"222 PROCEEDINGS OP THE AMERICAN ACADEMY 



XVIII. 



SURFACES OF THE SECOND ORDER, AS TREATED 

 BY QUATERNIONS. 



THE THESIS OF A CANDIDATE FOR MATHEMATICAL HONORS CONFERRED 

 WITH THE DEGREE OF A.B„ AT HARVARD COLLEGE, AT COMMENCE- 

 MENT, 1877. 



By Abbott Lawrence Lowell. 



Presented by Professor Benjamin Peirce, Jan. 9, 1S78. 



The surfaces of the second order, or Quadrics, as they are very 

 commonly called, present by far too great a field to be investigated in 

 every part in any single thesis. I have therefore chosen only a few 

 branches of the subject ; and I have been guided in the selection chiefly 

 by a desire to avoid, as much as possible, those portions of the subject 

 which have been the most thoroughly treated by Hamilton. With 

 this object in view, I have passed over entirely the vast field oi foci 

 and confocal surfaces, and have touched but slightly upon cyclic nor- 

 mals and asymptotic cones. I have been especially attracted to con- 

 sider the relations existing between the various conjugate lines and 

 planes of any quadric, and the general relations which the different 

 classes of quadrics bear to each other. 



It has also been my object to exhibit that variety of expression 

 which is so peculiar to quaternions, by approaching all questions from 

 more than one point of view. With this idea, I have studied many 

 of the cases arising under the self-conjugate function (fQ under both the 

 cyclic and the rectangular forms, showing how these forms give dif- 

 ferent expressions to the same result. And finally, considering it a 

 great advantage to be as general as possible in the treatment of any 

 mathematical subject, I have tried to keep the self-conjugate function 

 under the general form cfQ, without attending to the special forms of 

 the terms which compose it. 



The equation, ^Q<fQ = constant, 



where cpo is any vector function of q, represents in general a surface ; 



for if we write 



Q ^= xi -\- yj -\- zk. 



