90 Mr. Lubbock on the Limits upon the Earttis Surface 



Berwick Hill near Mason Dinnington, and Heddon-on-the- 

 Wall, and another traverses the country to the westward, 

 from the sea-coast in the vicinity of Howick and Warkworth 

 by the Helm-on-the-Hill, Netherwitton, Roadley, Shaftoe 

 Crags near Wallington, and Stamfordham. Basalt being evi- 

 dently the production of fire, and pervading in an irregular 

 manner rocks of almost every age, does not come within the 

 scope of these remarks; so that the similarity of organic re- 

 mains has at length become the geologist's chief guide and 

 reliance, in proving or endeavouring to prove the identity of 

 formations; but if I mistake not these exuviae will not always 

 warrant the conclusions drawn from their presence. 



XV. On the Limits upon the Earth's Surface within which an 

 Occultation of a Star or Planet by the Moon is visible. By 

 J. W. LUBBOCK, Esq. F.R.S.* 



A STAR is occulted at all those points which coincide at 

 **- any instant with the intersection of a cone circumscribing 

 the moon, and whose apex coincides with the star in question 

 and the earth's surface; and the star will be just occulted, or 

 appear to graze the moon's edge at those points which co- 

 incide with the intersection of a plane perpendicular to the 

 moon's orbit, and passing through the star and the centre of the 

 moon, with the earth's surface. In order, therefore, to de- 

 termine the limits within which the occultation is visible, it is 

 necessary to find the equation of this cone and plane. 



On account of the small diameter of the moon and the great 

 distance of the star or planet, the cone in question may be 

 considered as a cylinder whose axis coincides with the line 

 joining the centres of the moon and the star, which axis is 

 parallel to a line joining the centre of the earth and the star. 



Let F (x, y, z) = 0, F' (x, y, z) = 0, be the equations to 

 any curve, x = az + x fl y = bz -f y / the .equations to any 

 straight line, the equation to the cylinder whose axis coincides 

 with the line in question and has the curve {~F(x,y, z) = 

 F' (x,y, z) = 0} for its base, may be found by eliminating #, ?/, * 

 between the four equations {F (x, y, z) = 0, F' (x, y, z) = 

 x = a z + x / and y = b z + y t } and then putting for x f and y t 

 their values x a z and y bz in the resulting equation. 

 The following method however in some cases is perhaps 

 more simple. 



Let #' 3 -f y 2 = 1* be the equation to the cylinder in ques- 



* Communicated by the Author. 



tion 



