12 THE SATURNIAN SYSTEM. [§ 9 



+ I l^-2(z-bf] v-^ 2 [V^^H^' + H^"] 



if., /oi r __ \ i /-a j_i Q 3 ( r — e)1 z _^zl F 1 



+ VJ*\- + 7+fJ— 2 -/2o+* e+r J vt(H _ r) ._|_ (,_&)«] *v . 



and finally 



O ;./• (V" O'" Q!" _l_ O w \ 



The terms multiplied by { © are added to free the expression of Q!" from the dis- 

 continuity, with which it would otherwise be affected. Thus when 



/• = o 

 we have rj' 2 = 0, H, 2 ?; 3 = i ®> 



and the corresponding term of S2' 2 " would be 



V'O— L>) 2 V(-'-*) 2 



t© 



4 »• — e = — 6 ' 



which suddenly reverses its sign, and is balanced by the term which is added in the 

 corrected form. 



Also, when z = b, 



we have < = iff = 0, E^i£ = H^tf" = *©, /» = r, 



and the corresponding terms of I2 2 " are 



which suddenly reverse their signs, and are balanced by the opposing terms. 

 It is proper to observe, that in this latter case 



When / 20 = Q, 



we have *j 2 ' = & ©j H, s ^/ 2 ' = 0, 



so that in these cases the sudden reversal of sign does not affect the continuity of the 



function. 



The function H, a 7) r 2 " 



never vanishes, because •J} 2 " Cfin never become a right angle. 



9. The first term of I2 2 " is incapable of integration, but it may easily be made to 

 assume another form, which will be preferable for computation by means of quadra- 

 tures. For this purpose, denote it by the letter L, so that 



2® 

 U = -ii> 2 £\og(f 2 -\-z-b). 

 ' 



(274) 



