VOL. LXVIII.] PHILOSOPHICAL TRANSACTIONS. 363 



multiplied by C'i', m being indeterminate as before, then c""';^ — ac""'yi' = 

 d"-"'^^" X rc'"'Qi.i\ The fluent of the first member of this equation is evidently 

 of the form nc'^^y, the fluxion of which, viz. dc'""^ + Dmc^'yi being compared 

 with the former gives d = 1, and7« = — a; therefore, c-^'j/ := /c~"'.r /f^'ai', 



or y =1 c"" X fc~-'* X I cfOiX-. Let C'^Q.f = z, and c~^'''.r = v; then f c-'^'"'x 



c''*Qi' ■=. j zv •=. zv — / vz; but ^^ = 5^ 5 — ' supposmg that v and j; vanish 



at the same time; therefore vz —fvi = ^J^c^'^^i' — ^-^J^'^^'^^i' — }aS^ 



i"- /c''ai'. This value of ?/ is sufficient for the construction of the fluent. 



2a J -^ 



because the quantities j c"'ax, andj c"qlx depend on the quadrature of the 

 hyperbola; but if we would introduce into it the ordinates and abscisses of that 

 curve, we need only have recourse to the foregoing lemma, from which it appears, 

 that y = -ord. ax 1 ax abs. ax abs. ax I ax ord. ax. 



13. Let the co-efficient of y be now supposed affirmative, or let ^ + «V=®- 

 In this case imaginary expressions are introduced into the fluent, and the con- 

 struction by the hyperbola becomes impossible. For we have then, n == + 

 a ^ — I, from which, by proceeding as above, we get y = 

 c"- V - ' r^^ax ^/ - 1 Q y L_!l- /c«iV— iQ.r; hence also, by the lemma, y = 



sin. ax fax cos. ax — cos. ax f ax sin. ax. Here the quantities, I ax cos. ax. 



and fax sin. ax, are assignable by the quadrature of the circle, in the same 



manner as fax abs. ax, and I ax ord. ax, by the quadrature of the hyperbola; 

 but the method of investigating them, though an illustration of the principles 

 which we have laid down, is too well known to need to be inserted here. In 

 like manner might the fluents of innumerable fluxionary equations, compre- 

 hended under the general form Q=j/-f-^-|- .j + -4 + &c. be deduced, and all 

 of them would tend to prove that the arithmetic of impossible quantities is no 

 more than a method of tracing the analogy between the measures of ratios and 

 of angles. M. M. Euler,* and D'Alembert,-|- were the first to integrate such 

 equations as the preceding, and the method employed here differs from theirs 

 only by being better adapted to illustrate the principle which is common to 

 them all. 



* Nov. Com. Petrop. torn. lii. — Orig. + I'lieorie de la Lune. — Orig. 



3 A 2 



