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III. 

 A STUDY OF THE VECTOE PRODUCT Vi^adfi, 



By FEANK T. HITCHCOCK, Ph.D., 

 Massachusetts Institute of Technology. 



[COMMUNICATKII HY PR0FES8OK A. W. CONWAY, K.H..'!.] 

 [ReadJcNK U. Published Novemiikh 19, 11120.] 

 CONTENTS. 



I'AOK I PAOB 



1. I.NTROnUCTION, . . 30 0. PkOOK nv niRP.CT THAN8P011MATI0K, 35 



2. I'llOIlF THAT V^a$B + Via^S IH A 6. SyMMETUICAL rOllM OF TIIK FUNC- 



LINBAU VKCTOH FUNCTION OP i'aB, .'il TION ir, 36 



3. FoUM OP THK FUNCTION r, 32 7. PUOOK IIV USB OF V, .... 36 



4. OlTBHMINATION OF TBK BCALAB a, . 33 S. CoMI'AHIsoN WITH CaUTRHIAN MKTKUDS, 36 



1. Introduction. 



If <p and 6 are two linear vector functions, and if a and |3 are any two 

 vcct'jrs, the vector product YipaO^i possesses many properties dependent on 

 the iniiwrtant invariants discovered by the late Professor C. J. Joly ; in fact, 

 this product, being one of the simplest expressions which can be written down 

 containing two linear vector functions, appears well adapted to show the 

 meaning and application of Joly's invariants. 



The special problem I propose to study in this paper is suggested by a 

 relation long ago proved by Hamilton, who showed (Elements, Art. 350) that 

 the expression 



V^ali + Fa^/3 (1) 



is a linear vector function of Va^ ; is equal, in fact, to 



iin" - 4>') Vafi. (2) 



where m" is an invariant nf <p. 



1 propose to prove that the expression 



Vi,ae(i * V0af(5 (3) 



is a linear vector function of Va^i, and to study the form of this function. 

 It will be found to involve Joly's invariants of f and 6. 



