Vibrations in Solid and Hollow Cylinders. 335 



customary in most elastic problems — the differential equation 

 for the dilatation, Pochhammer obtains a differential equation 

 for the quantity 



du dw\ 



1 /du 



2\dz~ 



u and w being the displacements parallel to the radius r and 

 the axis z, and uses this quantity as a stepping-stone to the 

 values of u and w. On the other hand, I succeeded in 

 separating u and w so as to obtain at once two differential 

 equations, in one of which u appeared alone with the dilatation, 

 while the other contained only w and the dilatation (see (28) 

 and (29) later). 



§ 4. There are two other points in Mr. Love's Treatise to 

 which I should like to refer. In his art. 263 he substitutes 

 the term extensional for longitudinal, adding in explanation, 

 " The vibrations here considered are the ' longitudinal ' vibra- 

 tions of Lord Kayleigh's Theory of Sound. We have 

 described them as l extensional/ to avoid the suggestion that 

 there is no lateral motion of the parts of the rod." 



I am altogether in sympathy with the object which Mr. 

 Love has in view (1 expressed myself somewhat strongly on 

 the point in (A) p. 296, and (C) pp. 351-2), but I doubt the 

 wisdom of attempting to displace a term so generally adopted 

 as longitudinal. 



In the second matter I regret to find myself at variance 

 with Mr. Love. Referring to transverse vibrations in a rod, 

 he says on p. 124 of his vol. ii., " the boundary conditions at 

 free ends cannot be satisfied exactly ... as they can in the . . . 

 extensional (longitudinal) modes/'' In reality, however, the 

 boundary equations at a free end are not exactly satisfied in 

 the case of longitudinal vibrations either by Pochhammer's 

 solution or my own. The slip may be a purely verbal one on 

 Mr. Love's part, but his readers might be led to accept the 

 statement as accurate owing to a slight error in the expression 



for the shearing stress zr near the top of Mr. Love's p. 120. 

 We find there 



«r= 2^{ 7 A j- J (/c'r) + . . . }e«<7*+?0, 



where i= ^ — ±. 



The correct expression (compare Mr. Love's second boundary 

 equation on p. 118) is 



^ = ^{2 7 A ^-J (ft'r) + . . .}«to # +*>. 



2 A2 



