VECTOR ANALYSIS 97 



A = aeon + bcoi 2 + 



B = aco 2 i + bco 22 + cco 23 , 

 C = acosi + bco 32 + cco 3 3, 



and 



w = aS.A + bS.B + cS.C. 



4.617 If co is the general linear vector operator and co' its conjugate, 



coR = Rco', 

 co'R = Rco 



4.620 The symmetrical or self-conjugate linear vector operator has three 

 mutually perpendicular axes. If these be taken along i, j, k, 



co = iS.cOii +. jS.co 2 j + kS.cosk, 

 where coi, co 2 , co 3 are scalar quantities, the principal values of co. 



4.621 Referred to any system of three mutually perpendicular unit vectors, 

 a, b, c, the self-conjugate operator, co, is determined by the three vectors (4.616) : 



A = coa = aeon + bcoi2 + ccoi 3 , 

 B = cob = aco 2 i + bco 22 + cco 23 , 



C = coc = aco 3 i + bco 32 + cco 33> 

 where 



C0fcfc = Ukh, 



co = a5.A + bS.B + cS.C. 



4.622 If n is one of the principal values, coi, co 2 , co 3 , these are given by the roots 

 of the cubic, 



n 3 - w 2 (^.Aa + S.Bb + S.Cc) -}- n(S.&VEC + 5.bFCA + S.cVAB) 



- 6*.AFBC = o. 



4.623 In transforming from one to another system of rectangular axes 

 the following are invariant: 



S.Aa + S.Bb + S.Cc = co x -f co 2 + co 3 . 

 ,5aFBC -f- S.bFCA + 5.cFAB = co 2 co 3 + co 3 coi + C0ico 2 . 



.5.AFBC = coico 2 co 3 . 

 4.624 



COi + C0 2 + C0 3 = COii + C0 22 + C0 33 , 



C0 2 C0 3 + C0 3 COi + COiC0 2 = C0 22 C0 33 + C0 33 COn -f- COnC0 22 C0 2 23 C0 2 3i + C0 2 i 2 , 



COiC0 2 C0 3 = COiiC0 22 C0 33 + 2C0 23 C0 3 iCOi 2 - COnC0 2 23 - C0 22 C0 2 3 i C0 33 C0 2 i2. 



4.625 The principal axes of the self-conjugate operator, co, are those of the 



quadric : 



con* 2 + co 22 / + co 33 s 2 + 2^2^ + 2Uuzx + 2Wi&y = const., 



where x, y, z are rectangular axes in the direction of a, b, c respectively. 



