152 PHILOSOPHICAL TRANSACTIONS. [aNNO 1777. 



(= e + -J-) is less or greater than e; that is, according as the arc pq (whose 



sine is = — ) is less or greater than 90°. 



10. As an instance of the use of the preceding conclusions, Mr. L. applies 

 them in the solution of a very interesting problem, which he had not before seen 

 solved, viz. 



Suppose a given spheroid, ivhile revoking uniformly about its proper axis, with 

 a given angular velocity, to be suddenly urged by some perciussive force to turn, 

 with some given angular velocity, about a diameter of its equator; it is proposed 

 to explain the rotatory motion of the spheroid consequent to the impulse so received. 

 — In fig. 18, 19, let DOEQ be the spheroid, whose semi-axis co = cq is =: b, 

 and equatorial radius cd = ce = ;•; and supposing it before the impulse to re- 

 volve about its proper axis oa with the angular velocity c, measured at the dis- 

 tance r from the axis, let the poles (o and a) be suddenly urged by some percus- 

 sive force to turn about a diameter of the equator of the spheroid, with the an- 

 gular velocity (/, likewise measured at the distance r from that diameter. On 

 receiving such impulse, the spheroid will take a new axis of motion, which will 

 be a momentary one; suppose such new axis to be pcn-.* Then the particles of 

 the spheroid being urged (or having a tendency) to turn about that axis with the 

 angular velocity \^ (c" -|- d'), (which we will denote by e) their joint centrifugal 

 force will so urge the spheroid to turn about that diameter of the equator which 

 shall be at right angles to the momentary axis pcir, that the accelerative force of 

 the point d of the equator to turn it about the said diameter according to the 

 order of the letters oaE, will (as appears by what is proved in art. 1, and in the 



appendix following) be = — X ttt^' ^^~ ^ .z , ai according as b is less or 

 greater than r; and it follows from hence, and what is proved in art. 3 and 4, 

 that v, the angular velocity (at the distance r from c) with which the momentary 

 pole/; will change its place, will accordingly be = — X ., , -, or - X ^ ~ ^. 



Again, referring to our observation in art. 8, let u — e be to - x — — rJS^c 



value of v) as c to d, u being greater than e; or let e — m be to - x - , ~ ^^ as 

 c to d, u being less than e; whence, in both cases, we shall have the same ex- 

 presssion ( - X -rrj^ ^°'' ^'^^ value of m — e ; and consequently u, in botli cases, 



f-t yl 1,1 



will be := e -j — X -; — ,„• Conceive now a spherical sui'face without matter, 



e r'- -\- 0^ ^ ' 



* To find the position ot this axis see art. 1, by which the sine of the angle ocp (to tlie radius /) 

 dr 

 is found = — . 



