248 Proceedings of the Boyal Irish Academy. 



The portion of the differential coefficient arising from the second term of 

 the integral on the right and the corresponding term in J'^ (Xp), is asymp- 

 totically, for large values of A, the term in a Fourier expansion of x(r). 

 Such an expansion is, under the conditions stated, uniformly convergent, save 

 near the boundaries and the discontinuities in x I S'l^d its term is of order 

 not exceeding X"\ 



The portion involving the first term in the integral is of still smaller 

 order. 



Superficially, the boundary terms in (72) and its analogue seem to give 

 rise to a portion which is finite ; in reality, it is of order not exceeding X-^ ; 

 for its numerator is of the same order as 



(73) 

 The most important terms cancel, at the lower limit automatically, and at 

 the upper, in virtue of the approximate form of the equation determining X. 



Thus the terms in the development of x are at most of order A"'. 



The part of ^ which depends on x is obtained by multiplying each term 

 in the series for x by (cA)"' sin \ct. And, as before, this part gives a series 

 which, when differentiated term by term, once or twice, is uniformly 

 convergent, save, in the latter case, near certain values corresponding to 

 discontinuities in x pi'opagated and reflected. 



Consequently, it may be so differentiated, and therefore satisfies the 

 differential and boundary equations. 



And the two parts of together satisfy the initial conditions. 



Slight alterations in verbiage render this argument applicable to the case 

 in which \p'\ x i^^^y have integrable infinities. When the series for \p is 

 differentiated twice, term by term, we can still assert that the new series is 

 uniformly convergent except near certain values, which now include the 

 infinities of -ip", though the statement above as to the order of the terms no 

 longer holds. Multiplication by cos Xd shifts, just as above, the values for 

 which failure in uniformity occurs. The series for the corresponding parts of 

 (p, d(j)/dt, d(f)/dr are uniformly convergent everywhere in the range, (by 

 Dirichlet's test). Similarly for the portion of f which depends on x- 



The substance of the remarks of the preceding paragraph applies to 

 Part I., Art. 19. 



