THE INTERNAL CONSTITUTION OF BODIES. 461 



from this last equation, that is to say, that which is given by the for- 

 mula 



(2)' r{q-qo) = Qo'^Q,+ Q^ + Qi+ &c. 



This being premised let us return to the formula (5). As the inte- 

 grations indicated in the second member of this equation may, accord- 

 ing to what we have stated at the commencement of this paragraph, be 

 extended from r' = o to r' = cc, 6' = o to 9' = it, and 4/' = o to 4*' = 2 ir, 

 and as all these limits are independent of each other, observing that 

 we have in general 



o o 



P„r'. sin l3'rf9'rf\|/' = o 

 P„ F'. sin 9'rf9'rf4/' = o 



T-. 



V . 

 2n + I « 



ve shall find 

 F= S 



CO 4 * r /* 



+^ 2^rTT/^'V"/ 



o L «- r 



ar 



+ ''/' 



"a'.M .,"1 



ar 



/ 



Without actually making the substitutions of the expressions pre- 

 viously given for G, and latterly for F, in the equation (III)' for the 

 purpose of comparing the functions of the spherical coordinates of the 

 same degree which are to render it identical, we see that, as G, G^, G^, 

 &c., contain none of these functions, all the T^^ and V^ must be null, 



with the exception of T^ and V^, which answer to the value w = o, and 



represent two arbitrary constants. 



2i2 



