SEMINAR NOTES 39$
range of one dimension, a straight line, will no longer suffice, but a region of two dimensions, a plane, will be required for that purpose. The manner of variation of a complex quantity can then be repre- sented by assuming that a point / of the plane is determined by a complex value, z=^x-\-iy, in such away that its rectangular coordinates, in reference to two coordinate axes, assumed to be fixed in the plane, have the values of the real quantities x and_)'. In the first place, this method of representation includes that of real variables, for when once 2 becomes real, and therefore j';=o, the representing point/ lies on the axis of x. Next, the coordinates of the point / can vary inde- pendently of each other, just as the variables x zxvA y do, so that the point/ can change its position in the plane in all directions. Further, one of the two quantities, x and y, can remain constant, while only the other changes its value, in which case the point/ will describe a line parallel to the x- or j'-axis. Finally and conversely, for every point in the plane the corresponding value of s is fully determined, since by the position of the point / its two rectangular coordinates are given, and therefore also the values of x and 7.
"Instead of determining the position of the point/, representing the quantity 2 by rectangular coordinates x and y, we can accomplish the same by means of polar coordinates, for, by putting
r=:cos <^, and^^^ sin <^, we obtain
z = r (cos <^-\- i sin <^).
.... Hence we can also say that a complex quantity r (cos <^ -f- / sin <^) represents a straight line, of which the length is equal to r,
and which forms an angle <^ with the principal axis From the
property of complex quantities, that a combination of two or more of them by means of mathematical operations always leads again to a complex quantity, it follows that, if given complex quantities be represented by points, the result of their combination is capable of being represented by a point."
It is necessary to keep well in mind throughout the discussion the correspondence we have established, the values of our functions, the institutions {G, H, I, J, K, L, M) being expressed in terms of the variable desires (a, ^, i-, d, e,f). We shall speak as is usual of the variable 2, which may stand for any one of our variables or a combi- nation of them forming the complex variable z^=x-\-iy. Nor is the method purely artificial, since we are not comparing analogous things, but demonstrating relationships.
