VECTOR ANALYSIS. 61 



This equation expresses the well-known theorem that if the geometrical 

 sum of three vectors is zero, the magnitude of each vector is propor- 

 tional to the sine of the angle between the other two. It also indicates 

 the numerical coefficients by which one of three complanar vectors may 

 be expressed in parts of the other two. 



133. Def. If two dyadics 3> and are such that 



they are said to be homologous. 



If any number of dyadics are homologous to one another, and any 

 other dyadics are formed from them by the operations of taking 

 multiples, sums, differences, powers, reciprocals, or products, such 

 dyadics will be homologous to each other and to the original dyadics. 

 This requires demonstration only in regard to reciprocals. Now if 



That is, 3?' 1 is homologous to >P, if 3? is. 



134. If we call ty.Q- 1 or 3?- 1 . < I > the quotient of and 3>, we may 

 say that the rules of addition, subtraction, multiplication and division 

 of homologous dyadics are identical with those of arithmetic or 

 ordinary algebra, except that limitations analogous to those respecting 

 zero in algebra must be observed with respect to all incomplete 

 dyadics. 



It follows that the algebraic and higher analysis of homologous 

 dyadics is substantially identical with that of scalars. 



135. It is always possible to express a dyadic in three terms, so 

 that both the antecedents and the consequents shall be perpendicular 

 among themselves. ' 



To show this for any dyadic <, let us set 



p =$./>, 



p being a unit- vector, and consider the different values of p for all 

 possible directions of p. Let the direction of the unit vector i be so 

 determined that when p coincides with i, the value of p' shall be at 

 least as great as for any other direction of p. And let the direction 

 of the unit vector j be so determined that when p coincides with j, 

 the value of p' shall be at least as great as for any other direction of p 

 which is perpendicular to i. Let k have its usual position with 

 respect to i and j. It is evidently possible to express $ in the form 



We have therefore 



p'= 



and dp' { ai -f /3j + yk} . dp. 



