28 VECTOR ANALYSIS. 



It results from the principle stated in No. 35, that any vector 

 equation of the first degree with respect to p may be reduced to the 



But ap = d\'(\.p) + afJL'(im.p) + av'(v.p), 



where X', p! t v represent, as before, the reciprocals of X, /*, v. By 

 substitution of these values the equation is reduced to the form of 

 equation (1), which may therefore be regarded as the most general 

 form of a vector equation of the first degree with respect to p. 



41. Relations between two normal systems of unit vectors. If 

 i, j, k, and i', j', k' are two normal systems of unit vectors, we have 



a) 



and 



(2) 



(See equation (8) of No. 38.) 



The nine coefficients in these equations are evidently the cosines 

 of the nine angles made by a vector of one system with a vector of 

 the other system. The principal relations of these cosines are easily 

 deduced. By direct multiplication of each of the preceding equations 

 with itself, we obtain six equations of the type 



By direct multiplication of equations (1) with each other, and of 

 equations (2) with each other, we obtain six of the type 



By skew multiplication of equations (1) with each other, we obtain 

 three of the type 



Comparing these three equations with the original three, we obtain 



nine of the type 



'). (5) 



Finally, if we equate the scalar product of the three right hand 

 members of (1) with that of the three left hand members, we obtain 



^./)^!. (6) 



Equations (1) and (2) (if the expressions in the parentheses are 

 supposed replaced by numerical values) represent the linear relations 



