66 MR EDWARD SANG ON THE MOTION OE A HEAVY BODY 



and thus the velocity at M due to a descent from the level of C is exactly that 

 which the moveable point 7 has at the same place. 



16. We may now examine the velocity which this same heavy body would 

 have at any intermediate point as 7. The general expression for that velocity is 



v y = J {G y (NP — N7 2 )} = J {NZ . NE (NP — N7 2 )} 



but since 7 is the middle of the arc a (3, 



N7 2 = } {NZ 2 + Na . N/3 - Za . Z/3} 



subtracting this from the value of NI 2 and simplifying 



NE 2 . NZ 2 + NZ 2 . NZ 2 + 2NE . NZ . Za . Z/3 - 2NE . NZ . Na . N/3 



Ni 2 - N7 2 = 



4NE . NZ 



Now NE . Na = NZ . N/5 so that the continued product NE . NZ . Na . N/3 may be 

 written either NE 2 . Na 2 or NZ 2 . N/3 2 ; writing it once each way we obtain 



N1 2 _ N 2 „ NE 2 . Za 2 + 2NE . NZ . Za . Z/3 + NZ 2 . Z/3 2 

 7 ~ 4NE.NZ 



_ (NE . Za + NZ . Z/3) 2 

 4NE . NZ 

 wherefore 



, 7=7 {NE. KZ (NE.Za E+ KZ. Z ^ | =HNEZa + Nzzft; 



but we have seen that NE . Za is the velocity of (3 at the point /3, NZ . Z/3 that of 

 a at the point a, so that 



and thus, at every point of the circumference, the velocity of a body projected from 

 N with a velocity due to a descent from Z, and acted on by a gravitation having 

 its intensity represented by NZ . NE, is equal to the velocity of the middle of the 

 arc a /3. 



17. The motion of the body 7 round the circumference has for its conjugate 

 that of a fourth, which we may name 8 ascending from N to K, and thence 

 returning to N, while 7 rises from N to Z, and proceeding onwards, returns to N ; 

 the conjugation being analogous to that which connects the motions of a and (3. 



Hence, if we inflect the chord N£, a fourth proportional to NZ, NK, and N7, 

 we shall obtain the point at which the body 8 is found when 7 is at 7 ; and if 

 we make G§ a fourth proportional to NZ, NC, and G 7 , we shall obtain for the in- 

 tensity of the gravitation to which 8 must be subjected 



NZ : NC : : NZ . NE : NC . NE = G.. 



