SPHEROID. 



that the value of the general fruftum KAECL is ex- 

 preffed by ^pfrrx x — —- 



And if for/r be fubftituted its value abs, the fame 



taa — X s 

 fruftum will alfo be expreffed h^ ipbrsx x — 



Alfo, if for 3 a be put its value ^7^ 1 the laft ex- 



■^ bb — yy 



„ , ibb 4- yy 

 preffion will become ^prsx x IT^' o^ ' P ' ^ x 



(iir H — il' which, by writing z inftead of its value 



— , gives I psx X {ibr + yz.) for the value of the fruf- 

 tum, viz. the fum of the area of the lefs end, and twice 

 triat of the greater, drawn into one-third of the altitude or 

 diflance of the ends. 



And out of this lait expreflion may be expunged any 

 one of the four quantities b, r, y, z, by means of the pro- 

 portion b : r :: y : z. 



When the ends of the fruftum are perpendicular to the 

 fixed axis, then a is =r /, and the value of the fruftum 



becomes ^prrx x ^=^^-r- — for the value of the fruftum 



whofe e»ds are perpendicular to the fixed axis, its altitude 

 being X. 



And when the ends of the fruftum are parallel to the 

 fixed axis, a is = r, and the expreflion for fuch a fruftum 



fegment, refpeftively correfponding or parallel to/, r, the 

 femi-axes of the generating ellipfe, when parallel to the 



bafe of the fegment, and for "^ and r fubftitute their 



r 



F ^ RR + */- . r „ r ,1 



values -ir and j , the laid trultum will be ex- 



becomes -J pfx X 



3>-r — XX 



R 



2h 



Carol. 2. — If to or from \pfr r, the value of the femi- 



fpheroid, be added or fubtrafted ^pfrrx x '^ , 



aaa 



the value of the general fruftum K A C L, there will refult 



-J pfr rhh y. for the value of a general fegment, 



either greater or lefs than the femi-fpheroid, whofe height, 

 taken upon the diameter pafling through its vertex and cen- 

 tre of its bafe, is A = a + x. 



When a coincides with/, the above expreflion becomes 



iprrhh X — for the value of a fegment whofe bafe 



is perpendicular to the fixed axis. — And here, if we put 

 R for the radius of the fegment's bafe, and for rr its 



value J 2 t > the faid fegment will become ipRRh x 



2/- f,- 



And when a coincides with r, the general expreflion will 



-J. L 



become -ipfh h x for the value of the fegment 



whofe bafe is parallel to the fixed axis. And if we put 

 F, R, for the two femi.axes of the elliptic bafe of this 



prefledby4/FA x ^ , in which the dimenfions 



of itfelf only are concerned. 



Carol. 3. — A femi-fpheroid is equal to ?ds of a cylinder, 

 or to double a cone of the fame bafe and height ; or they 

 are in proportion as the numbers 3, 2, i. For the cy- 

 linder is = 4 nfr r = ',- nfr r, the femi-fpheroid := ? nfrr, 

 and the cone = ■; nfr r. 



Carol. 4. — When/= r, the fpheroid becomes a fphere, 

 and the expreflion ^fnrr for the femi-fpheroid becomes 

 ^nr^ for the femi-fphere. And in like manner,/ and r be- 

 ing fuppofed equal to each other in the values of the fruftums 

 and fegments of a fpheroid, in the preceding corollaries, 

 will give the values of the like parts of a fphere. 



Coral. 5. — All fpheres and fpheroids are to each other 

 as the fixed axes drawn into the fquares of the revolving 

 axes. 



Corel. 6. — Any fpheroids and fpheres, of the fame revolv- 

 ing axis, as alfo their like or correfponding parts cut off by 

 planes perpendicular to the faid common axis, are to one 

 another as their other or fixed axes. This follows from the 

 foregoing corollaries. 



Carol. 7. — But if their fixed axes be equal, and their re- 

 volving axes unequal, the fpheroids and fpheres, with their 

 like parts terminated by planes perpendicular to the com- 

 mon fixed axis, will be to each other as the fquares of their 

 revolving axes. 



Carol. 8. — An oblate fpheroid is to an oblong fpheroid, 

 generated from the fame ellipfe, as the longer axis of the 

 ellipfe is to the fliorter. For, if T be the tranfverfe axis, 

 and C the conjugate ; the oblate fpheroid will be = 

 ; n T ' C, and the oblong = 4 n C^ T ; and thefe quantities 

 are in the ratio of T to C. 



Coral. 9. — And if about the two axes of an ellipfe, be 

 generated two fpheres and two fpheroids, the four folids 

 will be continual proportionals, and the common ratio will 

 be that of the two axes of the cUipfe ; that is, as the 

 greater fphere, or the fphere upon the greater axis, is to 

 the oblate fpheroid, fo is the oblate fpheroid to the oblong 

 fpheroid, fo is the oblong fpheroid to the lefs fphere, and fo 

 is the tranfverfe axis to the conjugate. For thefe four 

 bodies will be as T^ T^C, TC , C^ where each term 

 is to the confequent one as T to C. 



To find the content of an univerfal fpheroid, or a folid 

 conceived to be generated by the revolution of a femi- 

 ellipfe about its diameter, whether that diameter be one of 

 the axes of the ellipfe or not. i. Divide the fquare of 

 the produft of the axes of the ellipfe by the axis of the 

 fohd, or the diameter about which the femi-ellipfe is con- 

 ceived to revolve : multiply the quotient by -5236, and 



T C * 



the produft will be the content required. That is, — j— 



d 



X .5236 =: the content ; T and C being the tranfverfe and 

 conjugate axes of the ellipfe, and J the axis of the folid. 



Or, 2. The continual produft of .5236, the diameter 

 about which the revolution is made, the fquare of its con- 

 jugate diameter, and the fqtiare of the fine of the angle 



made 



