rOL. XVI.] VHILOSOPHICAL TRANSACTIONS. ' 407 



But it is to be noted here, that the limit arising from the least y, is sometimes 

 negative, or less than nothing, viz. as whenever the greatest of the three per- 

 pendiculars exceeds GD, fig. 11. If this happen, the quantity + ^ may be 

 diminished to nothing by the limit prescribed from the middle y. But the de- 

 fect of a limit from the least y shows how great — r may be in the equation, if 

 there be three affirmative roots and one negative ; which if it exceed, there can 

 be only two, one affirmative and the other negative. And all this is demon- 

 strated from hence, that the above-mentioned limits of the quantity r, are the 

 differences between the squares of the line GD, and of the perpendiculars on 

 the curve of the parabola. 



But because of the perplexing cautions arising from the diversity of signs in 

 these equations, it is always better to take away the second term, and then to 

 seek out the number of roots and the signs, according to the rules already de- 

 livered ; especially if these quantities are not greatly different from each other. 

 And of these four affirmative roots, two are always less than \b, and two 

 greater; viz. if DG be less than AG, or \ph than -^h^ -|- 9- But three are 

 always less than -J-i, whenever the mean perpendicular, or that found from the 

 mean y, is greater than A G, or f Zj'^ y greater than 3 ^^ — py^ from the same 

 mean y. The fourth and greatest root is greater than the greatest y + \b, and 

 it is equal to the difference between b and the sum of the other three roots, and 

 therefore it is less than b. 



Perhaps those who more perfectly understand the nature of the parabola, may 

 be able to do all these things after a more compendious manner ; but there is 

 some reason to doubt whether all these quantities b, p, g, r, can be rightly de- 

 termined without the resolution of a cubic equation ; for whatever is done in 

 this affair by plane equations, exhibits not the true limits, but some approxima- 

 tions only. 



That the Obliquity of the Ecliptic and the Elevation of the Pole continue 

 unaltered. From some Observations of M. P. Wurtzelbaur. N° IQO, p. 403. 

 Whether the poles and axis of the earth be really fixed in the globe, or sub- 

 ject to be transferred from one place to another, is an old inquiry, though now 

 lately revived by Mr, Hook, in his ingenious essays on the great mutations and 

 catastrophies, which in all appearance have happened on the earth's surface. A 

 necessary consequence of such a translation of the poles, would be the change 

 of the latitudes of places, which would increase in those regions towards which 

 the poles approach, and decrease in those from which they recede : and under 

 the meridian 90° removed from that in which the poles shift, the latitudes con- 

 tinuing the same, the meridian line only would alter ; but no two places con- 



