VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. Q^^ 



greater than ^, it will be possible for yz to be equal to \q. Therefore, if 



x^ — qx, or r, is less than -^^t^, or^ is less than |-, the foregoing method 

 of resolving the cubic equation x^ — qx z=. r will be impracticable; bi\t if 

 .T^ — qx =. r, or r is greater than - -r^f , or - is greater than ''—, it will be prac- 



ticable. 



Art. lO. It now only remains to be proved, that while x increases, from beina; 

 equal to V q, ad infinitum, the compound quantity jt^ — ^x will likewise in- 

 crease from ad infinitum, without ever decreasing. Now this may be demon- 

 strated as follows. 



Art. 11. It is evident, that while x increases from being equal to \/ q 

 ad infinitum, both the quantities x^ and qx will increase ad infinitum likewise. 

 But it does not therefore follow, that the excess of x^ above qx will continually 

 increase at the same time. This will depend on the relation of the contemporary 

 increments of a? and qx: if the increment of x^ in any given time be equal to 

 the contemporary increment of qx, the compound quantity x^ — qx will neither 

 increase nor decrease, but continue always of the same magnitude during the 

 said time, notwithstanding the increase of x ; if the former increment be less 

 than the latter, the said compound quantity will decrease ; and if it be greater, 

 it will increase. We must therefore inquire, whether the increment of x^ in any 

 given time be greater or less than the contemporary increment of qx. 



Art. VI. Now if X be put for the increment which x receives in any given 

 time, the increment of oi^ in the same time will be the excess of {x -\- xf above 

 ot?, that is, the excess of c^ -\- 3x^x -{- 3xP -f i' above x^ ; and the increment 

 of qx in the same time will be the excess of ^ X (a? -|- x), or qx -|- qx, above 

 qx ; that is, the increment of x^ will be 3x\v -\- 3xx^ -j- x\ and that of qx will 

 be qx. Now, in the equation .r' — qx ■=■ r, it is evident that xx must be greater 

 than 9 ; for otherwise x^ would not be greater than ^.r, as it is supposed to be. 

 Consequently xx X x must be greater than qx ; and, a fortiori, 3x\v -f Sx.i^ -f- 

 P, which is more than triple of x'^x, must be greater than qx; that is, the in- 

 crement of x^ will be greater than the contemporary increment of qx. There- 

 fore the excess of x^ above qx, or the compound quantity x' — qx, will increase 

 continually, without decreasing, while a- increases from »/ q ad infinitum, a. e. d. 



Art. 13. It follows therefore, on the whole of these inquiries, that if the 

 compound quantity a-' — qx, or its equal, the absolute term ;•, be less than 

 ^2^, or- less than -^, it will be impossible for yz to be equal to -to, and con- 

 sequently the foregoing method of resolving the equation .r — qx = r will be 

 impracticable ; but if x^ — qx or r be greater than |^|, or ^ greater than |^ 

 it will be possible for ?/z to be equal to -\q, and consequently, the foregoing me- 



