VECTOR ANALYSIS. 29 



which subsist between one vector of one system and the three vectors 

 of the other system. If we desire to express the similar relations 

 which subsist between two vectors of one system and two of the other, 

 we may take the skew products of equations (1) with equations (2), 

 after transposing all terms in the latter. This will afford nine 

 equations of the type 



(i.j')K-(i.k')j' = (k.i')j-(j.i')k. (7) 



We may divide an equation by an indeterminate direct factor. [MS. note by author.] 



CHAPTER II. 

 CONCERNING THE DIFFERENTIAL AND INTEGRAL CALCULUS OF VECTORS. 



42. Differentials of vectors. The differential of a vector is the 

 geometrical difference of two values of that vector which differ 

 infinitely little. It is itself a vector, and may make any angle with 

 the vector differentiated. It is expressed by the same sign (d) as the 

 differentials of ordinary analysis. 



With reference to any fixed axes, the components of the differential 

 of a vector are manifestly equal to the differentials of the components 

 of the vector, i.e., if a, /3, and y are fixed unit vectors, and 



dp = dx a + dy ft + dz y . 



43. Differential of a function of several variables. The differential 

 of a vector or scalar function of any number of vector or scalar 

 variables is evidently the sum (geometrical or algebraic, according as 

 the function is vector or scalar) of the differentials of the function 

 due to the separate variation of the several variables. 



44. Differential of a product. The differential of a product of any 

 kind due to the variation of a single factor is obtained by prefixing 

 the sign of differentiation to that factor in the product. This is 

 evidently true of differentials, since it will hold true even of finite 

 differences. 



45. From these principles we obtain the following identical 

 equations : 



(1) 



= dna-\- nda, (2) 



(3) 

 (4) 

 (5) 

 (6) 



