56 VECTOK ANALYSIS. 



118. Since all the antecedents or all the consequents in any dyadic 

 may be expressed in parts of any three non-complanar vectors, and 

 since the sum of any number of dyads having the same antecedent 

 or the same consequent may be expressed by a single dyad, it follows 

 that any dyadic may be expressed as the sum of three dyads, and so, 

 that either the antecedents or the consequents shall be any desired 

 non-complanar vectors, but only in one way when either the ante- 

 cedents or the consequents are thus given. 



In particular, the dyadic 



aii+bij +cik 



which may for brevity be written 



is equal to 

 where 



y = ci + c'j+c"k, 

 andto iX+jfji + kv, 



where +ck 



'j +c'k 



119. By a similar process, the sum of three dyads may be reduced 

 to the sum of two dyads, whenever either the antecedents or the 

 consequents are complanar, and only in such cases. To prove the 

 latter point, let us suppose that in the dyadic 



neither the antecedents nor the consequents are complanar. The 

 vector 



is a linear function of p which will be parallel to a when p is perpen- 

 dicular to JUL and v, which will be parallel to ft when p is perpendicular 

 to v and X, and which will be parallel to y when p is perpendicular to 

 X and //. Hence, the function may be given any value whatever by 

 giving the proper value to p. This would evidently not be the case 

 with the sum of two dyads. Hence, by No. 108, this dyadic cannot be 

 equal to the sum of two dyads. 



