180 



Mr. A. McAulay. Octonions. 



[Dec. 12, 



one being equal and parallel to the perpendicular from the first 

 axis on the second, and the other equal and parallel to the same 

 perpendicular when it has been first rotated with the second rotation 

 and then reversed. To combine two twists, take two lines, 1 and 2, 

 such that half the first twist brings 1 into coincidence with the 

 shortest distance between the axes, and half the second twist brings 

 the shortest distance into coincidence with two. Then the axis of 

 the resultant twist is the shortest distance between 1 and 2 and the 

 twist itself double the twist about this axis, which will bring 1 into 

 coincidence with 2. 



The geometrical properties of the second order and third order 

 complexes, as given by Sir Robert Ball in " Screws," are estab- 

 lished. 



The octonion treatment of the motion of a single rigid body is con- 

 sidered at some length. The treatment is very analogous in many 

 parts to the quaterion treatment of the motion of a rigid body with 

 one point fixed. The variation in time of the pitch and position of 

 the velocity motor of a rigid body subject to no external forces is 

 considered. 



This leads to the consideration of a curve — called the " normal 

 cone curve " — related to the polhode. The polhode lies on a quadric 

 cone whose vertex is the centre of the Poinsot ellipsoid. The normal 

 cone curve lies on the quadric cone whose vertex is the same point, 

 and which is normal to the polhode cone. Defining a polhode as the 

 locus of points on a quadric at which the tangent planes touch a 

 concentric sphere, it is shown that a real polhode on a real quadric is 

 always a polhode on a second real coaxial quadric, and that the 

 normal cone curve is a polhode on each of two other coaxial quadrics 

 which are both real or both imaginary. The original polhode is 

 similar and similarly situated to the normal cone curve of the original 

 normal cone curve when the latter is regarded as a polhode. Several 

 reciprocal relations between the two pairs of quadrics are established. 

 If either of the original polhode quadrics is an byperboloid of two 

 sheets, the normal cone curve quadrics are imaginary, though the 

 normal cone curve itself is always real. When the four quadrics are 

 all real, not more than three of them can be ellipsoids, though they 

 may all be hyperboloids of one sheet. A figure drawn to scale is 

 given in which each pair consists of an ellipsoid and an hyperboloid 

 of one sheet, and in which each polhode is identical with the normal 

 cone curve of the other. The two polhode quadrics coalesce if, and 

 only if, one is a sphere. The general solution given breaks down in 

 certain limiting cases. These cases are examined, and all prove to 

 have simple geometrical properties. The methods adopted in these 

 particular operations are to all intents and purposes quaternion 

 methods, and they illustrate, what appears frequently throughout the 



