46G Mr. J. J. Sylvester on Successive Involutes to Circles. 



wonderful theory, this outlying and unexplored region of geo- 

 metry, in which the two great continents of algebra and arith- 

 metic trend towards and come into contact at more than one 

 point with one another, forms the subject of a communication to 

 be brought by the author of this Note before the Mathematical 

 Society of London, simultaneously and under the same roof with 

 Mr. Norman Lockyer's announcement to the Royal Society of 

 his equally, but not more surprising and certain to be prolific 

 discovery of the sun's unsuspected chromosphere, the analogue 

 of the ocean of forms of which the isolated power-forms 

 [((/> 2 — n 2 ) w ] correspond to the piled-up rose-coloured prominences. 



Athenseum Club, 

 November 23, 1868. 



Errata in No. 243. 



P. 295, footnote %,for finite solution read finite rotation. 



— 296, for s x =a(fil+b(ji read 5 1 =a^_+60. 



ft, ?+($* read ^ + (|) W. 



— 297, footnote, for G=s'—s'" ... 5 G'=s" — s"" . . . read 



G=s— s"+ ... ; G'=s'-s'" . . . 



— 301, dele clause commencing with the words "at such points the 



curve" and ending with the words " points of retrocession." 



— 303, in concluding line of fourth paragraph omit the words " either 



as an abortive loop or ". 



ought to be regarded as a number rather than as a negation of number ; for 

 the order of the system of equations is always lowered, not only by every 00 

 becoming equal to every y, but also by any number of a?'s or of y's beco- 

 ming equal to each other ; so that the order of the system sinking to zero, 

 in consequence of all the #'s and all the y's becoming equal, is only an ex- 

 treme instance of this general law. If we go to the wider case of alge- 



/(<£) 

 braical spirals, where p == -p^y the difference between the degrees of /and 



F being still an even integer 2m, where m is positive or negative, and re- 



( dp \ 3 

 quire p 2 +[-rr) to be made a perfect square, precisely the same method 



of solution is applicable as when F is of the degree zero. If we call the 

 degrees of /and <p k and k respectively, so that K — q—2m, we have to 

 make 



a?i+x 2 4- .... +w -£i-£ 2 ... -^ = m, 



2/1+2/2+ • • • +Vk-Vi -V-2 • • • ->V = m > 

 e+^=X+/n=i, where i takes all possible values, 



^+#2+ . . . +% e +y 1 +y,+ ■ • • +V\=k, 

 £i + £ 2 + . . . +fe,+7i-H7.+ • • .+V* = q. 

 Every such system of partitions give rise to a system of equations contain- 

 ing solutions of the diaphantine problem in question, i. e. the problem of 

 making r a rational function of <£. When the degree of p in $, i. e. k — q 

 (and consequently^) is zero, theorderof all the equation-systems undergoes 

 a marked depression. 



