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of C(X, Y)=0. When only scalar solutions are required, the 

 abstract equations to the first and last curve become qQ, $=0, 

 so that both of these curves coincide with OI produced both ways, 

 and the intermediate or connecting curve is the ordinary scalar radical 

 locus. 



Scalar Solid Geometry. The same theories apply with proper 

 modifications. The concrete equation 



with one abstract equation, /(#, y, )=0, gives a surface for its locus, 

 and with two abstract equations, 



/ 1 (^y,^)=0, / a (a?,y,*) = 0, 



a curve. From these equations all the ordinary theories are most 

 readily deduced. We may also take the concrete equation more 

 generally in the form 



OP=/; O, y, *) . OI+/ 2 (*, y, z) . OJ+/3 (x, y, z) . OK, 



the abstract equation being F(a?, y,*r)=0. We obtain the surface 

 by putting x l =f v y l =/ 2 , z^ =/ 3 , and eliminating x, y, z between these 

 equations and F=0, thus finding ^ (a? y y,, ^) = 0. The locus of 



OP=^ . OI+ yi . OJ+*! OK, (x 19 y v ^)=0 



must be limited by the condition that not only x, y, z, but also 

 / (*, y, z), / 2 (ar, y, z\ / 8 (a?, y, z) must all be scalar. If three 

 variable parameters a, 6, c are introduced, we require in addition 

 three equations of condition, F 1 =0, F 2 =0, F 3 =0, between x, y, z t 

 a, b t c, in order to eliminate all six and find $ (x l3 y lt 2' 1 ) = 0. 



Clinant Solid Geometry. Some precaution is now necessary to 

 indicate the plane on which the quadrantal rotation symbolized by i 

 is to take place. This is effected by introducing all three coordinate 

 axes into the concrete equations. Let the abstract equation be 



F(X,Y,Z) = 0. 

 Then, supposing 



this equation reduces to F^O, F 2 =0, where F x and F 2 are forma- 

 tions of the six scalars jp, <?, r, s, u, v. We now require two assignant 

 equations to determine the curves on which OX and OY are to be 

 taken. These may be given in the form of the single clinant equa- 



