Hitchcock — A Study of the Vector Product V(^ad^. 35 



5. Proof hy direct tran>\formntion . 



Thoi-e is lofifical satisfaction in piovinti; identities by the direct trans- 

 formation of one side into the other. While it would not liave lieen easy to 

 foresee at the start how to do so, we may now, as a recapitulation of the 

 main steps of the reasoning, prove the identity (40) in this manner. 



I'><.0f3 + Vda<p\i = O'-i^' [ V<i,ae^ + V{)a^\il, since d'HY = 1, 



= 0'-' [9' V,pad^ + «' r0«./,/3] by distributing 9' , 

 = e'-i[- m F/3W-V" + "' ViS-'',^^] by 21, 

 = B'-'[Vlo^+ F„e/3] by (27), 

 = e'-iK- ?']^«/3by(30i, 



= ri[r.-r]F<./3by(;i6), 



= [/'#-' - mii'-'^'Q'-'] Fo/3 by (28). 



6. Symmetrical form of the function w. 



Since, as already indicated, the form of tt in (38) is not symmetrical, it 

 must be possible to obtain this function by a method that shall treat ^ and 

 alike. Doubtless we shall not expect so compact a result. Eeturning to our 

 equation (14), we may develop both terms of the left side by Hamilton's 

 relation, applying it to 0, thus 



Vnpii = ,i>' VpOji + 6' Vp<l>j3 by (14), 



= f [(«/' - h') Vpfi - V0p(5] H- m"Vp,l>(i - Vep<i.(i - Vpbfii 



= m" [,(,' Vpji + Vpft>fi] - [<p' vepfi + vopfji] - [<p'iy Vpji + Vpeffi], 



where the last line is a mere re-arrangement. Now Hamilton's relation may 

 be applied to (p in the first two bracketed groups, and to ^'0' in the third 

 group. The first invariant of the function cfi'f)' may be called t". Then 



Vnp(i = m"[p"Vp(i - Vfpji] - lp"V9p(5 - V<j>0pl5] - [rVpfS - ve<i>pi3i 



where all the terms are vector products of some vector i^ito /3. Hence, since 

 /3 is any vector whatever, 



TT = m"p" - m"ip - 2^"Q + rbd + 0(p - t", (45) 



where ^ and enter in the .same manner. But, from Joly's paper already 

 referred to, the scalar t" is the same as Mi, the first invariant of fit/., whence 

 m"p" - t" = /j, another of Joly's new invariants, defined by 



hS\p.v = :S.S\{e,„i>v + .pudv). (46) 



"We may therefore write i'45i as 



TT = Z, - m"(tt - p"0 + (jiO + 0(p. (47) 



