1895.] 



Octunions. 



179 



are a conjugate set of n independent motors of (n) is constant. In 

 particular, putting 7 = 1 we get the theorem proved in " Screws," 

 that the sum of the reciprocals of n independent co-reciprocal motors 

 of (11) is constant. If A, B, C are any three independent motors of 

 a complex of the third order the pitch of SABC is constant, and if 

 A, B are any two independent motors of a complex of the second 

 order the pitch of MAB is constant. Or adopting the rotation above 

 the sum of d cot 0—e tan 0, and the pitches of A, B, C is constant in 

 the first case, and the sum of d cot 9 and the pitches of A and B is 

 constant in the second case. 



The most general forms of complexes of all orders, expressed when- 

 ever possible as consisting of reciprocal motors with axes along 

 mutually perpendicular intersecting lines, are given; as also the 

 reciprocal complexes in a similar form. 



The differentiation of octonion functions is considered. 



It is shown that physical problems may be treated in a manner 

 which, so far as appears from the present trials is in most cases prac- 

 tically identical with quaternion method. A symbolic rotor v is 

 denned, which has properties very similar to the vector y of quater- 

 nions. Integration theorems corresponding to the well known 

 quaternion ones are given. The octonion treatment of strain and of 

 intensities and fluxes is given at some length. The octonion formula?, 

 though bearing a somewhat more extended meaning than the quater- 

 nion formulae, are surprisingly similar in form to the latter. On the 

 whole, from this part of the paper, it cannot be said that octonions 

 prove more efficient than quaternions, though Avhat would appear 

 were the subject more developed cannot at present be said. 



The following are some of the applications made in the last 

 division of the paper : — 



A twist means a general displacement of a rigid body. A twist 

 can always be effected in an infinite number of ways by two rotations. 

 The following construction suffices to find any two such rotations. 

 Take any line 1 intersecting the axis of the twist perpendicularly. 

 Let 1 become 2 w r hen it is subjected to half the given twist. Take 

 any transversal 3 of 1 and 2. Then double the rotation that converts 

 1 into 3, followed by double the rotation that converts 3 into 2, will 

 effect the given twist. A right-about-turn means a rotation through 

 two right angles ; thus it is completely specified by its axis. A twist 

 can always be effected in an infinite number of ways by two right- 

 about-turns. The following suffices to obtain the axes in all cases. 

 The axis of the first right-about-turn is any line intersecting the 

 axis of the twist perpendicularly. The second axis is obtained from 

 the first by giving to the latter half the given twist. As is well 

 known, two equal parallel and opposite rotations combine into a 

 translation. The translation is compounded of two translations, the 



