324 Mr DAVIES on the Equations of Loci 



and he published, in the first volume of the Petersburg Commentaries, an 

 attempted demonstration of it *. JOHN BERNOULLI, in the Memoirs of 

 the Royal Academy, 1732 f, exposed the fallacy of this supposition, and 

 pointed Out the paralogism committed in the reasoning. He at the same 

 time gave a general expression for the length, and deduced from it a parti- 

 cular case, which actually did solve the problem of OFFENBERG f . By a 

 singular coincidence if, indeed, it were really accidental the attention 

 of MAUPERTIUS, NICOLE, and CLAIRAULT, were directed to the same 

 question, and they obtained the same kind of results. Their investigations 

 are printed in the same volume of the Memoires as that of BERNOULLI . 

 We perceive, by inspecting our equation (3), that the curve is algebrai- 

 cal or transcendental, according to the commensurability of sin R to sin r, 

 just as the case with the directing and generating radius of the epicycloid in 

 piano. To, ascertain the conditions of capability of rectification, we may 

 form the differential equation of the curve ; which is done by differentiating 

 (3). Thus we have 



, sin r _ sm(f>d(f) 



cosec <p (cos R + r cos f> cos 0) d <f) 



(4.) 



V (sin 2 R+r cos*>) + 2cosRr cosp cos(p cos*(p 



One essential condition of / ^/ sin *<p d 6* + dtp 1 becoming rectifiable, is, 

 that the quantity under the radical shall become a perfect square ; and, as 



* There is a copy of HERMANN'S dissertation inserted in JOHN BERNOULLI'S works, 

 vol. iii. p. 211. 



f Page 240 ; also his works, vol. iii. p. 220. 



Mem. de 1'Acad. 1732, p. 243. Opera Omnia, torn. iii. p. 223. 



MAUPERTIUS, p. 255 : NICOLE, p. 271 : CLAIRAULT, p. 289. It is rather singular 

 that Dr YOUNG, in his Catalogue, refers to the paper of MAUPERTIUS, but does not men- 

 tion those of NICOLE and CLAIRAULT, which follow them in the same volume. Dr YOUNG 

 mentions a paper on Spherical Epicycloids, by LEXELL, in the third volume of the Peters- 

 burg Acts ; but I have no means of consulting that series of Memoires, there not being a 

 copy in this city. 



