6 REPOUT — 1855. 



General Locus. — If x=p + {q and y=r + is, and f(x, y) be any function of 

 X and y, then f{jc, y) = X + iY, where X and Y are functions oi p, q, r, s, 

 which determines a point. If, moreover, we have given q^=q in order to 

 have only one variable p, and also (p{x, y)=.(p'{p, q, r, s)-\-\(p"{p, q, r, s)=0 as 

 any relation between x and y, then the whole system /(a;, y), q=^q , and 

 0(^, j/) = determines a curve, called the general locus, which is found by 

 eliminating p, q, r, s between the five equations X and Y= functions of 

 p, q,r, s, q — q^, ^' = 0, and ^" = 0, whence Y = a function of X, and the 

 required locus is the simple locus of X+iY and Y= function of X. 



Particular cases. — U q=^s^=0, and first, /(a?, y) =:^ + ir, we have a case 

 corresponding to the Cartesian rectangular coordinates. If, secondly, 

 y(a:, y)=p(cos a + isina) + /-(cos /3 + isin /3), we have the case of oblique 

 coordinates; while, thirdly, _/ (a.', y,):= r(cosj)+i sin ^) gives the case of 

 polar coordinates. 



Radical Loci of the equation <p{jr,y)=zQ for q=^qp- From q=qp, ^' = 0, 

 (t>" = 0, find *=s^ and r^-r^, and describe the simple loci of — 



1. p-\-^q and q=-q„ giving x from p, 



2. r + is and s = s giving y from r, while 



3. p-\-ir and r^^r gives r from p ; 



so that by setting off ph, both xh and yL are found. In the Cartesian case, 

 5'=s=0and the loci 1. and 2. coincide with the axis, while 3. is the ordinary 

 locus. 



Three Dimensions. 



Assume a known origin, axis, scale, and plane, called the cumbent plane. 

 On this plane draw a line determined by p + ^q. Through tliis line draw a 

 plane perpendicular to the former, called the sistent plane. On this set off 

 rh, where ?•=: some function of p and </, in the direction of the line already 

 determined, and then set off the line determined by r + is on the sistent plane. 

 The extremity of this second line determines any point in space. 



Simple Locus. — If p and q are independent, and s= a function of r, and 

 therefore of p and q, the points determined by p+'iq and r-\-\s lie on a 

 surface. If, in addition, q (and therefore r and s) be a function of p, the 

 points determined by the system lie on a curve. 



General Locus. — If x^^p + \q, y = r + is, and z=-u + \v, then /(.r, y) will 

 determine a line on the cumbent plane, and/'(^,5', r, s, z) aline on the sistent 

 plane drawn through the former. Assuming q a function of p and s a func- 

 tion of r in order to have only two variables, and (p{x, y, r) = as any rela- 

 tion between x, y, z, then finding/(A', y) = X + iY and /'(/;, q, r, s, £r)=R + iZ, 

 with (b{x, y, z) = 0' -\-i(p" = 0, where X, Y, Z, R, f', <p" are all functions of 

 p, q, r, s, u, V, or by virtue of the relations between q and p, s and r, func- 

 tions of p, r, V, V only, we find from ^' = and ^" = that « and v, and 

 hence X, Y, Z, R, are functions of j:; and ?• only. Hence by two eliminations 

 R and Z are found as functions of X and Y. The general locus is then the 

 simple locus of X + iY on the cumbent, and R + iZ on the sistent planes, 

 where R and Z are known functions of X and Y. 



Particular cases. — Taking (^=s = t; = 0, and assuming R so that RL is 

 always the length of the line determined by/(.r, y), we readily obtain cases 

 corresponding to the Cartesian rectangular oblique and polar coordinates. 



Radical Loci of ^(.t, y, z)^0 for q a function of jy, and s of r. Having 

 found u and v functions of j; and r as before, describe the simple loci of — 



1. p+'\q and^= function oi p ; and 2. r + 'is and s= function of r, both on 

 the cumbent plane, to find x and y. 



