363 Sir W. Rowan Hamilton on Qiiaternions. 



I — X, or in the exactly opposite direction of x — i, and deter- 

 mined by the condition 



A(x-«)=xp + /"<, (84.) 



we shall have also 



T(p-A)=6; (85.) 



and thus the equation (83.) of the ellipsoid may be regarded 

 as the result of the elimination of the auxiliary vector-symbol 

 X between the two last equations (84.) and (85.). But if we 

 suppose that this symbol K receives any given and constant 

 value, of the form 



A = //((-x), (86.) 



where h is a scalar coefficient, which we here suppose to be 

 constant and given, and if we still conceive the symbol p to 

 denote a variable vector, drawn from the centre of the ellip- 

 soid as an origin, the equation (84.) will then express that this 

 vector p terminates in a point which is contained on a given 

 plane parallel to that one of the two cyclic planes of the ellip- 

 soid which has for its equation 



Kp+pK = 0, (23.), art. 44; 



while the equation (85.) will express that the same vector p 

 terminates also on a given spheric surface, of which the vector 

 of the centre (drawn from the same centre of the ellipsoid) is 

 A, and of which the radius is =b. The system of the two 

 equations, (84 ) and (85.), expresses therefore that, for any 

 given value of the auxiliary vector A, or for any given value 

 of the scalar coefficient h in the formula (86.), the termi- 

 nation of the vector p is contained on the circumference of 

 a given circle, which is the mutual intersection of the plane 

 (84.) and of the sphere (85.). And the equation (83.) of the 

 ellipsoid, as being derived, or at least derivable, by elimination 

 of A, from that system of equations (84.) and (85.), is thus seen 

 to express the known theorem, that the surface of an ellipsoid 

 may be regarded as the locus of a certain system of circular 

 circumferences, of which the planes are parallel to a fixed plane 

 of diametral and circular section. 



57. One set of the known circular sections of the ellipsoid, 

 in planes parallel to one of the two cyclic planes, may there- 

 fore be assigned in this manner, as the result of a very simple 

 calculation ; and the other set of such known circular sections, 

 parallel to the other cyclic plane, may be symbolically deter- 

 mined, with equal facility, as the result of an entirely similar 

 process of calculation with quaternions. For if, instead of 



