PROCEEDINGS 



OF 



THE ROYAL IRISH ACADEMY 



PAPERS READ BEFORE THE ACADEMY 



I. 



ON THE SIMULTANEOUS FORMULATION OF TWO LINEAR 

 VECTOR FUNCTIONS. 



By FRANK L. HITCHCOCK, Ph.D., Massachusetts Institute of Technology. 



[Read January 22. Published May 25, 1917.] 



CONTENTS. 



1. Introduction, 



2. Possibility of the reduction to form ( 1 ). 



3. Typical form of $ and 6 when <p ' '0 



has a double axis, 



4. Geometrical application, . 



5. Typical form of <p and $ when (p' '6 



has a triple axis, .... 

 G. Restriction to self-conjugate func- 

 tions, . . . . . . 



page 

 1 



7. Typical form of (p and 6 when <p - >0 



has an infinite number of axes, 



8. Special case of self-conjugation, 



9. Typical form of cp and 8 when no 



function of the pencil has an 

 inverse, ..... 

 10. Application to quadratic vector func- 

 tions. .... 



1. Introduction. — In his valuable appendix to Hamilton's " Elements of 

 Quaternions " the late Prof. C. J. Joly exhibited, in a very elegant manner, 

 some of the properties of the most general pair of linear vector functions. 

 He pointed out that two such functions <j>p and dp may, in the general case, 

 be simultaneously expressed in the form 



fp = XSap + pSfip + vSyp ; Op = nXSap + bpSfip + CvSyp, (1) 



where the three vectors a, j3, y are the axes of 6'$' ' \ while A, n, v are the 

 axes, and a, b, c are the latent roots of <p0 " '. 



These function-pairs are important in many ways, and appear frequently 

 in Joly's writings. In the above formulation we note thai he lias expressed 



R,I,A, PROO., VOL. XXXIV., SECT. A. [1] 



