, the average time at the school of each scholar now attending it, is equal to half the average time during which each will have attended, when he leaves it.

b

If the children do not, according to the supposition we have made, remain all the same time in the school, a b is to be taken to represent the mean time of their continuance. The actual time of the continuance of any child will then be represented, not by a line intercepted between the two straight lines a q and bp, but between two curved lines, such as those shown by the dotted lines in the figure, which nevertheless are such that the area included between them, which area represents the sum of the times in the school, is equal to the area of the parallelogram a bp q, and nearly equal to twice the area included between the curved line extending

from b to p, and the straight line ap; which last area represents the sum of the times in the school of the children who at present attend.

Although the line ap does not necessarily divide the curvilinear area equally, as it does the parallelogram, so that this last conclusion is only approximately true, yet is the error probably such as will not materially affect the average, if the school be supposed to have reached, in respect to its numbers and attendance, a normal state; and if such averages be made to include a sufficient number of different schools, all of which have reached the normal state, it will be neutralized.

If the time of attendance be progressing, the averages thus taken will of course indicate that fact, but not the degree of progress; it will be greater than that shown by them.

It is obvious that the same method might be applied to determine the average duration of life in a community.