332 Notices respecting New Books. 



to another department of the general undertaking of which this 

 hook forms a part. The phrase "nach Abbe" which occurs in the 

 title is a little puzzling to non-German readers : in extending widely 

 the capabilities of optical construction and manufacture, a very 

 high degree of credit is due to Prof. Abbe and his collaborators at 

 Jena; but yet it seems strange to associate his name so markedly 

 with the title of a theoretical treatise whose contents would appear 

 for the most part familiar to many people who have possibly never 

 heard of his work, and whose knowledge of Optics was acquired 

 before his time. 



Prom the nature of the book, as forming an article in an 

 Encyclopedia of Physical Science, the subject is broken up some- 

 what into separate headings, with a view to facilitate reference. 

 If ever the time comes for a final gathering together of the threads 

 of this somewhat discursive subject into a compact form, it is to be 

 hoped that much attention will be paid to the geometrical methods 

 of discussion employed by the earlier English writers such as 

 Robert Smith and Thomas Young, which amalgamate so easily 

 with experimental requirements ; and that the capabilities of the 

 Hamiltonian method of Action as a basis for the analytical part 

 of the subject will be fully utilized. J. L. 



Anivendung der Quatemionen aufder Geometrie. Von 



Dr. P. MOLEKBROEK. 



This is the promised sequel to the Theorie der Quatemionen by the 

 same author, which we reviewed in November 1891. In his pre- 

 face Dr. Molenbroek replies to certain of our comments at that 

 time. He repudiates the description then given of his novel 

 interpretation of V — I a s an operator. Tet his own generalized 

 description is in these words : " This definition shows that under 

 the symbol sj — la there are included not only an infinite number 

 of arbitrary quaternions but also a similar number of right quotients 

 [that is, versors] whose indices are all perpendicular to a." And 

 thus, we still must believe, the operator \/ — 1 has the " singularly 

 felicitous but hitherto unsuspected power of adjusting its axis so 

 as to be perpendicular to any vector to which it may be applied !" 

 We had hoped to find in the present volume, which treats of 

 geometrical applications, a further development of Dr.Molenbroek's 

 pet creations — the Vehtorkreis, the Vektorkegel, and the Conisch 

 Bpaltender Quaternion. From plane triangles with their " circles " 

 and " points," to curvature and geodetics, we find many good 

 illustrations of the power and elegance of quaternions ; but of the 

 conically spreading quaternion which transforms a vector into a 

 conical sheet like a Japanese umbrella and then as with a fierce 

 blast turns it inside out or in some other fashion de-axializes the 

 original vector stem : — of this we find no mention. Our disap- 

 pointment is, however, more than balanced by the real quaternion 

 character of the book as a whole. A great many of the examples 



