§6-8] THE SATUKNIAN SYSTEM. 5 



6. When the cube can be rejected, as in § 4, the integration of the formula of that 

 section gives 



-£% = £% = f b -f=f 



b (p -\-r cos q>) 



pJo op Jo 



= 5/ + 1 r cos 9 log (/„ + Q — r cos 9). 



7. When the point is so near the axis of the ring that the square of r can be 

 rejected, the value of £2% is easily found to be 



QZ=-k f log CA°> + • - *) -/ ^jy+Li) 



(»-*) [./?'- r -^P + 2 r cos y log (/p + 9 )]. 



8. In order to obtain the last integral of £2, which is to be performed in reference 

 to 9 as the variable, elliptic integrals must be introduced ; and the following notation 

 is adopted, which does not differ materially from the ordinary notation : 



sin 6 = sin i sin 9, 

 Fi <p =f+ sec fi, 



FJ=F, (£'©), S? =$(*©). 

 If we also assume 



H i9 = (FJ — E; ) F coi 9 — F? E,,. y + H) 

 in which co i = i'0 — * 



we obtain from well known formulae of elliptic integrals 



cos 2 ^ sin?; cos q [" C sec_0 jnl _ jt 



V ( 1 — cos 2 i sin 2 t/) L J,! cos 2 qp + cos 2 1 sin 2 r t sin 2 gj ' J 



y/(l — cos 2 1 sin 2 ?;) 



s'sin 2 ;?) r /' sec ■ •> -rn\ tt 

 I I : 5 r^ sin" « 1? ■ ! = H; 1]. 



sv LLl + cot'-z/sin 2 ^ < } ' 



sin »; cos ri 



o 



We have also the following formulae of elliptic integrals, which are given in elementary 

 treatises : 



/ ^- = F, 9 — E, w — cot 9 cos (5, 

 y*^ (tan 2 J 9 sec (5) = 2 tan J 9 cos £ -f- F, 9 — 2 E, : 9, 

 /.(sin 2 ? secfi) == ^ | _.(F I 9 — ^-9), 

 y^ (sec 2 y sec fi) = F, 9 — sec 2 ? (E,- 9 — tan 9 cos fi), 

 f sec 3 fi = sec 2 i E ; 9 — tan 2 1 sin 9 cos 9 sec fi. 



By putting 



9' = i -© — i 9, 



(267) 



