300 Mr. J. J. Sylvester's Note on Successive 



From what precedes, and from the general theory of elimination, 

 it follows that in general the relation between , - and s is expressed 

 by a rational integral equation of the degree (2 + 1) in the former 

 and 2i in the latter. But this is subject to an obvious exception 

 in the case of z = l ; for then, calling G</> = #</>, we have 



j| =a<j>, s= *¥- + b, and r* = a*cj>* + a 2 =:2as+ (a*-2ab) ; 



so that the degrees in r 2 and s are here 1 and 1 in lieu of 2 and 

 2, as given by the general rule. 



As regards the polar equation to the general involute, 



it is obvious that, agreeable to the well-known case for the first 

 involute, 



dr G' 



^=sin" 1 ~+6= sin- 1 - + <£*, 

 as r T 



where G' and <£ are given by the solution of an algebraical equa- 

 tion of the 22th degree, and which will therefore usually be 

 incapable of expression in finite terms beyond the second invo- 

 lute. In the case of this involute the reducing equation is not a 

 general biquadratic, but a form involving only square and no cube 

 roots — being in fact reducible to a quadratic in <£ 2 , as will at once 

 be seen from the fact that we may write (a<£ 2 + Y) 2 -f-4a 2 <£ 2 = r 2 . 



ds 

 Since -77, i. e. G + G", is a quantic of the degree i or </>, we 



* (f) and G' will form 2i systems of values. Will they be all applicable to 

 the true involute, and how about the sign to be given to r? It must, I 

 think, be a matter of some delicacy and difficulty to answer these ques- 



a _i_ / t 2 a 2 

 tions. For take even the first involute, where #=sin -1 . - ~r \/ 



T a 2 



we know, as a matter of fact, that if the first term is made to decrease as 

 r increases, the positive sign of the square roots only must be employed, and 

 of course the negative sign if the first term increases with r. Were we to 

 reverse this rule, instead of the involute we should obtain what may be 

 termed the counter-involute ; i. e. a figure formed by points, each the op- 

 posite of every point in the involute in respect to its centre of curvature. 

 Or, again, if a pair of parallel rulers were made always to touch a circle at 

 opposite points, and the under parallel to roll round the circle, whilst 

 each point in this line describes an involute, each point in the one above 

 would describe a counter-involute. Or, again, if a string, by aid of a pin, 

 were unwrapped back upon itself from a circle, the extremity would describe 

 the extraneous curve. From this last observation it would seem as if the 

 forced intrusion of a foreign curve into the polar equation of the involute 

 resulted from the impossibility of affixing an absolute sign to the length of 

 an arc — the condition of drawing a tangent always equal in length to the 

 varying arc of* a curve admitting of satisfaction without breach of continuity 

 in two distinct modes. 



