300 



Profs. W. E. Ayrton and J. Perry. [Feb. 14, 



vol. i, §§ 588-608, the general behaviour of wires subjected to forces 

 at their ends is investigated, and we might have obtained equations 

 (1) and (2) from the general result of that investigation given in 

 Article 607, but we have preferred to deduce these two equations from 

 first principles. 



The general expression worked out by M. de St. Yenant for A 

 for a prism of rectangular section involves an infinite series, and 

 its use would give rise to mathematical expressions of an unwieldy 

 character in an investigation like the present. We have, there- 

 fore, decided to consider our strip of which the new spring is to 

 be made as having an elliptic section. "We may mention, how- 

 ever, for the benefit of students, that an examination of the expres- 

 sion given by Thomson and Tait, "Nat. Phil.," vol. i, for the tor- 

 sional rigidity of a prism of rectangular section has led us, we 

 believe, to detect two errors in it ; one is a misprint of nab 2 for nab s , 



- ) ought to be ( - ) — 

 it) a \irj a 



A friend of ours who has been kind enough, at our request, to check 



the investigation agrees with us in thinking that there are two errors, 



but considers that the second is not in the coefficient but in the fact 



that \ has been used in the formula in place of a. We still are of 



opinion, however, that the error is in the coefficient, because when we 



employ our two corrections, we obtain — 



where a and b are the length and breadth of the strip, and this formula 

 we find agrees with the experimental results of M. de St. Venant, 

 and also with the calculated numbers given in Thomson and Tait's 

 " Natural Philosophy,"* § 709, for the torsional rigidity of a square 

 shaft ; whereas neither the formula for A given in Thomson and Tait's 

 book, nor the formula as corrected by our friend, will do this.* 



Reverting to the elliptic section, if the principal semi- diameters 

 of this ellipse are b, measured in the osculating plane of the spiral, 

 and a measured perpendicular to this plane, the values of the two 

 rigidities are — 



A=^-landB: M 



a~ + lr 4 



where N is the modulus of rigidity of the material, and E is Young's 

 modulus. 



Hence substituting, equation (1) becomes — 



* Since this paper was pi'esented Mr. R. T. GTlazebrook has been kind enough to 

 go through the investigation, and he confirms our correction of the formula as 

 given in Thomson and Tait's "Natural Philosophy." 



