82 Proceedings of the Royal Irish Academy. 



where Ui{ijl, Y') denotes the vahie of ni(;u) when each y, v, is replaced by the 

 corresponding Y', dY'/dt. This equation is obtainable directly, and by its 

 aid the normal coordinates which constitute the coefficients of the separate 

 terms in the series of residues which is equivalent to the right-hand member 

 of (16) may be obtained in the usual manner by integration. 



The analogous equation applicable to the problem of Art. 12 differs only 

 in having in the first term dx in the integrand replaced by rdr, and in having 

 the second and third terms multiplied by a, b respectively. 



The forms of the analogous equations in the corresponding problems of 

 Heat-Conduction should be obvious. 



Again, in the problem of Art. 13, the form of the right-hand member of (46) 

 suggests the equation analogous to (50), which may be written in the form 



a{c-f+ c\v + v') + gvv') <j,<p'dx + aU, {v, Y'„)^„/An(v) + 6ri,(v, 7'j)^*/5„(v) = 0, 



(51) 

 in which v may be either vi or v^, and v either v, or v'i. This equation, also, 

 may be verified directly. 



And, to obtain the corresponding equation for the problem of Art. 14, we 

 again simply in (51) replace in the first term dx by rdr, and multiply the 

 second and third terms by a, h respectively. 



In the case of each of the si.K problems alluded to, I have verified that 

 the solutions obtained by the two methods agree. It must be borne in mind 

 that, in the Fourier solutions given above for the problems of Arts. 13, 14, 

 each term occurs twice. 



