168 General Analytical Theorems [CH. vn 



If we write also 



~ 



-47T 1 



the resultant force parallel to the axis of Y will be 



Y = - (^ + mPyy + nP yz ) dS, 



and there is a similar value for Z. The action is therefore the same (cf. 

 159) as if there was a system of stresses of components 



p p p p p p 



J xx > * yy > J zzi J yz> * zx j -* xy j 



given by the above equations : i.e. these may be regarded as the stresses of 

 the medium. 



194. It remains to investigate the couples on the system inside S. If 

 L, M, N are the moments of the resultant couple about the axes of x, y, z, 

 we have 



L = jjfp (yZ- zY) dxdydz + JS (jo- (yZ - zY) dS 



8 2 F\ / 8F 8F 



, , 



+ - 6 + m ^ + n -5- ) (y -= -- z -=- 



STT ]] \ dx dy dz / \ 3^ ty 



F 9F 



Now 



f[[dV d ( 3F 8F\ , 



* 1 1 1 "5~~ ^r V ^r -- ^ ^~~ ^^ y 



./ JJ 3a; 8a? V^ 82: dy J 



3F/ 8F 8F\ . ff;8F/ 8F 8F\ , 

 ^- (y -^--z^-} dS- III ^- [y-= -- ir-s-joS, 

 8^ V^ 8^ 9/ JJ a V 82: S/y 



so that 



,,. 8 / 8F 8F\ 8F 8 / 8F 8F\ 



L = 1 1 H ( ?/ ^r- - z ) + I v -5 -2-5 



8^ 8y/ 



, . , 8F 8F 



+ dzdz [y ^ 



1 v [[fjdV 8F 8F \/ 3V 8F 

 - ^ S 1 1 U - + m ^- + n -r- ) { y -5 * -5- 



8?r JJV 9 9y 8^/V 82- 8y 



8F^ dV t 8F\ / 8F 8F\ 



^- +m^- +n^- }(y-~ z^-) 



dx dy dz J V dz dy J 



