166 Sir William Thomson on the 



are thus found : and summation, after the manner of Fourier, 

 for different values of m, with different amplitudes and dif- 

 ferent epochs, gives every possible motion, deviating infinitely 

 little from the undisturbed motion in circular orbits. 



The simplest Subcase, that of i=.0, is curiously interesting. 

 For it .(50), (51), (52) give 



J|(g) _ —¥ (ma) - g. 



qJo(q) ma$ (ma) ? • \ / 



(54) 



and 



s/XmW + q 2 ) 



The successive roots of (53), regarded as a transcendental 

 equation in q, lie between the 1st, 3rd, 5th . . . roots of 

 J (q) = 0, in order of ascending values of q, and the next 

 greater roots of J f (q) at 0, coming nearer and nearer down to 

 the roots of J the greater they are. They are easily calcu- 

 lated by aid of Hansen's Tables of Bessel's functions J and 

 J l (which is equal to J f ) from q = to q = 20*. When ma 

 is a small fraction of unity, the second member of (53) is a 

 large number ; and even the smallest root exceeds by but a 

 small fraction the first root of J (^) = 0, which, according to 

 Hansen's Table, is ,2*4049, or, approximately enough for the 

 present, 2*4. In every case in which q is very large in com- 

 parison with ma, whether ma is small or not, (54) gives 



2coma . . , 



n= approximately. 



Now, going back to (6), we see that the summation of two 

 solutions to constitute waves propagated along the length of 

 the column gives : — 



r= —psm(nt-mz); r# = T + rcos (n£ — mz)-A 



k = iv cos (nt—mz); p— +vr cos (nt—mz). J ^ ' 



The velocity of propagation of these waves is n/m. Hence, 

 when q is large in comparison with ma, the velocity of longi- 

 tudinal waves is 2<oa/q, or 2/q of the translational velocity of 

 the surface of the core in its circular orbit. This is 1/1*2, or 

 § of the translational velocity, in the case of ma small, and the 

 mode corresponding to the smallest root of (53). A full ex- 

 amination of the internal motion of the core, as expressed by 

 (55), (13), (48), (15) is most interesting and instructive. It 

 must form a more developed communication to the Royal 

 Society. 



* Republished in Lommel's BesseUche Fundionen, Leipzig, 1868. 



