[ 60 ] 



IV. 



ON THE SIMPLEST MODE OF EEPRESENTING A EEAL CON- 

 TINUOUS LENICAR OETHOGONAL TEANSFORMATION (WITH 

 CONSTANTS ADDED) BY MEANS OF EOTATION AND 

 TRANSLATION OF A RIGID SCHEMA IN A EUCLIDEAN 

 MANIFOLD OF n DIMENSIONS.* 



By REGINALD A. P. EOGEES. 



[Rend Deckmbkii 12, 1921. Published Aphil G, 1922.] 



CONTENTS. 



PAGE 



I. — Theohem and Preliminaky Analysis. 



1. Statement of theorem, ... 60 



2. Meaning of a Euclidean );-nianifold. 



Flats, 61 



3. Orientation of flats, ... 61 



4. Mutually normal flats, ... 62 



5. Co-ordinates of a flat, ... 62 



6. Plane-normal flats, ... 63 



7. Complete systems of plane-normal 



flats, 63 



8. Number of co-ordinates of complete 



systems 63 



II. — Displacement and Rotation. 



9. Eq\iations for displacement of a rigid 



schema, ..... 64 



10. Condition of continuity, . . 64 



11. Rotation and rotatory flats, . . 64 



12 



13 



Number of co-ordinates specifying 



a rotation, . . . . . 65 



14. Rotation is in circles round an 



Sn-2, and parallel to an S2, ■ • 65 



15. General formula for rotation, . . 65 



16. Co-ordinates of a complete system of 



rotations. Proof of theorem, . 66 



1 7. Canonical form f or displarement equa- 



tions, 67 



III. — Reduction to Canonical Form. 



18. Invariant equation for determining 



angles of rotation in a given 

 motion 68 



19. Independent invariants of a motion, 69 



20. Determination of rotatory flats, . 69 

 21.22. Choice of origin; the central 



axis, . . . . . . 71 



23. Invariant of translation (n odd), . 72 



I. — Theorem and Pkeliminaey Analysis. 



1. In a plane the motion of a rigid schema from one position to another 

 can be effected by a single rotation round a point in space of three 

 dimensions by rotation round a straight line, together with an independent 

 uniform translation along the line (a screw). 



* The principal feature in this paper ■which I believe to be original is the co-ordinate 

 method of argument used in §§ 7-16. The other sections are for the most part explana- 

 tory and verificative, and are included for the purpose of making the paper more 

 intelligible and perhaps more interesting than it would be otherwise. See Note, p. 73. 



