April 30, 1897.] 



SCIENCE. 



701 



■exponential theorem. I have demonstrated that 

 in space e''y:e''^e'' + '', and the demonstra- 

 tion shows conclusively that the Hamiltonian 

 ideas about the addition of vectors require to be 

 revised. Although I have asked quaternionists 

 to point out any error in the demonstration, no 

 •error has been pointed out. 



The Hamiltonian principle that a unit-vector 

 may be identified with a quadrantal versor re- 

 quires to be modified. The conception of a 

 line does not involve the idea of an angle, 

 whereas the conception of an angle involves the 

 idea of two lines. The question reduces to the 

 following : Can a line be conceived apart from 

 an initial line? The answer appears to be yes, 

 for Hamilton did not succeed in his endeavors 

 to extend algebra to space until he abandoned 

 the idea of an initial line and considered all 

 three axes as equally real. The vector and the 

 -versor are complementary ideas, and just as a 

 vector is expressed in terms of rectangular co- 

 ordinates which are in their nature vectors, so 

 a versor is expressed in terms of rectangular 

 quadrantal coordinates which are in their na- 

 ture versors. 



On the other hand, a vector cannot take the 

 place of the versor. To ignore the versor and 

 more generally the quaternion is the mistake 

 made by writers who confine space-analysis to 

 vector-analysis, which is merely a branch. The 

 very name vector-analysis implies a restricted 

 view of space-analysis. The versor is the prop- 

 er idea in spherical trigonometrical analysis, 

 and in a modified form expresses the rotation of 

 a rigid body. It leads up to higher ideas which 

 express elliptic and hyperbolic angles and the 

 motion of a body which is not rigid. 



In mathematical analysis the product of two 

 quantities having the same direction is positive, 

 while that of two quantities having opposite d'i- 

 Tections is negative ; consequently the square of 

 a quantity is always positive. Consistent with 

 this the reciprocal of a negative quantity is the 

 negative of the reciprocal. Now, are all the 

 quantities considered in algebra or the Carte- 

 sian analysis scalar quantities, or are they in 

 some cases partial vectors ? If in any case they 

 are partial vectoi'S (that is, component of a 

 vector) then, in order to be consistent, the 

 square of a vector in space must be positive 



and the reciprocal of a vector have the same 

 direction as the vector. 



The order of writing of the terms of a sum or 

 the factors of a product should conform, as far 

 as possible, to the order followed in mathemat- 

 ical analysis. There the natural order of writ- 

 ing is followed, from left to right, and, as in a 

 determinant, from top to bottom. But in books 

 on Quaternions, for example, Hathaway's 

 Primer, p. 49, we have the Hebrew order of 

 writing. This abnormal order of writing was 

 adopted from the idea that a product of qua- 

 ternions supposed an operand and that the 

 operand ought to be on the right. As a matter 

 of fact, in the expression for the rotation of a 

 versor the operator is written both before and 



behind. 



Alexander Macfarlane. 



SCIENTIFIC JOURNALS. 



JOURNAL OP GEOLOGY, APRIL-MAY. 



Professor Chamberlin continues his gla- 

 cial studies in Greenland, giving a description 

 of the Bowdoin glacier. This is a tongue of 

 the great inland ice-cap which descends from 

 the north into the head of Bowdoin Bay. On 

 the west it is confluent with the Tuktoo and 

 Sun glaciers. The Bowdoin glacier has a length 

 of six or eight, and in its lower part a breadth 

 of about two miles. It has a descent of 2,000 

 to 3,000 feet, and is notably crevassed. It dis- 

 charges icebergs of considerable dimensions, 

 the discharge varying greatly with the season. 

 The west side does not present the usual verti- 

 cal scarp, and this is thought to be due to the 

 fact that the ground which should act as a re- 

 flecting plane is covered by protuberances from 

 the Tuktoo glacier. The stratification and 

 basal loading of the ice is much the same as in 

 the glaciers previously described, though the 

 debris does not rise so high. The bowlders 

 were usually more rounded, and this rounding 

 is of such a nature as to imply very considera- 

 ble wear. This considerable rounding, the 

 small amount of debris and its low position in 

 the ice are especially significant in view of the 

 fact that the Bowdoin is one of the larger 

 tongues of the great icecap. 



Dr. Henry Washington describes the Rocca 

 Monfino region in the fourth of his Italian 



