1893.] On the Whirling and Vibration of Shafts. 365 



We are now able to pass from any particular redundant generating 

 function to an equivalent generating function which involves nl 

 undetermined quantities. Assuming these quantities at pleasure, we 

 obtain a number of different algebraic products, each of which may 

 have its own meaning in arithmetic, and thus the number of arith- 

 metical correspondences obtainable is subject to no finite limit. 



This portion of the theory is given at length in the paper, with 

 illustrative examples. 



Incidentally interesting results are obtained in the fields of special 

 and general determinant theory. The special determinant, which 

 presents itseJf for examination, provisionally termed " inversely 

 symmetric," is such that the constituents symmetrically placed in 

 respect to the principal axis have, each pair, a product unity, whilst 

 the constituents on the principal axis itself are all of them equal to 

 unity. The determinant possesses many elegant properties which are 

 of importance to the principal investigation of the paper. The 

 theorems concerning the general determinant are connected entirely 

 with the co-axial minors. 



I find that the general determinant of even order, greater than two, 

 is expressible in precisely two ways as an irrational function of its 

 co-axial minors, whilst no determinant of uneven order is so expres- 

 sible at all. 



Of order superior to 3, it is not possible to assume arbitrary values 

 for the determinant itself and all of its co-axial minors. In fact of 

 order n the values assumed must satisfy 



2 n~ + n 2 



conditions, but, these conditions being satisfied, the determinant can 

 be constructed so as to involve nl undetermined quantities. 



IV. " On the Whirling and Vibration of Shafts." By STANLEY 

 DUNKERLEY, M.Sc., Berkeley Fellow of the Owens College, 

 Manchester. Communicated by OSBORNE REYNOLDS, F.K.S. 

 Received July 13, 1893. 



(Abstract.) 



It is well known that every shaft, however nearly balanced, when 

 driven at a particular speed bends, and, unless the amount of deflec- 

 tion be limited, might even break, although at higher speeds the shaft 

 again runs true. The particular or " critical " speed depends on the 

 manner in which the shaft is supported, its size and modulus of 

 elasticity, and the size, weights and positions, of any pulleys it 

 carries. 



