VOL. LIII.] PHILOSOFHICAL TRANSACTIONS. 2\ 



3d. If the last term of it be equal to nothing, and the signs of the terms of the 

 equation be not continually changed from -j- to — and — to -f" ; then either 4 

 or 2 roots of the given equation will be impossible, according as 2, and not more 

 of the last terms of the given equation are equal to nothing, or the contrary. 



Prob. III. Let T, y, v, be the abscissa, ordinate, and area of a given curve; 

 and let y" + {a -\- bx) X y-' + (c + (te + ex'') X y"^ + (/+ gx + hx"" + kx^) 

 X 3/""' + &c. = : to find whether the area (v) can be squared or not. 



Suppose the equation to the area to he if -\- {a -{- sx -\- car*) v"~' -\- {d -{• ex 

 + ¥x^ + Ga^ + Hx*) X r;"-* + (1 + kx + lx^ + Ma;^ + nx* + ox* + px^) X 

 v*"' &c. ^ O: consequently it will be nyv'^' -\- {n — l) X (a + bx + cx^) yv''* 

 + (n— 2) X (d + EX + Fx^ + Gx^ + HX* B + 2cx) v'-' + (e + 2fx + aGx** 

 + 4hx^) X yv"-' + &c. ) _ 

 XV "-* + &c. 5 ~ 



If from these equations, by the known methods, v be exterminated, there 

 will result an equation expressing the relation between x and y. Then the co- 

 efficients of this equation must be equal to the coefficients of the given equation 

 y + {a + bx) y" -' + {c -{- dx -{■ ex') 3^" " - + &c. = 0; and if the quantities 

 A, B, c, &c. can be hence determined, the curve is quadrable, for it is v" -\- (a-\- 

 Bx + ex*) X V - ' + (D + EX -|- PX^ + GX* + HX*) X t^" " ^ + &c. = O; 

 otherwise, it is not quadrable. 



Exam. Let the given equation be i/^ + x* — 1 =0, and suppose the equation 

 to the area be y* + d -|- ex -j- ^^^ + go^ + hx* = 0; then will 2vy -f- e + 2f 

 X -f- 3g x^ + 4hx* = 0; hence reducing these two equations into one, to ex- 

 terminate V, and there results the equation y'^ + 



i6h'j° -I- 24HGjr' + (i6hf 4- 9g') j-* + (8eh + 12fg) x^ + (6ge -I- 4f') j» + 4rEJ + £ »_ 

 ~~" 4 X (hi< + gx* + fx' + ex + d) * 



„ , r .■ i6h»x« + 24HGX' + (16hf + 9g') ** + (8eh + 12fg) x' + (6oe + 

 But the fraction^ 4 x (hx* + gx^ + fx^ + ex + d) '^ ^ 



♦y')x'+ 4FEx-t E» Qygj^j to be X* — 1 ; and consequently 



4h = i6h'' 



4g = 24HG 

 4f — 4h = i6hf + Qg* 

 4e — 4g = She + 12fg 

 4d — 4f = 6ge + 4p* 



— 4e = 4fe 



— 4D = E^ 



But, by the method of finding common divisors, it appears that these equa- 

 tions are contradictory to each other ; and consequently the curve is not gene- 

 rally quadrable. 



Theobem. Let x, y, v, be the abscissas and ordinates of the curves abcsefo 



