328 Mr DA VIES on the Equations of Loci 



In the set (15), both equations a,, and & are fulfilled by the same value 

 cos r 0, or r = . . 



.4 



There is hence, in this case, a possibility of rationalizing equation (6), 

 whilst R remains perfectly arbitrary. Here, the element of the length 

 of the epicycloid is 



. sin ' 



sin R 



.(18.) 



This is, in fact, the solution of JOHN BERNOULLI, and agrees with his 

 result. We shall return to this expression presently. 



In the set (16), we have equation e ltl fulfilled only by sin R 0, or 

 R zz n TT ; and b,,,, d llt are also fulfilled by the same. But a tll and c ltl are 

 not fulfilled by that value of R : and hence, on the hypothesis of R + r 

 being taken, r cannot be rendered indeterminate by any value whatever 

 ofR, 



In the set (17), too, we have the absurdity 1 zz 0, given by equation a, T . 

 This, then, is also impossible. 



Hence we have proved that JOHN BERNOULLI'S is the ONLY solu- 

 tion possible, of finding a spherical epicycloid accurately rectiftable : so 

 long, at all events, as the tracing point is in the periphery of the rolling 

 circle. 



In the numerical evaluation of the integral of equation (18), we have to 

 recur to some considerations respecting the conditions of the formation of 

 (11, 12, 13, 14). It will be recollected that these are the expansions of 

 cos Rrhr, in which we have worked as if R were the greater radius- vec- 

 tors, or, in other words, as if the direction were the tropic most remote from 

 the pole of reference P. But as the results of all the operations we have 

 hitherto performed are the same as if we had employed cos R + r instead of 

 cos R r, our conclusions apply equally to one case as to the other. We 

 may, however, consider that the above expression (18) is adapted alike to 

 both cases. The double sign attached to (18) indicates that the length 

 may be estimated either way from the origin of the arc, that is, the same 

 way as we have estimated 6, or the reverse. Put the length zz L : then 



+ cos <p + c 



sin R 



(19.) 



