CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECI'l.ls 



443 





f>. lii'il urtin, i of tin- 



The evaluation of the expressions (20) and (21) is facilitated by changing the 

 variables (U, V, W), (U',V, W) to (X, Y, Z), (X,, Y,,Z,) in accordance with equations 

 (16) and (17). The Jacohian of transformation is easily found to be (m + m')~*, and 

 the limits remain as before, viz., oo and + . The two integrals are now of the 

 form 



A jjj jjj 



,' Q ( Vo ) ^ (X, Y, Z, X,, Y,, Z,) ( 1 + F+ F') dX dY dZ dX, eZY, dZ, 



where ^ is an integral polynomial in the given variables ; F and F' are also 

 polynomials in these variables. Evidently only those terms in the product of >/< and 

 (l + F+F') which are of even degree in each of the six variables separately will give 

 any result upon integration. It is easy, though tedious, to pick out these terms ; 

 evidently if -0- is of odd degree in the six variables combined, only those terms 

 of F and F' whose a-coefficients have an odd number of suffixes will need to be 

 considered ; similarly, if >/r is of even degree, we need only consider terms in F which 

 have a u , a ia , ... for coefficients. Having picked out these terms we are left with a 

 Dumber of integrals of the form 



: '*o(v ): 



1 dX dV dZ dX, dY, dZ,. 



The integration with respect to X,, Y,, Z t can be carried out immediately most 

 conveniently by changing to polar co-ordinates. We do the same also in the case of 

 the variables X, Y, Z changing to the variables V , 0, <f>. The integration with 

 respect to the latter two variables is simple, and there remains an integral of 

 the form 





J1J' 



where a = 



hmm! 



,/ > 



, or %hm' according as Q(V ) or A has the suffix 12, 11, or 



m + m 

 22 respectively. 



We shall consider this more particularly later ; at present it is sufficient to denote 

 it by a symbol. As we shall only need to consider the three cases n = 2, n = 3, n = 4, 

 it i. perhaps, most convenient to denote it by P, R, S respectively, instead of 

 adopting a more general notation. To distinguish between the different cases arising 

 f'n un the various functions Q ( V ) occurring in the integrand, we add the same dashes 

 and suffixes to P, R, S. Thus we have integrals P' ia , P' n , P",,, and so on, 

 corresponding to the cases when in the above integral n = 2 and 1} ( V ) has the 

 special forms Q' 12 , Q' n , Q" n , and so on : similarly for R and S. 



Though the execution of the processes indicated is rather lengthy, it is quite simple 

 and straightforward, so that, without entering into the details of the calculation, we 



3 L 2 



