NATURAL PHILOSOPHY. 263 



The latter surface (No. 2) is generated by a straight line resting on the 

 two profiles, and moving parallel to the vertical plane which passes through 

 the axis of the road. It also is a hyperbolic paraboloid, though a different 

 one from the former. 



The French engineers adopt the latter hypothesis. "We have seen, how- 

 ever, that the former is the more probable one. 



But fortunately the difference is really very slight; for a very small 

 change in the latter hypothesis will make its result identical with that of 

 the former. Conceive the straight line Avhich rests on the two profiles to 

 move on them in such a way as always to divide them proportionally. The 

 surface thus generated (No. 3) is identical with No. 1. 



This last conception is also more probably correct than No. 2 even if 

 we suppose the engineer to consider longitudinal straightncss since he is 

 more likely to extend his imagination from all parts of one profile to all 

 parts of the other, than in lines perpendicular to the profile on which he 

 stands. 



II. We shall therefore now proceed to investigate the content of a solid, 

 bounded on one face by a warped surface, generated on the first hypothesis 

 the other faces being planes. 



We will take the case of an Excavation ; that of an embankment being 

 the same inverted. 



We will begin by considering the sides of the excavation to be vertical ; 

 and will afterwards discuss the more usual form. 



[The mathematical discussion is here omitted. The result of the integra- 

 tion shows the required volume of the solid to be expressed by this simple 

 and symmetrical formula : 



i C(* + V>'} (a + *) + (*' + ^ (<J + *')] 



In it / is the length of the solid ; b and b f are the breadths of its two ends ; 

 g and h the side depths at one end ; and g' and // those at the other end. 



It is next shown that the expression for the volume obtained by treating 

 the solid as a prismoid can be transformed into the above formula.] 



We thus arrive at the practical conclusion that the familiar " Prismoidal 

 formula " can be applied with perfect accuracy to such solids, having one of their 

 faces a warped surface, generated as in our first, or third, hypothesis. 



The general adoption of these views would enable beginners to economize 

 much time and labor, since they would no longer feel themselves under the 

 necessity of taking their cross sections so near together that the ground 

 between them should be approximately plane, but could take them as far 

 apart as the ground " varied uniformly," no matter how much or how far 

 that might be. 



MOLECULAR AGGREGATION OF CRYSTALLINE SOLIDS. 



Robert Mallett, the well-known English physicist, in a recent publication, 

 affirms that in the " molecular aggregation of crystalline solids, the crystals 

 always arrange and group themselves with their principal axes in lines per- 



