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I. STRINGHAM. — ORTHOGONAL TRANSFORMATIONS IN ELLIPTIC. 337

Thus régressive multiplication ofp bj the quaternion q produces iden-tically the sarae resuit as the triparametric. orthogonal transformation (cos <h, X, p-, v) applied to the point (ce, x,y, z) (1). This is only a par-ticular case of the identity of resuit obtained through multiplication by quaternions and transformation by matrices of the fourth order. The general quaternion form of the orthogonal transformation with six. parameters is qpcf, where q and q' are unit quaternions, but otherwise inde-pendent of one anolher. But the discussion of this phase of my subject lies outside the proper scope of this paper.

11. The general conclusion to which this investigation lias led may be summed up in the following statement (2) :

The orthogonal transformation with six. independent parameters, when interpreted geometrically, represents a motion, without distortion, of elliptic or hyperbolic space in itself, by which no points remain fixed ; but it is at once decomposable into the product of a pair of complementary commutative transformations : (a, b, c,f, g, h ; © = o), (/', g', h', a', b', c’\ 0' = o), whose parameters satisfy the conditions :

— = — = — = = a ~ b ~ c ~ f g h*

af -4- b g + c h = a'f + b'g'-¥- c'h' = o,

and which represent rotations about a pair of axes that remain alternately fixed. The équations of these axes are respectively

ax -+- by 4- cz = o, I w — A# —B y — Gz — o (the I-axis),

and

fx-\-gy-srhz = o. J(v-t-A« + B/ + G^ = o (the J-axis), and the angular displacements of the two rotations are respectively

.9. -K'

<I) = cos-1 —g— » <t> = COS-1

12. The following dualistic relations, here announced without proof, may be verified without difficulty :

(!) Cf. Study, loc. cit., p. 351 ; Hausdorff, loc. cit., p. 200.

(2) Cf. Lindemann, loc. cit., § 3 et 4.

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