676 PHILOSOPHICAL TRANSACTIONS. [aNNO 1780. 



the equation x^ -{- qx = r was practicable or not, would be to inquire, whether 

 it was possible in all cases, that is, in all magnitudes of the known quantities q 

 and r, for 3yz to be equal to q, or for yz (or the product or rectangle of the 2 

 quantities y and z, whose sum is equal to x) to be equal to -^q, and, if it was 

 not possible in all cases, but only in some, to determine in what cases it was pos- 

 sible, or what must be the relation between q and r to make it possible. 



j^rt. Q. Now, in order to determine this question, it would be proper and 

 natural to observe, that the quantity j/z, or the product of the 2 quantities y 

 and z, whose sum is supposed to be equal to x, can never be greater than the 

 square of half that sum, that is, than the square of 4:X, or than ■^a-'x, by El. 

 2, 5, but may be of any magnitude that does not exceed that square. There- 

 fore, if -^q is greater than -^xx, it will be impossible for yz to be equal to it ; 

 but, if -^q is either equal to, or less than -^xx, it will be possible for yx to be 

 equal to it, and if -^q is exactly equal to -^xx, z will be exactly equal to y, and 

 each of them equal to one half of x. We must therefore inquire, what is the 

 magnitude of x when -j^q is equal to -^xx. Now when -^xx is = ^q, then xx will 

 be = — , and x =■ -^ : therefore, when .r is less than — -^, it will be impossible 



3 V 3 V 3 



for yz to be equal io^q ; but when x is greater than — ^, it will be possible 

 for yz to be equal to -^q. 



But when :r is = 2^|, x^ will be = |^' and qx will be = "^^j or g^, 



and consequently x' - qx will be = ^-^ - -^- = ^-. 



Therefore, if it be true, as we shall presently see that it is, that while x in- 

 creases from being equal \.o s/ q (which is evidently its least possible magnitude) 

 to any other magnitude, the compound quantity x^ — qx, or the excess of z* 

 above qx, will also continually increase from O (to which it is equal when x is = 

 V q, or XX is =: q) to some correspondent magnitude, without ever decreasing ; 

 it will follow that, when x is less than — ^, the compound quantity x' — qx will 



be less than ^\ ; and when x is greater than -|, the compound quantity 

 x^ — qx will be greater than -^7^ ; and, e converso, if the compound quan- 

 tity x^ — qx is less than J , x will be less than ~—~\ and if the compound 



quantity x^ — qx is greater than -/^J, oc will be greater than — -|. Consequently, 

 if the compound quantity a?' — qx, or its equal, the absolute term r in the equa- 

 tion x^ — qx z= r, is less than ■——, or- is less than |^, it will be impossible 



for yz to be equal to -^q ; but, if x^ — qx, or r, is greater than — ^ , or ^ is 



