322 Prof. W. B. Morton and Mr. L. J. Close on Hertz's 



potential flog^ + r) dm. Hertz showed that 4z7TfiX (dis- 

 placement) is the resultant of the two vectors 



(1) -slope{2tf> + (l-2<r)x}, 



(2) (l — a)(j), parallel to the normal, 



where fx= rigidity, cr= Poisson's ratio. 



From this the strain-components can be obtained in any 

 chosen system of co-ordinates, e. g. polars (r </>) or cylindri- 

 cals (ot z <j>) . It was found most convenient to use a rather 

 unorthodox combination of these, the cylindrical components 

 being expanded in polar series. The advantage of the 

 ('sr z cf>) system lies in the relation to the fixed direction z. 

 The expressions for the components contain the potentials 



v, ^=6, — - . — -j?. These are expanded as zonal har- 



monic series in powers of r/a and a/r. 



Partial differentiation with respect to m is replaced by 



=r- and —. — 7^7, , and so ultimately power-series are obtained 

 ^r sm#B# J L 



whose coefficients involve P n , P w ', cos 6F n , cos#P n '. Having 

 tables of P ?l we can tabulate the other functions, for a chosen 

 value of 6 and sufficient range of n, by using the " recurrence- 

 formulae. " 



Numerical values for the strain-components were thus 

 obtained for a succession of values of r, less than a and 

 greater than a, by using the ascending and descending series 

 respectively. When these were plotted against r it was 

 found easy to connect the two portions of the curves across 

 the value r = a. This was done for (9 = 0, 30°, 60°, 90°. 

 Poisson's ratio was throughout the work taken as ^, the value 

 used by Hertz and Fuchs. From the strains the cylindrical 

 components of stress were calculated, and finally the magni- 

 tudes and directions of the principal stresses at points lying 

 along the four directions from the centre of contact. The 

 results are shown in figs. 1, 2, 3, 4. 



3. Graphs of Principal Stresses. 



On these diagrams the abscissa is distance from the origin, 

 the radius of the circle of contact being taken as unit. The 

 greatest principal pressure p 1 is that which passes into 

 the normal pressure at the origin. Its value there is taken 

 as the unit for ordinates ; it is one and a half times the 

 average pressure exerted over the area of contact. The 

 other principal pressure in the plane through the axis is ;j> 2 , 



