192 Dr. C. Cliree on Applications of Physics 



The complete value of the potential V is given by 



V=-±ga-h* + ^?>g(rla) i a i a i l(2i + l) + ^r i V i f Gi*. . (45) 

 In ordinary circumstances we are supposed to be given the 

 unstrained surface, with full information as to the force 

 system, and it is customary to regard the surface equations as 

 applying to the unstrained surface. In the present instance — 

 and I daresay as a rule in practice — the forces depend to some 

 extent on the disturbed form of the body. It is thus con- 

 venient, to say the least of it, to suppose that the surface 

 equations apply in the present instance to the disturbed sur- 

 face. This implies nothing more serious than the replacing 

 the ordinary definition of strain, viz. 



(final length— initial length) /(initial length), 



(final length— initial length) /(final length). 



The two definitions are equivalent so long as it is justifiable 

 to apply the mathematical theory, which assumes the square 

 of a strain negligible f. 



§ 25. The problem whose results I am about to use was 

 more general than the one at present before us, inasmuch as 

 the surface was not assumed to be naturally spherical. The 

 notation employed in its solution was also somewhat different, 

 the potential being given in the form 



V=-i^-V 2 + 2«/vW (46) 



Thus in utilizing the results we must put 



V«=%ra-V(2t + l)+V//a, (47) 



In the general problem Vi w y as unrestricted, but I contented 

 myself with giving the two arbitrary constants a^Y iy a^Li 

 explicitly in terms of oiVi and ga~*<u. The expressions for 

 the displacements freed from arbitrary constants were given 

 (I. c. equations (13) to (15), pp. 280, 281) only for the case 

 when 



Y^dga-i/W+l), 



or when V/ in (47) is zero. 



It is easy, however, to add the terms containing V/. For 

 if in the equations (11) and (12) of p. 280, 1, c, we substitute 

 for Vi the right-hand side of (47) and multiply up by 0^, 

 we notice that V/cr* appears with the same coefficients as V 

 possessed in the earlier equations (32) and (33) (I. c. p. 264), 

 which determined the unknowns Y* and Z] — treated in that 



* See Prof, G. H. Darwin, Phil. Trans. 1879, Part 1. 

 t See Phil. Mag. Sept. 1891, pp. 246-7. 



