208 ON THE DETERMINATION OF THE ATTRACTIONS 



It will now be easy to obtain the value of V correspond- 

 ing to 



'jfo'X, ...... a. 1 ) ...(28) 



without integrating the formula (1) No. 1, where F is the cha- 

 racteristic of any rational and entire function. In fact it is easy 

 to see from what precedes (No. 9), that we may always expand 

 F in a finite series of the form 



5,fc' + 6.# + &c. 

 after a?/, a? 2 ', &c. have been replaced with their values (7). 

 Hence, we immediately get 



p' = v\ { b.ti + b& + b& + &c.} ............. (29). 



By comparing the formulae (2G) and (27) it is clear that any 

 term, as & r $/ for instance, of the series entering into p, will 

 have for corresponding term in the required value of V, the 

 quantity 



2 y , , , , . h-"dh 



' g " 



j5T being a particular value of H satisfying the equation (17) 

 and immediately deducible from < by the method before ex- 

 plained. 



12. All that now remains, is to demonstrate that 



V'V0 = VV'< ........................ (31), 



whatever </> may be. For this purpose let us here resume the 

 value of y<, as immediately deduced from the equation (2") 

 No. 7, viz. 



_ 



trap <ra 



. ...(32), 



