38 Mr. Hopkins on the Structure of the 



the portion A B A' B' (No. 12) being supposed to be acted on by an elevatory force 



C 



of greater intensity than the remaining portion. Let F denote the force acting on 

 this latter portion, and let that on the former be equal to the sum of F and an 

 additional force (/). If F alone acted on the whole area, we should have the case 

 of fig. 1 1 ; and the question is, how would the fissures of that case be modified by 

 the contemporaneous action of/, in addition to F, on the oval AB A' B' ? Now if 

 / acted alone on that portion, and F did not act at all, either in that or the other 

 part of the district, we should have the case of fig. 1 0, and therefore it is obvious 

 that the modification produced by it would be such as to make the lines of fig. 1 1 

 approximate more or less to those of fig. 10, in the manner represented in fig. 12, 

 the direction of a fissure through any point situated like P being intermediate 

 to those in which it would proceed through that point in the two preceding cases 

 respectively. 



This conclusion follows immediately also from the consideration that the direc- 

 tion of the longitudinal fissures at any point will be perpendicular to the direction 

 of greatest tension at that point^. Thus, if the force acted uniformly throughout 

 the whole space A C C, the direction of greatest tension at the point P would be 

 q q' perpendicular to the axis A A' ; and if the force acted on the oval alone, the 

 direction of greatest tension would be in some such direction as rr'. Consequently, 

 in the actual case proposed, it is obvious that the greatest tension will have a 

 direction p p', intermediate to the two former directions, and therefore also, the 

 direction of the fissure will be intermediate to those which would be formed in the 

 previous cases respectively. 



* This may admit of exceptions, but will be true whenever there is a greater tendency to form longi- 

 tudinal than transverse fissures. At certain points, not remote from the extremities of an elongated oval, 

 the tension along r r' may be a minimum ; in which case the direction of maximum tension will be per- 

 pendicular to r r', and the transverse would probably be formed first ; but in such case the longitudinal 

 one, if formed at all, must be perpendicular to the transverse one, and the conclusion of the text would 

 still hold true. 



