26 report — 1877. 



Thus, generally, 



LW+f)^+^»- 1 )+( 2 ' i - 3 >1^ + i 4(B - 2)2 - 1 !vr i J Es 

 [va4io^+(.-i)4a+v)^5+{^- 1 ><«-^+M^^2 Fl 



_ (-1)"- 1 sin 2 ''- 3 j- cos .r l"" 3 ' 2 f «^ _ m + , ( 2n-S)(2n - 5) + (2b - l)p* } sin 2 x 



(l+ i rsiir«)' 1 "* 2 "" 3 



+2p*sin*#], 

 and 



+ {l+4( M -2)[(»-2> 2 +(2 M -3)(l+/)]}^^z- 2 _jE 1 



, h-n-o/"7 2 \ lA-Hai—A-s^ 1 + (l_9/*+/> 2 )sin 2 .r 



= C_lV'-i2*-' ! -( h ) — — ; u , siir /t+1 .rcos.r, 



1 ' h=o \ J 2»- 2 (l+p 2 ) M_ *" S (l+/-sin 2 .r) A+ 5 



[V ( W)^ +4 i /{(2 rt -3)(l+/)+(«-2)F} ^— , 

 + {l + 4(n-2)[( W -2)(l+/) + (2«-5)^]}~^ I) ]F 1 



1 . o /»-2\ l»-*-3 I 2 1 i-1 | 2 : 2»-2A-3 



= (-l)'- 1 2* = r ( A ) ~ L ir ir^— ° * {2+(2^-5>H[(2«-2A- 3) 



-{(2«-2/ J -3)(2«-2A-5)4-(2M-2/t-l)/}sin 2 .r+2/) 4 sin'.c]. 



3. If the integrals E,, F^ E 2 , F 2 are taken hetween the limits and \n, so that 

 E and F become complete elliptic integrals, it is easily seen that the formulae lose 

 the terms sin x cos x <$>{x), as the factor sin X cos x vanishes at each limit, and the 

 other factor is never infinite. 



On Cubics of the Third Class with Tliree Single Foci. 

 By Henry M. Jeffery, M.A. 



1. There are four cases for consideration, according as the number of foci at 

 infinity is three, two, one, or none. 



If three foci are real and at an infinite distance, the line at infinity (£=i;=0) is 

 an acubitangent, 



