196 PHILOSOPHICAL TRANSACTIONS. [anNO 1787. 



multiplication, division, &c. can be found such functions of the above-mentioned 

 quantities, that some may become greater than others, and thence may be de- 

 duced the propositions above-mentioned. 



7. It may not be improper in this place to adjoin a few observations on finding 

 the limits of some quantities in which others contained in given equations become 

 negative or affirmative. 



1. Given an equation involving 1 unknown quantities, x and 3/; the limits of 

 the quantity y, between which the quantity x will become affirmative or negative, 

 may be deduced from the following principles. The quantity x passes from 

 affirmative to negative or from negative to affirmative, either through nothing or 

 infinite ; or from 2 impossible roots it passes to affirmative or negative through 2 

 or more equal roots ; and, vice vers^, from affirmative or negative to 2 or more 

 impossible roots through 2 or more equal roots. Find therefore the values of y, 

 when X becomes = O, or infinite ; and also all the cases in which 2, &c. values 

 of X become equal, that is, when its roots become impossible ; and thence can 

 be deduced the limits of the quantity 7/, between which x becomes affirmative or 

 negative. 



2. If a? = - be an affirmative quantity, then p will be affirmative or negative, 

 according as o. is an affirmative or negative quantity, &c. Assume therefore 

 p = O and Q = O, and from the roots of the resulting equation can be deduced 

 the cases in which x becomes an affirmative quantity. 



3. If more n unknown quantities, ar, z/, z, v, &c. be contained in a given equa- 

 tion; then, by the preceding method, find the limits of z, f, &c., between which 

 X becomes an affirmative or negative quantity, and let the quantities denoting the 

 limits contain not more than n — 1 unknown quantities : from the above-men- 

 tioned quantities or equations expressing the limits, find others denoting their 

 limits, which do not contain more than n — 2 above-mentioned quantities, and 

 so on. 



4. Often from the substitution of the limits of given quantities can be acquired 

 the limits of the remaining one x. Find all the greatest values of the quantity x 

 contained between the above-mentioned limits, and thence can be deduced the 

 limits sought. 



5. If there are given m equations involving w -|- 1 or more unknown quanti- 

 ties ; then sometimes with, and sometimes without, reducing them to others in- 

 volving fewer unknown quantities, can be found by the preceding method limits; 

 and from comparing the limits so acquired can sometimes be deduced the limits 

 sought. 



6. If a given function of the unknown quantities x, y, z, &c., is asserted to be 

 contained between given limits, when other functions of the above-mentioned 



