The logarithm is then obtained by use of the formula d l d2 l d3 2 log e (x+d) = log e x-f- - x2+3 x3 - &c., in which of course the object is to render dlx as small as possible.
Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that the expression i d x +j - + k dz could be, like ix+jy+ kz, effectively expressed by a single letter.
The result of this superposition is a seven-year period, which makes 6 periods of the 428-day term (4 2 8 d X 6 = 2568 d = 7 years, nearly), and 7 periods of the annual term.
