VECTOE ANALYSIS. 37 



at the point considered above the average of its values at six points 

 at the following vector distances: ai, ai, aj, aj, ak, dk. Since 

 the directions of i, j, and k are immaterial (provided that they are 

 at right angles to each other), the excess of the value of u at the 

 central point above its average value in a spherical surface of radius 

 a constructed about that point as the center will be represented by 

 the same expression, %a?Wu. 



Precisely the same is true of a vector function, if it is understood 

 that the additions and subtractions implied in the terms average and 

 excess are geometrical additions and subtractions. 



Maxwell has called V.Vu the concentration of u, whether u is 

 scalar or vector. We may call V.Vu (or V.Vo>), which is proportioned 

 to the excess of the average value of the function in an infinitesimal 

 spherical surface above the value at the center, the dispersion of 

 u (or co). 



Transformation of Definite Integrals. 



73. From the equations of No. 65, with the principles of integration 

 of Nos. 57, 59, and 60, we may deduce various transformations of 

 definite integrals, which are entirely analogous to those known in the 

 scalar calculus under the name of integration by parts. The follow- 

 ing formulae (like those of Nos. 57, 59, and 60) are written for the 

 case of continuous values of the quantities (scalar and vector) to which 

 the signs V, V., and Vx are applied. It is left to the student to 

 complete the formulae for cases of discontinuity in these values. The 

 manner in which this is to be done may in each case be inferred from 

 the nature of the formula itself. The most important discontinuities 

 of scalars are those which occur at surfaces : in the case of vectors 

 discontinuities at surfaces, at lines, and at points, should be considered. 



74. From equation (3) we obtain 



fV(tu).dp = tf'u" - t'u' =fuVt.dp +ftVu.dp, 



where the accents distinguish the quantities relating to the limits of 

 the line-integrals. We are thus able to reduce a line-integral of the 

 form fuVt.dp to the form ftVu.dp with quantities free from the 

 sign of integration. 



75. From equation (5) we obtain 



where, as elsewhere in these equations, the line-integral relates to the 

 boundary of the surface-integral. 



From this, by substitution of Vt for o>, we may derive as a 

 particular case 



p = -ftVu.dp. 



