ALONG THE CIRCUMFERENCE OF A CIRCLE. 61 



: : NA : NE : : NE : NZ : : &c. ; and seek the ratio of the two velocities, viz., of a 

 at the point a, and of (3 at the point (3. 



5. Putting V B for the initial velocity of (3, and vp for its velocity at the point (3 

 we have, — 







V B 2 : Vj3 2 : : BN : BH : : BN : BN-NH 







: : NZ . BN : NZ . BN-NZ . NH 







::NF 2 : NF 2 -N/3 2 . 



But we have also 









V A 2 : V 2 B : : NE 2 : NF 2 , 



wherefore 





v 2 • v „* : : NE 2 -Na 2 : NF 2 -NB 2 



a • a 



Now, from our 



construction, 







NE 2 : NZ 2 : : N« 2 : N/3 2 



wherefore 





NE 2 : NZ 2 : : NE 2 -N« 2 : NZ 2 -N/3 2 



but 





NZ 2 - N/3 2 = Z/3 2 , 



wherefore 





NE 2 -Na 2 : NE 2 : : Z/3 2 : NZ 2 



and similarly 





NE 2 : NA 2 : : NF 2 -N/3 2 : NZ 2 -Na 2 ; 



or, 







NE 2 : NF 2 -N/3 2 : : NA 2 : Za 2 . 

 Compounding these ratios we obtain, — 



whence 



NE 2 -N« 2 : NF 2 -N/3 2 : : NA 2 . Z/3 2 : NZ 2 . Za 2 , 

 v a . Vp : : NA . Z/3 : NZ . Z« . 



6. Let us now suppose that the body a moves through an exceedingly minute 

 distance, represented by aa, and let us make the proximate chord N/3', in the 

 same ratio to Na' as before ; then, since N« may be held equal to Na and N& to 

 N/3, we have aa' : b(3 : : Na : N/3. 



The minute trigons aaa and b(3(3f are similar, respectively, to aNZ and /3NZ, 



wherefore 



aa' : aa! : : NZ : Za 



6/3' : jSjS' : : Z/3 : NZ . 



By compounding these three ratios we obtain 



aa' : /3/3' : : Na . Z/3 : N/3 . Za 

 : : NZ . Z/3 : NF . Za . 



7. On comparing the lengths of the arcs aa' and /3/3', and also the velocities 

 with which they are passed over, we find that the minute intervals of time are in 

 the ratio 



NZ NF 

 time in aa' : time in /3/3' : : r— - : z—z - : : NZ : NE . 



NA NZ 



