Notices respecting New Boohs. 333 



are reproduced, with very slight changes of notation, from Pro- 

 fessor Tait's well-known treatise, and, we are sorry to say, repro- 

 duced without acknowledgment. In his preface Dr. Molenbroek 

 refers to Professor Tait only by way of criticism, accusing him of 

 finding an integrating factor where none such can be. The truth 

 is that quaternions give a solution where ordinary mathematics 

 fail — although it would probably baffle even a Hamilton to give 

 a geometrical interpretation of log Uq. 



Occasionally Dr. Molenbroek advances along a path of his own, 

 as, for example, in his approach to the quaternion equation of quadric 

 surfaces. His transformation of the ordinary Cartesian equation 

 (which is assumed) into quaternion form may well send a shiver 

 down the back of a disciple of Hamilton. 



Having neglected (as was poiuted out in our former review) to 

 discuss the operator \7 in the Theorie, Dr. Molenbroek has to make 

 a digression in the present volume so as to establish its elementary 

 properties. In the preface he gives as an excuse for the former 

 neglect the statement that Tait devotes only one liue to v i n the 

 theoretical part of his book. Still, Tait has it ; and in the 3rd 

 edition (1890) devotes four pages to it in the chapter on Differen- 

 tation. But even in the very first edition (1867) of his treatise 

 (in which no hard-and-fast line was drawn between the theoretical 

 part and the applications) Tait gives ample evidence of the import- 

 ance of v ; an d we still think that a more recent writer, wise in 

 the accumulated experience of his predecessors, should have had 

 something to say concerning the theory of this remarkable differ- 

 entiating operator. Dr. Molenbroek himself employs it with effect 

 in his discussion of partial differential equations. It plays a 

 conspicuous role in the theory of orthogonal surfaces. It is pre- 

 eminently the force-function operator. It is the heart of Green's 

 theorem, the key-note of spherical harmonics. But we now learn 

 on Dr. Molenbroek's authority that its value in physical applications 

 is much over-rated (sehr uberschatzt). "We shall wait with interest 

 his promised demonstration of this statement, which will probably 

 be a unique feature of a succeeding volume. Meanwhile it is 

 interesting to note that the last chapter of the present volume, 

 which discusses with considerable elegance Hamilton's theory of 

 rectilinear rays, ends with the equation, S . ryr=0. 



It is satisfactory to find in Dr. Molenbroek an enthusiastic 

 admirer of the quaternion and all that it involves. Adhering more 

 or less closely to the methods and notation of Hamilton, he has 

 made an honest endeavour to familiarize German reading students 

 with the beauty and power of the system. We trust that his 

 efforts will be rewarded and that in Leyden, at any rate, will grow 

 up a school of Hamiltonian workers who will effectively help in 

 developing the infinite resources of quaternions. 



