to the Vibrations of a Circular Membrane. 55 



in the interval.'* The property just established seems to 

 allow the proof to he completed. 



As regards the latter part of the statement, it may be 

 considered to be a consequence o£ the well known relation 



J» + lW=" J»(*)-Jn'(*) (2) 



When J n vanishes, J n+1 has the opposite sign to J/, both 

 these quantities being finite *. But at consecutive roots of J n , 

 J J must assume opposite signs, and so therefore must J n +i> 

 Accordingly the number of roots of J„ +1 in the interval 

 must be odd. 



The theorem required then follows readily. For the first 

 root of J n +i must lie between the first and second root of J n . 

 We have proved that it exceeds the first root. If it also 

 exceeded the second root, the interval would be destitute of 

 roots, contrary to what we have just seen. In like manner 

 the second root of J w+1 lies between the second and third 

 roots of J„, and so on. The roots of J n+ i separate those of 



The physical argument may easily be extended to show in 

 like manner that all the finite roots of JV (z) increase con- 

 tinually with n. For this purpose it is only necessary to 

 alter the boundary condition at r=l so as to make dw/dr = 

 instead of w = 0. The only difference in (1) is that z^ now 

 denotes a root of J«'(^) = 0. Mechanically the membrane is 

 fixed as before along = 0, # = /3, but all points on the circular 

 boundary are free to slide transversely. The required con- 

 clusion follows by the same argument as was applied 

 to J n . 



It is also true that there must be at least one root of 

 J' n +i between any two consecutive roots of J,/, but this is 

 not so easily proved as for the original functions. If we 

 differentiate (2) with respect to z and then eliminate J» 

 between the equation so obtained and the general differential 

 equation, viz. 



J/+Jj»' + (l-J)j»=0, .... (3) 



* If J n , J M +i could vanish together, the sequence formula, (8) below, 

 would require that every succeeding order vanish also. This of course 

 is impossible < if only because when n is great the lowest root of J ;i . is of 

 order of magnitude n. 



t I have since found in Whitaker's ' Modern Analysis,' § 152, another 

 proof of this proposition, attributed to Gegeribauer (1897). 



