)) 



Note on Successive Involutes to a Circle. 295 



Polarized portion of the 

 heat radiated at an 

 Substance. angle of 35°. 



Tinned iron .... 27*6 per cent. 



Copper 22*4 „ 



Aluminium .... 28'5 



Mercury 32*0 



Transparent glass . . 10*4 „ 

 Black glass . . . 12*4 „ 



Rape-oil 5*64 „ 



Colophony .... 7*26 „ 

 White wax .... 7*3 „ 

 Glycerine .... 5 '61 

 Paraffine .... 5*0 



>•> 



)) 



From these results we are justified in concluding that all sub- 

 stances, whether solid or liquid, radiate from their flat surfaces 

 heat which is partly polarized when it makes an angle of about 

 35° with the surface. 



i 



XXXIX. Note on Successive Involutes to a Circle*. 

 By J. J. Sylvester-}-. 



T is surprising that the families and groups of families of forms 

 capable of being educed by successive involution from a 

 circle should not have attracted the attention of geometers. I 

 find, if any, not more than a passing allusion to their existence in 

 Dr. Salmon's ' Higher Plane Curves/ in the f Integral Calculus' of 

 Mr. Todhunter, or in the memoirs of the late Dr. Whewell in the 

 Cambridge Philosophical Transactions ( vols, viii andix.), although 

 these latter are exclusively devoted to the study of various curves 

 of mechanical and kinematical origin by aid of the so-called in- 

 trinsic equation, which is in fact the natural expression of, and key 

 to, the properties of such like curves. And yet this form of equa- 

 tion almost instantaneously furnishes the general polar equation 

 to the entire system of circular involutes, and exhibits at once 

 their leading properties J". 



* The germ of this Note was communicated to the Mathematical Sec- 

 tion of the British Association at the Norwich Meeting. 



+ Communicated by the Author. 



X Professor Rankine and Mr. Merrifield have made a useful application 

 of the second involute of the circle to the calculation of the stability of the 

 finite solution of vessels. In the Turkey carpet under my eyes whilst this 

 is being written, I perceive graceful and complicated figures of winding 

 and intersecting scrolls and convolutions, which render it, I think, not at 

 all improbable that the successive involutes of the circle would furnish 

 or suggest many patterns available for decorative purposes : the enormous 

 variety of each kind of involute, which of course increases with the order 

 of derivation, adds to the probability of this conjecture. 



