ALONG THE CIRCUMFERENCE OF A CIRCLE. 453 



EG = V(CE . ED), GF = i (CE + ED), obtain a third trigon EGF ; and we 

 can continue this series of trigons indefinitely. The ratio of CE to ED is much 

 nearer to a ratio of equality than is AC : CB ; EG : GF is still nearer, and after 

 a very few steps the ratio of say GI to IH becomes, sensibly, an equality. By 

 continuing the progression in the opposite direction, that is, by making- 

 Ad = CB + ^/(CB 2 - CA 2 ), AC = CB - ^(CB 2 - CA 2 ), we obtain trigons 

 more and more scalene, the ratio of disparity of the two sides increasing with 

 greater rapidity at each step. Hence of the general formula 



dt.J'Ig = j (§2 _ r2 giu g2) , 



we can by continuing this series either way, render s equal to r, or greater or 

 less than r in any enormous degree. The integration in these three cases must 

 be considered separately. 



In the first place, let us suppose that s, represented by cA, is infinitely small 

 in comparison with r, represented by Ad. Here the arc S can never exceed a 

 certain infinitesimally small limit, so that it may be held to be equal to its sine, 

 and thus the formula becomes 



dtj(?g) = j (§2 _ r2 g2) , 



which is easily integrated, as in the familiar case of isochronous motions. 



In the second place, when s, represented by Ad, is many times longer than 



r, or Ac, the velocity of the moving point being proportional to gd, may be 



S 

 held as constant ; in which case the integral becomes t J (2g) — - ; this limit 



has been used in the previous paper. Both of these limits belong to the 

 inverse progression, which leads directly to the result already explained. I 

 shall, therefore, now direct attention to the third case, in which r and s have 

 been rendered equal to each other. 

 The equation now becomes 



dtj(2g) = rV(I _ ginS2) = ^.secS.^S, 

 which has for its integral 



tj{2g) = I log tan (| + §) ■ 



so that if a heavy body be projected from the nadir-point of a circumference, 

 with a velocity just due to a free descent along the diameter, the time in which 

 it describes a given arc is proportional to the meridional part on Mercator's 

 projection of the sphere corresponding to a latitude homologous with the half 

 of that arc. 



VOL. XXVI. PART IT. 6 B 



