VOL. XXX.] PHILOSOPHICAL TRANSACTIONS. 311 



8. From the foregoing it is easily collected, that all the curves abd are similar, 

 and similarly posited about the given point a, their like sides being proportional 

 to the parameters «. 



Solution of the id Part, viz. the finding the cutting Curves. 



g. From art. 2, it is ?; : ir :: a" : or . But bc : bh :: z; : k. Hence bc : bh :: 

 aP'.af. But, by the condition of the problem, bc is a tangent to the required 

 curve ebf. Therefore if we now take ah (z) and bh {oc) for co-ordinates of 

 the curve ebf, the curve itself eb being called r; then, by the direct method 



of fluxions, r : — i :: (bc : bh ::) aP-.ocf. Hence — = —- .. 



10. In the curve abd suppose the equation 



z = — —■ to be transformed into the equation k = a.v -t + bx -^ -\- &c. 



not affected by radical signs. Then, by returning to the fluents, 



^ = ,7+1 ^— + §^1^ -^- + 5^TT^ -I^+ ^^- ^^^'^ "° "^^ coef- 

 ficient is introduced, because, by the conditon of the problem, x and z must 

 begin together. Here, instead of 

 - substituting its value ~ by art. Q, then z=z -^ax—. -\- ~~ bx — ' + 



&c. which is a fluxional equation of the first order belonging to the required 

 curve EBF. And this is reduced to a simpler form in finite terms in the follow- 

 ing manner. 



1 J . Let r flow uniformly, and a being a constant quantity, make 



ZLf = ^; then this value of ~. being substituted in the equation last found, 

 and multiplying the equation by -, it will be transformed into this, 



'"+7"'" = A.f. + ^s%hc. = :;^. This last appears from the 



analogy of the two series Ai — + &c. and a^ ^ + &c. Hence, substituting 



for s and s their values, derived from the equation — ' = S-j there will arise the 

 equation nx^kz — xxkz — nxxk'^ — xj^x'^ = O; which is reduced to first fluxions 

 in the following manner. 



12. In the last term — xxx^, instead of 'd\v writing its value — - zz, and then 

 dividing the equation by i, there arises nx\ — xxz — nxxz'-\- xxz-=. 0. Which 

 equation multiplied by a?""'', is the fluxion of the equation — xx"" z + a7"i =: 

 c?'-"r, the quantities d and r being constant. Therefore this equation, or 



