VOL. XXXVII.] PHILOSOPHICAL TRANSACTIONS. 525 



It appears in every hypothesis, that the section of the fluid is an algebraic 

 curve, except only that of the attraction towards y or towards c being in the 

 simple inverse ratio of the distance; for if it be only m = — l, the fluent for 

 the section will be, ^^^±^1 log. {bh + ibhr^rr) + ^^^^--ZMl _-^ = 



xc_ , hb + ihhr + rr _ _ pr'' + ' fbhr fhhrr pa _/ab_ _ faM_ 



And if only n = — 1, then it will be 



3- (ii + 2Mr +(•?■)*"'+ i , ^ 1 fhhr fhhrr v(a + b) , , fab 



faa , r !i(M + 26>^;- + r;)^'" + 4 /Mr , fhhrr , ^c 



-2(^TT)' or pa log. - = ^,p^y^ +V + V +;rTT - 



"c 2c ■ 



But if both m= — \ and w = — 1 together; then it will be 

 •_<i±i> log. (H, + .Mr + rr) + p. log. ,■ - BL^ - J^^ = 



= '-^ log. (« + *)• + P« log. » - j^, - Jif^. or 



TC , bb + 2/'/^/- + rr , , r y*Ar . fhhrr fab faa 



— og. — \- pa log. - = h -z '^ '-^- 



2 ^ cc ' " o a c 2c c 2c 



If there be desired the equation for the section of the fluid by perpendicular 

 co-ordinates, making ce = x, and de = ?/, there will result these two equations, 

 rr = XX + yy, and hr = y, by means of which r and h will be exterminated 

 from the equations above found; and there will result for the general case, 

 2 (n + 1 ) Tra" {bb + iby -\- yy + xx) J- + 4 + 2 (m + 1 ) pd" {xx + yy) i-" + 4 _ 

 Iqfa" bC" ~' y — qfa" C" - ^ yy = 2 (»i + 1 ) ncf c'" + ' + 2 {m + 1 ) /jc" + ' c"" — 

 Iqfa" + ' Z'c'" " ' — qfa" + - C" " '. 



And in like manner, in the cases of ?« ^ — \, n^ — 1, will be found the 

 equations for the perpendicular co-ordinates. 



As we have found the curve paq, so also, mutatis mutandis, may the curve 

 paa be found. For then, if the gravity at a towards y be given and =: tt, and 

 the gravity at a towards c =p, and the centrifugal force at a =/"; also ca = a, 

 cy = b, eg = r, gl = hr, and yg = y^ (bb — ibhr -f rr), there will be found 

 the gravity at g towards c, arising from the attraction towards y, 



„ _ ^{bh - )■) . {bb - Ibhr + n - )4"' - 4 

 ■^ - (i - «)- 



There will also be had for the gravity at g towards c, p = ^. 

 Also the part of the centrifugal force towards c,/" = — ■ [ J^ " . 



