nric'iicoCK A Sludij oi' llie Vector Product V<l}aO/3. 37 



Joly's iin'aviaul /■., which is the siiuplesL ul' his new invariants, becomes 



Our identity i4Sj is equivalent to three scalar identities, of which the first 

 must sullice. It is 



(Pairti + Piitti + Pistt,) (Qai&i + QA + QiA) - (/'aifti "I- /'32a2 + P-^/h) 

 {QJh + Q,A + Q'Jh) + (0...,«1 + Q22«2 + Q.3«3) (P3,6, + PzA + P:M 



- (Q3,a, + Q,,a, + Q3aa3] (/^>&, + P22&2 + PrA) = [P.M.« + Q,2Pz:> + P33Q,, 



+ Q33P,, + PuQ.z+QuP.2- P^^Q,,- Q.J\,- P,.Q.,- QrJ\- /'2iQ,2- Q.J'n) 



{ajh - n-A) - [Qii + Q-n + Qx<]lPn{(iA - aA) + P'u {njj, - aA) + P^^MA 



- a-A)] - [Pu + -P22 + ^'33] [Qu [aA - aA) + Q2,{aA - a,h) + Q^iaA - aA )] 



+ Qn [Pu (aA - «3&2) + -^21 {aA - aA) + P.n {aA - a-A)] 



■^ Qii[Pn{aih- aA) + -P22 (rts&i - ffi&.i) + Pn{aA-aA)] 



+ ft, [Pi3 {aA - aA) + P23 («3&i - aA) + P-xi (a A - a-A)] 



+ ^u [Qn (a^h- aA) + Q21 {a A - aA^ + ft, (a A - a A)] 



+ P21 [Qn («2&3 - aA) + ft2 '\«3&. - a A) + Qi2 (aA ~ aj},)} 



+ -^31 [(3,3 («2&3 - «3&2) + Q-a (aA - aA) + Qii (aA. - aA)] 



Here tiie last three lines express in Cartesian form one componcnl of the 

 vector ^'Q' Va(3. It is clear that vector language and processes justify them- 

 selves not alone by tlieir compactness, bi\t by a two-fold lucidity : the 

 vectorial expression for any quantity indicates both what it is and what 

 may be done with it. 



B.I. A. PKOC, VOL. XXXV, SECT. A. [5J 



