Mr. J. J. Sylvester on Successive Involutes to Circles. 459 



Solid product. 



Na * + C 5 lP(n _ ^Nal C*H90\ , C^Ol 



JNa +^ c 2 H 5 / U - C 2 IP/ 0+ Na/ + C 2 !! 5 / ' 



On the addition of water to the solid product we should have 

 caustic soda, alcohol, polyvalcral, and valerianic ether, in the 

 proportions which the above-described experiments have indi- 

 cated. 



Folyvaleral . . . nC 5 H 10 0. 



The properties of this very interesting compound will be more 

 fully described on a future occasion. It is an oily liquid of a 

 very pale yellow colour (whether this colour is essential or not I 

 cannot tell). Boiling-point about 21 5° C. It has a very power- 

 ful and very peculiar smell. " Limited oxidation " appears to 

 give only valerianic acid. Its spec. grav. in the liquid state is 

 about 090 at ordinary temperatures. 



London Institution, 



November 1868. 



LVIL On Successive Involutes to Circles. — Second Note*. 

 By J. J. Sylvester-}-. 

 [With a Plate.] 



SINCE the appearance of the former Note on this subject, I 

 have enjoyed the inestimable advantage of securing the 

 cooperation of my all-accomplished and omni-capable friend Mr. 

 Spottiswoode, to whose kindness and skill my readers are indebted 

 for the beautiful figures given in the annexed Plate, which I shall 

 proceed briefly to describe, and which, as far as I can learn, offer 

 the first examples of the actual visible representation of any de- 

 rived involutes of the circle beyond those of the first order. I 

 propose, for want of a better word, provisionally to give the name 

 of Cyclodes (suggested by Professor Cay ley) to these spirals. 

 They may be considered a genus of a more general class of 

 spirals which I propose to name algebraical spirals, defined by 

 the condition that the perpendicular on the tangent from a cer- 

 tain fixed point (which may be termed its pole) is a rational 

 algebraical function of the angle of contingence; so that a cy- 



* The thought foreshadowed in the concluding paragraph of the former 

 note leads to the following theorem. 



Let/, <fi, yjr be quantics in a., /3 ; F the unicursal function obtained by 

 elimination of cc, j3 between 



*—/* y=<$>> z=^-> 



A x . F the discriminant of F regarded as a quantic in x and 1 ; J((/>, y\r) the 

 Jacobian of <p, \|/-; R the result of eliminating <fi, \^ between 



y = cf>, z=f, J(</>, ^)=0; 

 Q the product of all the homogeneous linear functions of y, % which vanish 

 at the double points of F; then I say (and the proof is all but self-evident) 

 A*.F = R.Q 2 . 



t Communicated by the Author. 



