mere particular cases of Quaternions, where most of the distinctive features have disappeared; and that when, in the treatment of any particular question, scalars have to be adopted, the Quaternion solution becomes identical with the Cartesian one. Nothing therefore is ever lost, though much is generally gained, by employing Quaternions in preference to ordinary methods. In fact, even when Quaternions degrade to scalars, they give the solution of the most general statement of the problem they are applied to, quite independent of any limitations as to choice of particular coordinate axes. There is one very desirable object which such a work as this may possibly fulfil. The University of Cambridge, while seeking to supply a real want (the deficiency of subjects of examination for mathematical honours, and the consequent frequent introduction of the wildest extravagance in the shape of data for "Problems"), is in danger of making too much of such elegant trifles as Trilinear Coordinates, while gigantic systems like Invariants (which, by the way, are as easily introduced into Quaternions as into Cartesian methods) are quite beyond the amount of mathematics which even the best students can master in three years' reading. One grand step to the supply of this want is, of course, the introduction into the scheme of examination of such branches of mathematical physics as the Theories of Heat and Electricity. But it appears to me that the study of a mathematical method like Quaternions, which, while of immense power and comprehensiveness, is of extraordinary simplicity, and yet requires constant thought in its applications, would also be of great benefit. With it there can be no "shut your eyes, and write down your equations," for mere mechanical dexterity of analysis is certain to lead at once to error on account of the novelty of the processes employed. The Table of Contents has been drawn up so as to give the b student a short and simple summary of the chief fundamental formulae of the Calculus itself, and is therefore confined to an analysis of the first five chapters. In conclusion, I have only to say that I shall be much obliged to any one, student or teacher, who will point out portions of the work where a difficulty has been found; along with any inaccuracies which may be detected. As I have had no assistance in the revision of the proof-sheets, and have composed the work at irregular intervals, and while otherwise laboriously occupied, I fear it may contain many slips and even errors. Should it reach another edition there is no doubt that it will be improved in many important particulars. COLLEGE, EDINburgh, July 1867. P. G. TAIT. CONTENTS. Elementary geometrical ideas connected with relative position, § 15. De- finition of a VECTOR. It may be employed to denote translation, Expression of a vector by one symbol, containing implicitly three distinct numbers. Extension of the signification of the symbol The sign + defined in accordance with the interpretation of a vector as representing translation, § 19. Definition of -. It simply reverses a vector, § 20. Triangles and polygons of vectors, analogous to those of forces and of Any vector whatever may be expressed in terms of three distinct vectors, p = xa+yB + 28, The Commutatire and Associative Laws hold in the combination of vectors represents the plane through the origin parallel to a and 8, § 29. The condition that p, a, 8 may terminate in the same line is is the condition that the extremities of four vectors lie in one plane, Examples with solutions, § 31. Differentiation of a vector, when given as a function of one number, where s is the length of the arc, dp is a vector tangent to the curve, parallel their quotient is a number, §§ 45, 46. When they are not parallel the quotient in general involves four distinct numbers-and is thus a QUATERNION, § 47. A quaternion regarded as the operator which turns one vector into another. It is thus decomposable into two factors, whose order is indifferent, the stretching factor or TENSOR, and the turning factor or Representation of versors by arcs on the unit-sphere, § 53. Versor multiplication illustrated by the composition of arcs, § 54. [Digression on Spherical Conics, § 59*.] Quaternion addition and subtraction are commutative, § 61. Quaternion multiplication and division are distributive, § 62. Composition of quadrantal versors in planes at right angles to each other. Calling them i, j, k, we have ¿2=)2=k2= −1, ij=―ji=k, jk-kji, A unit-vector, when employed as a factor, may be considered as a qua- drantal versor whose plane is perpendicular to the vector. equations just written are true of any set of rectangular unit-vectors The product, and quotient, of two vectors at right angles to each other is Ka= = -α, Every versor may be expressed as a power of some unit-vector, § 74. Every quaternion may be expressed as a power of a vector, § 75. The Index Law is true of quaternion multiplication and division, § 76. Quaternion considered as the sum of a SCALAR and VECTOR. |