OBSERVATIONS ON AVEkiAGINO SURFACES, BOUNDED ON ONE OR ON TWO OPPOSITE SIDES BY A CURVE.
In the mensuration of superficies bounded by curves, the com mon method of taking the average, or mean, of any number of ordinates, by adding all the ordinates together, and dividing by their number, and multiplying the quotient by the length, is extremely vague.
For let the figure proposed to be measured be A EFTIK A, divided by the equidistant ordinates a 1, C H, D a. Let A K be denoted by a, B I by b, c by c, h by d, E F by e ; and let the common distance, A B, or B C, &c., be denoted by m; and let each of the areas, A a KA,BCIIIB,CDGIIC,DEFO 0, be computed according to the method of measuring a trape zoid ; then the sum of the contents will be the area of the whole. space. A K F H K A, provided that the lines, K I, I H n, F, be straight, and very nearly equal to the true area, when the figure is bounded on one side by a curve ; being in excess e The sum of these areas is a b + c + m ; that is, if the half sum of the extreme ordinates be added to the intermediate ones, and the sum multiplied by the common distance, the product will give the area : and it is evident that this will always he the case, whatever be the number of ordinates.
Therefore, let a, b, c, d, &c., to p and q, be any series of ordinates whativer, whereof a is the first term, and p and q the two last ; then the area of the curvo-rectilinear space will be generally expressed by + b + c + d, &c. to p) 2 X m ; now, let 1 = the length ; then if n be equal to the number of ordinates, n — 1 will be the number of + spaces, or areas ; therefore + b + c + d, &e. to p)
X m = +6+c+d,&c. to p)x1=-. n — 1 a + q + 2 b + 2 c + 2 d, &c. to 2 p and the value of 2 a + b + c + d, &c. to p + q the average method, viz.
a + + e + d, &c. to p + q X m X (n. -- 1) = x 1; then putting one of these equal to the other, there will result + q n x a =a+b+c+d, &c. to p + q.
It appears, therefore, that there can be no equality except when the half sum of the extreme ordinates, multiplied by the whole number of terms, is equal to the sum of all the ordinates ; which, therefore, must be in arithmetical progres sion, or amount to such.
The method of averaging will therefore be very uncertain ; as the half sum of the two extreme terms, multiplied by the number of ordinates, can be taken in any ratio, with respect to the sum of the ordinates.
The difference between the average method and the true method, would be the very same as the difference between a + g a ÷ b + c d, &c. to p + q / X 2 X (n — 1) and / X n x (n — ; for if from two unequal quantities equal quantities be taken away, their difference will still be the same ; and if to two unequal quantities equal quantities be added, the difference between the sums will still be the same.
If equimultiples be taken of unequal quantities, the differ. ence between the products will be the same multiple of the difference ; and if any aliquot part, or equisubmultiple, be taken of two unequal quantities, the difference will be equal