450 



MR EDWARD SANG ON THE MOTION OF A HEAVY BODY 



Fig. 4. 



Fig. 6. 



a right angle, at which instant ABC has reached a maximum value. After 

 that, CAB still continuing to increase, ABC must decrease and become zero 



just when CAB becomes 180°. Afterwards, when 

 CAB has become reverse, or greater than 180°, 

 ABC appears on the other side and reaches its 

 maximum value on that side when CAB = 270°. 

 Thereafter ABC decreases to become zero just 

 when CAB = 360°. 



Thus the continuous growth of the angle 

 CAB may typify the continuous motion, while the 

 reciprocating angle ABC typifies the oscillatory 

 motion of the heavy body. Seeing, then, that the 

 general phases of this arrangement represent the 

 leading characters of the two kinds of motion, 

 we may inquire somewhat more narrowly into the 

 resemblance. For this purpose we shall put ACB, 

 fig. 9, for one form of the changeable trigon, and 

 imagine its form to be altered by turning the arm 

 CA into the closely approximate position Ca, so 

 that ACa becomes the decrement of the angle 

 ACB, while ABa is the increment of ABC. These 

 changes being supposed to be infinitesimally 

 minute, the arc Aa may be regarded as a short 

 straight line perpendicular to CA. Draw CP and 

 ae perpendicular to AB ; then the minute trigon 

 Aea is similar to CPA, whence Aa:ae: :AC: AP. Now the angle ACa is 



Fig. 7. 



Fig. 8. 



Fig- 9. 



Aa 



expressed by -^ , and ABa by j-j. , wherefore 



AC«:AB.::£:£::*°:g::AB:AP. 



Hence, if we regard ACa as the differential of the exterior angle ECB, we 

 have 



d . ECB : d . ABC : : AB : AP , and consequently 



d . ECB : d . CAB : : AB : PB , 



so that the differentials of the three angles ECB, CAB, and ABC are propor- 

 tional respectively to AB, BP, and PA. If then we suppose the angle at A to 

 be generated with a velocity varying as the distance PB, that at B will be 

 generated with a velocity proportional to AP, and the exterior angle at C with 



