52 



where 



DEFINITE INTEGRALS. 



[INT. III. 



l g = (# s - ^_j) 2 + (y g - y s _-$ + (z s - 

 Hm ^-^_ 1= ^ = co 



and take s for the independent variable, the above integral 

 becomes 



cos (<fo, *) . ds, 



=/(/ J) ds = 

 in which there can be no ambiguity. 



In like manner if we divide the area of any surface S into 

 parts, multiply the area of each by the value of a point-function 

 f(p) at some point on that part, and sum for all the parts, the 



FIG. 9. 



limit approached by the sum, if any, as both the dimensions of the 

 parts approach zero, is called the surface -integral of f(p) over the 

 given portion of surface and denoted by 



jjf(p)dS. 



Here if we multiply f(p) not by the area of the part of the 

 surface S, but by its projection on the XF-plane, we reduce the 



surface-integral to the double integral 1 1 / (p) dS xy already 



treated, with the exception that the point-function depends not 

 only upon x and y but upon the surface If, as is generally the 

 case, several regions of the surface project upon the same part of 

 the XF-plane, the integral must be interpreted in an analogous 

 manner to that used in the case of the line-integral. If n is the 



