24 VECTOR ANALYSIS. 



28. This principle may be used in the transformation of more 

 complex products. It will be observed that its application will 

 always simultaneously eliminate, or introduce, two signs of skew 

 multiplication. 



The student will easily prove the following identical equations, 

 which, although of considerable importance, are here given principally 

 as exercises in the application of the preceding formulae. 



29. 

 30. 

 31. 



32. 

 33. 



34. [aX/3].[/3xy]x[yXa] = ( 



35. The student will also easily convince himself that a product 

 formed of any number of letters (representing vectors) combined in 

 any possible way by scalar, direct, and skew multiplications may be 

 reduced by the principles of Nos. 24 and 27 to a sum of products, 

 each of which consists of scalar factors of the forms a./3 and a./3xy> 

 with a single vector factor of the form a or ax/3, when the original 

 product is a vector. 



36. Elimination of scalars from vector equations. It has already 

 been observed that the elimination of vectors from equations of the 

 form 



is performed by the same rule as the eliminations of ordinary algebra. 

 (See No. 9.) But the elimination of scalars from such equations is at 

 least formally different. Since a single vector equation is the equi- 

 valent of three scalar equations, we must be able to deduce from such 

 an equation a scalar equation from which two of the scalars which 

 appear in the original vector equation have been eliminated. We 

 shall see how this may be done, if we consider the scalar equation 



which is derived from the above vector equation by direct multipli- 

 cation by a vector X. We may regard the original equation as the 

 equivalent of the three scalar equations obtained by substituting for 

 a, /?, y, S, etc., their X-, Y-, and Z-components. The second equation 

 would be derived from these by multiplying them respectively by 



