go PHILOSOPHICAL TRANSACTIONS. [aNNO 1704. 



wind, but some of them also thrown up or exhaled into the clouds. In the 

 said small particles of water are conveyed the above-mentioned small animalcule, 

 far up into the land ; and when the ground becomes dry, they contract them- 

 selves into an oval figure, and the pores of their skin are so well closed that they 

 do not perspire at all, by which they preserve themselves till it rains, when they 

 open their bodies, and enjoy the moisture. 



The Solution of a Problem, proposed by Mr. John Bernouilli, in a French Jour- 

 naly Feb. 1703. By the Rev. Mr. John Craig. N° 289, p. 1517. Trans- 

 la ted Jrom the Latin. 



Problem. To find innumerable curves, that shall be of equal length to a pro- 

 posed geometrical curve. 



Solution. Let iv, s denote the co-ordinates of the given curve, and cc, y 

 those of the curve sought : then from the condition in the problem it will be, 

 ti;'* + ** = •^ + y^' Put .f = w; — m'z\ then will ^ = */ s"^ '\- 2mivz —m'^z*: in 

 this equation, for s letits value besubstitutedas expressed by «;,«/, and determinate 

 quantities ; and for z let such a value be assumed, composed of w, iu, and deter- 

 minate quantities, as that the fluents of i' and y may be found. And thus will 

 be determined a: and y, the co-ordinates of the curve sought. 



Example 1 . To find a curve equal to a parabolic line. Let 2a be the latus- 

 rectum of the parabola: then is 2as = t^% and s^ = a-^ww^, and therefore 

 y •=^ V a-^'^wV -f 2mwz — m^z^. To find the fluent of this, assume m'z = 

 a-^'w^w'^ hence x =. w —a-'Ww, andy z= w \^ 3 a-V — a -%'*; the fluents 

 of which, by ibe methods already known, are 



Example 2. To find a curve equal to the circular arch. Let a be the radius 

 of the circle ; then is 5 = ^ a'^ — w^; hence s^ = , ^ , , and therefore 



y = \/ ^ ^ 4- 2mwz — rr^z^. To find the fluent of this, assume mz = 

 1^; theni = w — -^, and y = "~ ° ^ m^; the fluents of which, by the 



common methods, are found x = w — r^, and y = — r-j— ^ (^ — w;^ 



Example 3. To find a curve equal to that of an ellipse. Let 2r be the 

 latus-rectum, and 2a the transverse axis : then is 



s =s \ ; hence s^ = ~, and v = V -7 n + ^tww-j — ir^z". 



