7 6 Dr. Waring’s Obfervations 
rally be true ; for the equations may have fome roots the fame, 
but not all. 
Thefe obfervations may be applied to more equations. 
4. From («) given equations a = o, b = o, c — o, See. can 
eafily be deduced others dependent on them, by finding any 
direct algebraical functions of the above-mentioned equations, 
that is, <p (a, b, c, &c.), which will always = 0; and in like 
* 
manner, from the relation between any lines being given, can 
be deduced innumerable relations between the above-mentioned 
lines, and other lines dependent on them. 
Partlll. 1 .Ratios, which are fuppofed greaterorlefs than others, 
can eafily be transformed into equations, which contain affirma- 
tive and negative quantities : for example, let the ratio a : b be 
be 
greater than the ratio c : d , then will - = - k ; if it be lefs, 
then will ^ = c - + k, where k denotes an affirmative quantity ; 
and, vice verjd , if ~ = c - — k, then will the ratio of a : b be 
greater than the ratio of c : d , &c. 
2. If one quantity (a) is affirmed to be greater than another 
b , for a in the given equations fubftitute its value b-\-k\ if 
jefs, for a write b — k, where k denotes an affirmative quan- 
tity. 
3. Reduce the equations, fo as to take away their denomina- 
tors, and the demon ftration of the propofition will often very 
eafily follow. 
p p' 
4. Let £ = ^and Id- ; and if P and Q be affirmative, 
let P' and be affirmative ; and, vice verfd 9 if negative, ne- 
gative ; then, if k be affirmative, will k' alfo be affirmative ; 
the fame alfo may be affirmed, if P and Q have both con- 
trary 
