Roche — The Quadratic Vector Function. 95 



3. Using the ordinary definition of ^r, ^ FA;it = F<I>'X<t>';ii, we get for 

 our function easily 



^ = ki\px + ki^\pi + k-i'xPs + hhxiz + hhx^^ + ^i^iX"^ 

 and X = ^, Xi + AsX2 + hxz- 



4. We may get these results as follows : — 



ai2 = 0iXi2 + '>ni(j)2(t>r\ «2i = ^aXis + wiz^i^r' {"v. § 2a ahove), 

 and from § 1 we have, on substitution for aij and a^ in Ai, 



m = S^i^mi + 'Siki'kilipiXii + mi^,"^!"'] + 5 ^z'/^i [1^2X12 "'' '"^2^i^2~^] 



+ k,k2h[(j)iX23 + ^2X31 + ^3X12]- 

 Grouping terms, we get on reduction 



m = (ki(j)i + k2<j)2 + k^rpa) (ki'\pi + k^'xp, + ks'^ps + kik^xn + khxis + ^'3^1X31) = *^^! 

 as above. 



5. Writing this result 



m{ki<j)i + fe^2 + ^'3^3)"^ = SZ-fi/zi + "^kjusxzi = S^i'wJ-i^r' + 2/^2^3 Y2s, 



and expressing that the invariants of both sides' are equal, we get another 

 batch of identities on comparing the coefficients of ^'s, and incidentally can 

 form the symbolic cubic of xn- We can write down at once 



m of nii(jtr^ is m^, m! of vii<pf^ is niimi", ni" of m,(/)j"' is m/, 

 and 



fty of mitpf'^ and mjcpf^ is mjaji, 



we shall denote by at^y), ai^yyi and x*®) the a-invariants and X'f'iiictions 

 of mi(f>f^ and Xv ^^^ the symmetrical invariant of these and mjipp by (^y(,y). 



(2A;i'm, + 'S.{ki^k^ait + ki^k^a^C) + kjije^di^^'^ = 



%ki^mi^ + liki^^mis + kik,[%k2%^d2m^)] + hhl'S.k^'h^dnin}] 



+ h\ [S^'l^^2'^12(31)] + hVC-^h^ [dii3 + (Zci2)(23)(31)] + h' [S^'l^2''^3(^(12)(23)l] 



+ As' [5AiA2''fcj(^(i2)(23)2] + h^ [2Ai't./A-3(Z(i2)(23)3] + 22 [h'h'mian + h%^ai2mi] 

 + 22 [^i^A-jaifiz) + k/kitt^uii] + 2'S,ki%kianir. + ^[ki*h''a(nu + h-^k'a^un] 

 + k,%%''S,a^is^i + 2/;i'fe^A;3a(i2)(23) + 2Aife'yfc3''ff(23)(i2). (A5) 



[2Ai^mi + 2 (ki%2ai2 + k^k^a2\) + £^123] \kim"i + kim'\ + kzm"^ = 



2/fc2^A;sC2(23) + ^ki'h^Cs^ir, + ^kih^kiCi^t^^zu + 'S,kiHniin"i + %ki^ki^m\.i + %kiki^c^t 

 + Mi%k,cn23h (B5) 



where c^ is the symmetrical 5-in variant of -m-i^f' and m2^2"' and c,.j;j the 

 corresponding function for mi^r' and X23 (v. Ei). 



