Theory of the Contact of Elastic Bodies. 321 



in some of its features. The subject was further developed 

 by Huber* for the contact of two spherical surfaces, in 

 which case the potentials can be expressed in finite forms. 

 For this simple case the question of the correct forms of the 

 lines of principal stress was settled by Fuchsf, who obtained 

 the result by carrying out a laborious process of arithmetical 

 integration. 



The present work had its origin in an endeavour to apply 

 the method of expansion in zonal harmonics in order to 

 obtain numerical values for the stresses, etc., in Huber's 

 case of axial symmetry. The formulae of Huber contain an 

 elliptic co-ordinate, the parameter of the spheroid through 

 the point which is confocal with the circle of contact. This 

 is inconvenient for purposes of calculation. On the other 

 hand, a good deal of labour was found to be needed in 

 evaluating the series to which we were led, so it is doubtful 

 whether one method has an advantage over the other in 

 this respect. Tt does not, therefore, seem worth while to 

 reproduce the details of our analysis ; but some of the results 

 obtained, with regard to the magnitudes and directions of 

 the principal stresses, may' be of interest as adding a little 

 to what is already known. We give graphs for the stresses 

 along lines running from the centre of contact in directions 

 making angles 0, 30°, 60°, 90° with the normal. 



A further note is added on the limiting forms assumed by 

 the lines of principal stress at a distance from the contact. 

 It is shown that the characteristic features of these curves, 

 as discovered by Fuchs, can be simply deduced from the 

 solution given by Boussinesq of the problem of a body acted 

 upon by a pressure concentrated at a point on an otherwise 

 free plane boundary. 



2. Outline of Method. 



Let the axis of z be the inward normal to one of the 

 bodies at the centre of the circle of contact. Suppose a 

 distribution of matter on this circle equivalent to an oblate 

 spheroid of vanishing axis, the whole mass of the distribution 

 being equal to P, the normal force with which the bodies 

 are pressed together. This makes the surface-density at 

 distance r equal to 



3P(a 2 -r 2 ^/27ra 3 , 



where a is the radius of the circle. Let <£ be the ordinary 

 inverse potential of this distribution and ^ the logarithmic 



* Huber, Ann. d. Phys.vol. xiv. p. 153 (1904). 

 t Fuchs, Phys. Zeitschr. vol. xiv. p. 1282 (1913). 



Phil. Mag. S. 6. Vol. 43. No. 254. Feb. 1922. Y 



