yOL. XLIX.] PHILOSOPHICAL TRANSACTIONS. 2i 



Lemma 3. To determine the Ratio of the Motion of the Whole Earth, to the 

 Motion of the Material Stratum over the Interior Globe of the Earth. — Let c 

 represent the centre of the earth (fig. 2,) ck a portion of any diameter of the 

 equator, edgk a section of the earth perpendicular to the diameter ck, and to 

 the plane of the equator ; this' section and all the sections parallel to it are si- 

 milar ellipses, as is well known. From the centre k of the ellipse edg draw the 

 radius ke in the plane of the equator, and it will be the greater semiaxis of the 

 ellipse, and the radius kd perpendicular to it will be the less semiaxis ; draw also 

 the two radii km, kw very near each other, and with the centre k and radius 

 KD, describe the circle one cutting Km, km, ke, in h, h, e ; and with any ra- 

 dius Kr describe the arc rn cutting km, kot, in r, w, and the arc st very near the 

 arc rn, cutting km and Km in s and t. Now because the areola rstn, while the 

 earth revolves about the axis ck, has a velocity proportional to the distance Kr, 

 its momentum will be proportional to Kr X rs X st or ' hence the 



momentum of the whole areas kmw will be proportional to — ^ . Make 



Hv perpendicular to kd, and if the greater semiaxis ke be supposed but little to 



exceed the less semiaxis kd, it will be km = nearly, therefore — — 



KH^ •'^ 3kh 



^ ^'_^_i_ J ^ therefore the sum of the momenta of all the areas 



3 KH 



kmw, that is, the momentum of the whole section, will be proportional to the 

 circumference drawn into -I-kh^ -f- ^kh X Ee. Now let ca be equal to the 

 greater semidiameter of the earth, cb the less semidiameter, and ab their dif- 

 ference ; let Kk be a very small particle of the axis ck, and let c denote the cir- 

 cumference of the equator; then, because kh:ke:: cb: ca, and Ee: ab::ke: 

 CA, the momentum of the speroidical portion whose thickness is k^, terminated 

 by two parallel sections, that is, the circumference dhd drawn into kA X (i-KH^ 



X .,, 1 ,. , , C X cb' X KE^ X k/c . ex CB^XKE^X AB X Kk 



-{- J-kh X Ee,) will be proportional to -^- 1 ^^^^ ; 



therefore the sum of all these momenta, or the momentum of the whole sphe- 

 roid about the axis ck, will be denoted by —r^ 1 — , or by 



— X CA X X CA x_AB .£ ^ (jej^Qte the circumference described with the ra- 



16 ' 32 ' 



diuscB. Hence the momentum of the interior globe, whose radius is cb, will 



be expressed by -p'T(i^ X cb^ : therefore the momentum of the interior globe is 



to the momentum of the whole earth revolving about the axis ck, as cb^ to ca 



X CB -f- -|CA X AB, or as ca— 2ab to ca -f- 4-ab nearly ; and the momentum- 



of the matter incumbent on the interior globe of the earth, to the momentum; 



of the whole earth, as 5ab to 2ac nearly, q. e. i. 



CoROL. By the same mode of reasoning, if the circumference of the circle 



described with the radius cb be revolved about its own diameter, since the motion of 



