macfarlane: algebra of physics 399 



But there is a complementary part which may be denoted by 

 Sin £ 2 ; and it is 



2 Sin {AB + AC + AD] 



+ 2 Sin {BC + BD} 

 + 2 Sin CD 

 This vector part by reading the columns is seen to be 4 times the 

 directed area of the triangles AB, (A +B) C, {A+B + C) D; and 

 the resultant is 4 times the maximum projection of these areas. 

 The third power of the complex is formed after the same alge- 

 braic principle, giving, for a trinomial, 



£ 3 = A 3 + B s + C 3 + 3{A 2 B + A 2 C + B 2 C) 

 + 3{A£ 2 + AC 2 + BC 2 } + 6ABC. 

 The conjugate third power is much more complex, viz: 



£££ = A* +B 3 + C 3 



+ A 2 B + A 2 C + B 2 C + A£ 2 + AC 2 + BC 2 

 + 5A 2 + CA 2 + OB 2 + £ 2 A + C 2 A + C 2 B 

 + ABA + A(M + 5(75 + BAB + CMC + CBC 

 + ^£C + ACJ3 + BCA + £AC + CAB + C54 

 The last six terms form a determinant in vectors. 



Cyclic complex of vectors. In a cyclic com- 

 plex the vectors have a common point of 

 application, and the order is determined by 

 the order in which the free extremities occur 

 in a cycle. Let R denote such a complex; 

 for example R = A + B + C (fig. 9). The 

 square is formed from the algebraic square 

 by inserting cyclic order : thus F"/q 9. 



R 2 = A 2 + B 2 + C 2 



+ 2 AB + 2BC + 2 CA. 

 This differs from the corresponding line complex in having CA 

 instead of AC. Hence 



Cos R 2 = A 2 + B 2 + C 2 



+ 2 Cos AB + 2 Cos £C + 2 Cos CA. 

 So far as unit is concerned, these terms break up into the three 

 pairs 



A 2 + 2 Cos AB, B 2 + 2 Cos £C, C 2 + 2 Cos CA. 



