[ 523 ] 



n . 



n is even the fluent o£ mz z can be found by help of 



r" I r* — 22 

 the circle. 



Therefore if the value m be taken according to the different 

 values of n fo that z may become equal to r, innumerable fo- 

 lutions w^ill be had. 



By thefe and other obvious methods innumerable folutions 

 may be obtained. Curves may be obtained that fliall pafs through 

 neither B nor C, or through B and not through C. 



The curves above obtained are, except the circle, fpirals. 



4. If it be required thai the curves be fuch that the relation 

 betvi^een the abfciifa C D = ;f, and the ordinate D« = y may be Fig. 2. 

 algebraical ; fuch curves may be found, but not fo readily as the 

 above. When the curve is algebraical z is an algebraical fundion 



of X and y, whence, becaufe z = — — and z = i— , it follows that 



s, a cs, a 



/, a and cs, a are algebraical fundtions of z. 



I . Let s,a = — , then cs,a = ^^ — 

 r r» 



* z I 

 therefore a = - x , 



r \J I — z^ ^'r» — z^ 



This 



3U2 



