RocHU — The Quadratic Vector Function. 99 



the ^'s are transformed. 



where K^^, etc., are the minors of the modukis of transformation of the a's, 

 The transformation is, of course, eontragredient. 



12. If P"f{(j>i^iif,saia2a3p) = /{4>'i<^'z'l^'-iaiaia<ip), where P is the modulus 

 of transformation, / is a concomitant. It is an invariant if p is absent, 

 and an absolute invariant if m = 0. Invariants may be classified according 

 to the value of n. 



13. If we transform ^ [pp) of § 7, by this substitution we get 



p^l^ipp) = 4.'{ppi 



where the dash denotes transformed constituents. So i// (pp) is a concomitant 

 of our function. It contains p to the second degree. Expanding 



p = ix + jy + kz, 



and operating with V- we have 



V'^ (pp) = 2 (c^.a, + <^,o, + cf>,a,). 



Here p is absent, so V'-ip (pp) is an invariant. 



Put v'i.(pp) = e(pp), 



PVY(w) = PO = V=^' (/,/,) = 9'. ■ 



14. Similarly, if we examine the function 



2 Fill Vpi^ip = ^4>ipSaip - pSaitpip, 

 we get for the second 2 



V'p'S,Sui<j>ip = 4(</)iai + </>2aj + "^sOs), 

 expressing p in the trinomial form. So that 



V*2 Vai Vp<jiip = - 2 [</)ini + ^,02 + <^3a3] = - 0. 

 V= [^ (pp) + « (pp)] = 0, 



where u) (pp) = %VaiVp4>ip, so (■ip + uj)(pp) is harmonic. 

 V'l// (pjo) = - is" an invariant, and 



Pw (pp) = w' (pp). PV-w (pp) = - P6 = V-oj'{pp) = - 6*'. 



15. As another example, take 2Fai</>ip, 



2ra'i^'ip = PSra,<^,p, 

 so again ?i = 1, 



V2Fai<^ip = 2zFai<^i* + %jVa.'i>;_j + Mra,4>Jc=p, 



E.I. A. PBGC, VOL. XXX., SECT. A. ("IG 1 



