Camh'idge Philosophical Society. 377 



take place ; and by lateral, that in virtue of which lateral or transverse 

 vibrations take place. 



The forces due to the several terms of h are obtained by means 

 of the following simple considerations. 



Let AB be any line in a perfectly uniform medium, and conceive 

 the medium to be divided into elementary slices by planes perpen- 

 dicular to AB ; let OM{=x) be the distance of any slice PP' from 

 any particular point O of AB, and suppose this slice to suffer a dis- 

 placement equal to — cx'^ (c being a constant) in the direction OAB, 



and the other slices to be similarly displaced. Then it is evident 

 that the medium suffers by these displacements a uniformly increasing 

 expansion in the direction OB, and a uniformly increasing condensa- 

 tion in the direction OA ; the rate of increase both of the expansion 

 and condensation being c. Now in all known substances, whether 

 solid, fluid, or gaseous, a disarrangement of this kind would bring 

 into play on the slice O a force along the line AB proportional to 

 the rate of increase c, i. e. a force Ac, A being a constant depending 

 upon what we may call the direct elasticity of the substance. 



Again, suppose that the slice PP' receives a displacement — cx'^ 



in the direction OC perpendicular to AB, and the other slices similar 

 displacements. Then the line AB will become curved into a para- 

 bola A'OB', and all the lines of the medium parallel to AB will be 



similarly curved, the radius of curvature being equal to — and per- 

 pendicular to AB. Now in all known substances* a disarrangement 

 of this kind would bring into play upon the slice O a force in the 

 direction OC proportional to the curvature c, i. e. a force Be depend- 

 ing upon what we may call the lateral elasticity of the substance. 



Lastly, suppose that M.V^=y, and that the point P of the medium 

 receives a displacement cxy parallel to AB, and the other points 

 similar displacements. Then the slice PP' will, in consequence of 

 this kind of displacement, turn through an angle tan~'(ca;) into the 

 dotted position, and the other slices will suffer similar rotations, 

 those on the other side of O, such as QQ', turning the opposite way. 

 Now it is easy to see that a disarrangement of this kind produces a 

 uniformly increasing expansion in the direction OC, and a uniformly 

 increasing condensation in the direction OC, the rate of increase 

 both of the expansion and condensation being c. But the expansion 

 and condensation here described are quite different from that pre- 

 viously noticed ; since it is produced, not by displacements parallel 

 to C'C, but by lateral displacements, i. e. perpendicular to C'C. On 

 this account all that we can assert without further investigation is, 

 that the force brought into play upon an element at O by this dis- 

 arrangement acts along the line C'C, and is proportional to c, i. e. 

 equal to Cc, where C is some constant evidently depending in some 

 way both upon the direct and lateral elasticity of the medium. 



* Fluids and gases possess lateral elasticity as well as solids, only in a 

 comparatively feeble degree. 



