112 Proceedings of the Royal Irish Academy. 



the lowest numerical value is that for which r is unity. For any finite 

 value of /, the value of the real part of p, or p + vP + vn"^, and therefore, 

 a fortiori, that of ]?, is numerically greater than in this ease. This may be 

 proved as follows. 



Considering, firstly, the real values of jp', if we write, as in Art. 15, 

 S = F + iQ, and integrate equation (20) from - a to y, we obtain 



v{PdQldy - QdPldy] = //3 ['' y{P' + Q')cly. (59) 



Since P' + Q' is not changed by changing the sign of y, the right-hand 

 member is essentially of opposite sign to I between ± a, except that it is 

 zero at ± a ; consequently so is the left-hand member. Hence we may 

 infer that between every two real zeros of P, provided y = ± a be not one 

 of them, there lies one zero of Q, and between every two of Q, with the 

 same exception, there lies one of P. From the forms assumed by (16), (17), 

 when p is real, evidently of the two functions P, Q one is odd, the other 

 even ; we will choose P even, Q odd. Then Q vanishes when y is zero ; 

 it seems to be the case that for given values of I, n, in the disturbance 

 which has the smallest numerical value of 'p, with this exception, neither 

 P nor Q can vanish for any other values than ± a ; if, however, this be 

 not the case, we have just proved that as y increases from zero it will reach 

 a zero of P before another of Q ; and thus in any event a zero of P not 

 later than another of Q. When y is zero it results from (59) that if P be 

 taken positive as it may, dQjdy is of sign opposite to that of /, and thus as 

 y increases from zero, Q also has its sign opposite to that of I. Consequently 

 in the equation which (16) now becomes, viz. : 



vdP/df = pP - (5lyQ, (60) 



the first term on the right is negative, and the second positive. Thus the 

 variation of P, until it becomes zero, is analogous to that of the displacement 

 of a particle v subject to a force to a fixed point, which force is less than 

 the displacement multiplied by - p' ; and the particle starts from rest. The 

 time which elapses until the particle reaches the centre is greater than 



IT f — iy\2 



Therefore, m the problem which is the subject of discussion, the value 

 of y for which P first vanishes — a value which, as we have seen, cannot 

 exceed a — is greater than 



TT / — v\a 



i.e., -/> vTT^I^a^ 

 ^ \ P J 



Thus the result is established for real values of p)'- 



