ALONG THE CIRCUMFERENCE OF A CIRCLE. 



69 



Hence the following very simple construction : — Having made NS the fourth 

 part of the arc NK and drawn a horizontal line through N, join OS and produce 

 it to meet the horizontal tangent at T ; draw TU 

 perpendicular to OT, meeting ON produced in U ; 

 lastly inflect NF double of NU ; then the time of 

 descent along KN is to that of descent from F to N 

 in the ratio of OU to ON. 



By repeating this operation in regard to the arc 

 NF, we should obtain a much smaller arc, and 

 thence again one still smaller, and so on ; the 

 periodic times in these successive arcs bearing 

 known ratios to each other. Now it is obvious, 

 from a glance at the figure, that these arcs, which 

 we may denote by 2B , 2B 1? 2B 2 , &c, form a very 

 rapidly decreasing progression ; and that the ratios 

 of which ON : OU is the first, approach at the same 

 time to a ratio of equality ; hence the time of descent along KN may be deduced 

 from the time of oscillation in an exceedingly minute arc. A very familiar in- 

 vestigation shows that the time of oscillation in a small arc is almost independent 

 of the extent of the arc ; but instead of founding on this well-known proposition, 

 I prefer to deduce the truth of it from our present considerations. 



25. If we suppose the arc NF of figure 2 to be very minute, the height NA, 

 which is inversely proportional to NZ, must become very great in proportion to 

 the diameter NZ, and hence the velocity at the point Z must be nearly equal to 

 that at N, the two being in the ratio of ZE to EN ; and the time in which a body 

 descending from A describes the circumference must always be greater than that 

 in which another moving with a uniform velocity equal to that at N would de- 

 scribe the same circumference. Putting g for the actual intensity of gravitation, 

 the velocity acquired by falling from A to N is 



V A = */(2g . AN) 



so that, as ttNZ is the length of the circumference, the value of T A must be 



NZ 



greater than 



ttNZ _ //NZ\ /NZ //NZ\ 



<s/2<7 . AN - W V ty ) ' sl AN ~ ^ si \~2g~ ) ' 



NZ\ NF 

 NZ 



while, since the velocity at Z is to that at N as ZE to NE, the same time must 

 be less than 



//NZ\ NF .,_, 



V {~w) m ■ sec Nzr - 



26. Now we have seen that 



T a : Tp : : NF : NZ 



