[ 73 ] 



lY. 



THE ASSOCIATIVE ALGEBEA APPLICABLE TO HYPEB- 

 SPACE. Br CHAIILES JASPER JOLT, M.A., F.T.C.D. ; 



AndreTFs' Professor of Astronomy in tiie University of Dublin, 

 and Royal Astronomer of Ireland. 



[Read June 27, 1897.] 



TABLE OF CONTENTS 



Art. Ar' 



1. Products of units. 32. 



2. Curvature and torsion. 33. 



3. Deviation of a cui-ve into the Fourth 34. 



Dimensidn. 35. 



4. Angle bet\^ een a pair of planes, of 36. 



spaces. 



5. M— 1 affections of a curve in «-space. 37. 



6. Moving axes. 38. 



7. Analogues to circle and helix. 39. 



8. Change of system of units. 40. 



9. Canonical reduction of homogeneous 



functions of the units. 41. 



10. Eeduction of a quadratic. 42. 



11. On the function i|/p = T'lqop. 



12. Its imaginary axes. Condition that 43. 



\pp = 0. 44. 



13. Special cases of reduction. 45. 



14. Different kinds of cuhics. 46. 



15. Linear functions derived from a 47. 



homogeneous function. 48. 



16. Conjugate (X) of a function. 49. 



17. Inverse (or reverse) (/) of a function. 50. 



18. J=IK^KI. 



19. Formulae for parts of products. 51. 



20. Conditions tliat qKq = Kq.q. 52. 



21. Cases of quadratic and cubic. 53. 



22. Conditions that qEq = Eq .q = 54. 



scalar, 55. 



23. Reqmre a quadratic to he a quater- 56. 



nion. 



24. Case of a cubic. 57. 



25. Conditions that qlq = Iq . q. 58. 



26. ConAhionXha.t qlq = Iq.q=: scalar. 59. 



27. Conditions combined. 60. 



28. V = qpq-^. 61. 



29. Condition for P + Kv = 0. 



30. Condition for P + ip = 0. 62. 



31. Conditions for P = V(i)P. 63. 



Case of P odd in units. 



Eefiection of a vector.' 



Rotation in three dimensions. 



Dual representation. 



Operator of orthogonal transforma- 

 tion. 



Its structure. 



Regarded as a linear function. 



Calculation of roots and axes. 



Rotations in hyper-perpendicular 

 planes ; 



Independent one of another. 



Finite displacement of a body in odd 

 and even spaces. 



Canonical form ei^ ( ) e"«2. 



Dynamical equations. 



Angular momentum. 



Permanent motions. 



Series of linear functions. 



Systems of ■«'renclies. 



Co-reciprocal sj'stem. 



Canonical system of fundamental 

 screws. 



Change of origin. 



Centre of system. 



Formation of Invariants. 



Axis and pitch of a -wrench. 



Screw motions in odd spaces. 



Difficulty in finding analogue to 

 pitch in this case. 



Quadrantal versors. 



F functions. 



Yarious properties of three functions. 



Construction of a P function. 



New method ia the Theory of Sub- 

 stitutions. 



Simplifications. 



Final i-eduction. 



