RpBliRTS — On the Equation of the Tangent. 65 



As a result we find that equation (8) becomes divisible by h, and can be 



written in the form 



(Ay + GnA<p)h + GjiAA'yii+ 9A*(2B + </b) + A'A'^) = 0. (11) 



To show that (10) is divisible by h, we form the following table, aud 



write 



$' = <p + h(aJB - ?>A(b„0 + &,)) - b a Alr, 



A' - A = //((',,/( +, Ai), 



Q'-Q = h(qfi+Q 1 ), 



A$'-A'4> = h:-h0i o <l>+b a A-) + A[a i) B-3Ab u + b,)]-<j ) A l <,, 



.. J = 3BA, AZABu 



30 = J - AB X , 



A' = 2{QA 1 -AQ l ), 



x + 3* = — — e = — — \ab, + a:-k- ibj\ 



2^B 2yB ( ) 



We now find, on dividing by Ji, 



(A + hA t + Ira:, tf + lSA^B \- h (a f + b,A 2 ) 



- A . [a„B - 3A <b o + 5,)] -<p.A t \ 

 + 3A{'a k + A,) + SAB [a 7? - SA b 9 + &,) - 6 ^A] 



+ 3.4 3 [a.^Jv + /i. [^,^0 + a^ + §,^1, + g^] } 

 + 6pA\<l> + h a B - 3a %0 + J, ] - M^l = 0. [12 



This equation then is the quadratic in 9' which determines the parameters 

 of the two points in which the tangent at again meets the curve. Now. 

 if we let 0' = 6, or /( = 0, in this equation ; then one value of 6' will equal 6, 

 aud the tangent becomes an inflexional tangent to the curve. The last term 

 then of the equation when put equal to zero, gives us the equation of the 

 lines joining the points of inflexion on the curve to the node. This result 

 when cleared of radicals can be written in the form 



imB'BQ, + 4JIRQ] ~ A*Q* = 0. (13) 



where 



11 = -iBD - 3AI, 



£2 = J(QJ - KB) + -LB . (AD n - l) Q) + \ A-R'\ 



- K- = m\I/A- - 2D n AQ - Wf\, 



where I) = a a 2 - «,'-, 



B 12 = a q, + <>_■!„ - 20,0,, D' m ffpff, - qf, 



/ = ye" . (aj> 3 - 2a,6. f a 2 i,). 

 Now this is an equation of the twentieth degree, and as there are eighteen 

 inflexions the equation must contain A as a factor. 



