To find the pth moment, when uo, u l, u 2, ...
In cases other than those described in § 82, the pth moment with regard to the axis of u is given by Pp = XPrA where A is the total area of the original trapezette, and S 2 _ 1 is the area of a trapezette whose ordinates at successive distances h, beginning and ending with the bounding ordinates, are o, x1P -1A, x2 P-1 (AI+AI),.
Y where K-=4, X qth moment with regard to plane y =o, Lm yn X pth moment with regard to plane x =o, and R is the volume of a briquette whose ordinate at (x,.,y s) is found by multiplying by pq x r P - 1 ys 4-1 the volume of that portion of the original briquette which lies between the planes x =xo, y =yo, y = ys.
Gergonne had shown that when a number of the intersections of two curves of the (p+q)th degree lie on a curve of the pth degree the rest lie on a curve of the qth degree.
Pth power of n, the series itself being called the power-series.
If we know p and N, n is called the pth root of N, so that n is the second (or square) root of n 2, the third (or cube) root of n 3, the fourth root of n 4,.
The pth root of a number (§43) may, if the number is an integer, be found by expressing it in terms of its prime factors; or, if it is not an integer, by expressing it as a fraction in its lowest terms, and finding the pth roots of the numerator and of the denominator separately.
A number of this kind is called a surd; the surd which is the pth root of N is written ¦JN, but if the index is 2 it is usually omitted, so that the square root of N is written, /N.
More generally, if we have obtained a as an approximate value for the pth root of N, the binomial theorem gives as an approximate formula p,IN =a+6, where N = a P + pap - 19.