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XXVIII. On the Coefficients in certain Series of 

 BesseVs Functions. 



To the Editors of the Philosophical Magazine. 

 Gentlemen, — 



IN your December number, pp. 947-948, Mr. P. W. 

 Bridgman refers to " an apparent oversight in the 

 usual method o£ determining the coefficients in certain series 

 of BessePs functions," which he illustrates by reference to a 

 paper by me read to the Cambridge Philosophical Society in 

 1887 and printed in their Transactions, vol. xiv. Owing 

 probably to lack of familiarity with recent pure mathematical 

 terminology, I am uncertain whether I fully understand 

 Mr. Bridgman's conclusion. 



He seems to admit the accuracy of the differential equation 



d 2 v 1 civ __ v d 2 v _ () 

 d? rdr ? cW ' 



and of the solution I gave — novel I think at the time — 

 v = 2E sinh kzJifkr) in my notation. What I understand him 

 to question is the applicability of this solution to the torsional 

 problem I supposed it to apply to, viz. the case where the 



shearing stress zO^ndvjdz is given over the end z=l of a 

 right circular cylinder of radius a (the end z = being fixed) 

 as an arbitrary function f{r) of the axial distance. By 



taking J 2 (/ta) — I was able to satisfy the condition r# = 

 over r = a, and " following the usual methods" — i. e. making 



use of the fact that I rJi(kr)J i (k J )') dr vanishes if k and Id 



Jo 



are two different roots of J 2 (/k.-a) = — I supposed the ex- 

 pansion of f(r) in J\'s possible over the end z = l, thus 

 determining the constants E. In this I believed myself to 

 follow the example of the ordinary text-books. 



Mr. Bridgman says my solution is clearly in error, for 

 when applied to the ordinary case f(r) = r the determining 

 equation makes the constants E all zero, and " gives the 

 absurd result that a cylinder experiences no torsion under a 

 torsional force varying as the distance from the axis." If, 

 however, the E's all vanish, then the stresses vanish as well 

 as the strains ; thus the result on the physical side is simply 

 that a cylinder under no stress is under no strain, a result 

 which to me presents no absurdity. The real fact is that it 

 never occurred to me that my new solution should include 

 the simple ordinary case of the text-books v = rz. For this 



