THE AXES OF A QUADRATIC VECTOR. "'^^^'^•Chi. 



By Frank L. Hitchcock. 



Received Mar. 2, 1921. Presented Mar. 9, 1921. 



CONTENTS. 



1. Introduction 331 



2. The A's as functions of the axes 332 



3. Identities on which depends the simpHfication of the A's .... 333 



4. Simplification of the A's 335 



5. AppHcation to cases of coplanarity of axes 336 



6. Properties of a quadratic vector possessing a central axis .... 339 



7. Vectors with four sets of coplanar axes 341 



8. Other special t^-pes 343 



9. The converse problem 343 



10. Application to irrotational vectors 345 



11. Conditions that the curl shall be of the form r5p 346 



12. Interpretation of the equations 348 



13. Case where three given axes form a rectangular system .... 350 



14. Application to consecutive chemical processes 351 



1. Introduction. 



The well-known linear vector function ^ permits of several kinds of 

 extension or generalization. We may, for example, have a function 

 linear in each of two vectors, as (p{p, a) ; or we may have F[p] ciuad- 

 ratic in a single vector p. These two concepts evidently merge into 

 one another, for ^(p, a) becomes a quadratic function of p when we 

 let cr equal p. 



In a former paper ^ a study was made of various types of vectors 

 quadratic in a single vector p. The following results will be funda- 

 mental to the present discussion: 



(A) An axis of Fp is a direction jS such that FjS is parallel to (5 or 

 else null. A quadratic vector has in general seven axes. If six axes 

 lie on a quadric cone there are an infinite number of axes, and con- 

 versely ; the \'ector function is then rrducihie. The number of distinct 

 axes may be less than seven, since an axis may be of multiple order. 



(B) In general any quadratic vector may be written as the sum 

 of two terms 



evj 1 Hamilton, Elements of Quaternions, Chap. II, Section 6; Gibbs- Wilson, 



O^ Vector Analysis, Chap. V. 



'^ 2 Proc. Amer. Acad. Arts & Sci., 52, No. 7 (Jan. 1917), pp. 369-454. 



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