Solution of a Problem in Fluxions. 287 



rdv^ — d^r 

 same as 5 ' and SP^ xQT2=r''<?u2,). The use of 



the equation (E) is to determine the curve when the force 

 F is given, and it is obvious that it requires F to be a func- 

 tion of r, since it involves the integral ^¥dr -, also (H), (K), 

 and (I), are for the purpose of determining the law of force 

 for a given point in a given curve ; either will answer, but 

 that one ought to be used which will accomplish the object 

 most expeditiously. It appears to me, however, that (I) will 

 generally be found to be quite as easy in practice as either 

 of them : an example of this may be given in the case of 

 the logarithmic spiral, the centre of force being at the cen- 

 tre of the spiral ; in this case 4- is invariable ; since r always 



cuts the curve at the same angle, .*. by (I) —^ =F, 



1 

 .'.F varies as Tj for different points of the curve, (which 



agrees with prop. 9th, sec. second, Prin,), if 4-= a right an- 

 gle, the cosec. ■4'=1, and the spiral becomes a circle, and 



=F=const. for the same circle, and for different cir- 



f 3 ' 



dv^r^ 

 cles substituting for c'^ its value "jyl"' it becomes 



~di^)=y =F, by puttmg-^-=V=the velocity, (which 



r 

 agrees with Prin. sec. second, prop. 4th, cor. Isl.) Again, by 

 taking the finite difference of (D), relatively to c'^ and F, 



d^r Dc'2 



regarding r and ji^as constant, I have — - — =DF,(D be- 

 ing the characteristic of finite differences,) if Dc'^ is consid- 

 ered as constant, DF varies as— (which in prop. 44th, sec. 



9th, Prin.) I shall here leave the subject, as I suppose I have 

 said enough and perhaps too much already. 



CORRECTIONS. 



Page 284, 6th line from bottom, dele (z) and insert (2). 

 u u 3d « " « (y) " " (4). 



" " bottom line, insert dv after sin. in numerator thus, sin. edv. 

 Some smaller corrections, not deemed important, have been omitted.— JSd. 



