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tion A (X, Y, Z)=0, which reduces to the two scalar equations 

 A^O, A 2 =0. Now take OR^/, (X, Y, Z) . OI+/ 2 (X, Y, Z) . OJ. 

 Then, since i.OI=OJ, because the assumption of the two axes 

 determines the rotation to be in the plane IO J, we can reduce this to 

 OR=R 1 . OI + R 2 . OJ, where R lf R 2 are formations of the six scalars. 

 By virtue of the equations A x = 0, A 2 =0, F x =0, F 2 =0, which 

 will give q, s, u y v in terms of p t r, the line OR is perfectly determined 

 by the assumption of p and r. Next take 



so that ORj is a unit radius in the direction OR. Put 

 OP=/ 3 (X, Y, Z) . OR,+/ 4 (X, Y, Z) . OK, 



which reduces to OP=R 3 . OR. + P,, . OK, where R 3 , P 3 are forma- 

 tions of the six scalars, because i . OR X =OK on the plane RjOK. 

 The locus of P will now manifestly be a surface, the concrete equation 

 of which becomes of the usual form on putting for ORj its value. 

 Determine P x , P 2 by the equations 



P, . V(R 1 2 +B 2 2 )=R 1 B 3 , P 2 V(R 1 2 +R 2 2 )=R 2 R 3 > 



and we find 



0?=^ . 01+ P 2 . OJ+P 3 . OK. 



Putting #=P 1} y=P 2 , =P 3J and eliminating the six scalars between 

 these three equations and A t =0, A 2 =0, F a = 0, F 2 =0, the locus 

 becomes that of 



OP=# . OI+y OJ+* . OK, (a?, y, ^)=0, 



which is limited by the conditions of scalarity. 



After illustrating this theory by an example, the theory of inter- 

 section, when the elimination gives clinant values to determine the 

 points, is discussed. The general theory is further illustrated by the 

 determination of equations to loci, leading to very simple and ex- 

 tremely general modes of finding families of surfaces. The problems 

 of the transformation of coordinates and of radical loci are shown to 

 be precisely analogous to those of plane curves. 



It will be evident that these investigations merely open out a new 

 field for algebraical geometry, of which it is impossible to foresee the 

 extent. 



