SURFACES OF DISCONTINUITY 601 



(and much the larger) part of the varied surface. This con- 

 tributes to the variation of SaDs the amount S^i^i + C2) 8NDs. 

 But around this large portion of the varied surface there 

 will be a narrow "rim" bounded by the varied position of 

 the perimeter and by the line obtained by projecting the perim- 

 eter of the original surface on the varied position of the 

 surface. Between an element Dl of the varied perimeter and 

 the corresponding element of this projection there is a distance 

 8T, where 8T is the "component of the motion of this element 

 which lies in the surface and is perpendicular to the perimeter." 

 Thus this rim can be conceived as consisting of elements of area 

 dTDl, and we obtain in this way the second contribution to 

 dJ'aDs, viz., SabTDl; thus we see that 



/ cbDs = / o-(ci -f C2)8NDs + / adTDl 



f/da- da \ ^ 



J \oa>i OC02 / 



These two (condensed) terms of the original condition of 

 equilibrium [600], viz. — SyWv + fabDs, are the two which 

 offer the most trouble in being transformed into a convenient 

 form. When we replace them in [600] by the expressions just 

 obtained we can rewrite the condition [600] in the form 



ft Wyf + ft SDrj' 



+ fW + gz.') 8Dm,'^ + /(mi" + gzx") 8Drm"^ 



+ /(mi" + gzi') ^Dm,' 



+ /(m/ + gz2') bDm-r + finz" + gz-n 8Drm"^ 



+ finz' + gz^') dDm2^ 



dx' dy' 



fj, 8x" + „ , 

 dx dy 



+ /{k+S^^" + S-^^"+S^.'7o»' 



