On the Quadrature of Surf aces and the Rectification of Curves. 209 



DLnoUnl^* 11 - " T imUm /° rCS Whcn the re P ulsi ™ acts 

 which i . JS T\ ??>' ES bef01 ' e ' We aW at a co » clu sion 

 crttal nf Tl \ d f h ? ex P. enment i for the repulsion of a 

 o r ;l f c «bonate of lime is a maximum when the repelling 



that e th a e Ct e l r t n§ ^ , aX1S i° f thG C1 ^ Stal - Heilce i ^ *>& 



thai ofS^of lr b ° nate ° f ^ CaMOt bG ^ — aS 



minatnX 1116 ? fle f on !^ presented themselves to my 

 mind on the evening to which I have referred. I now submit 

 them to you as a fraction of that thought which yomla 

 memo,r upon this great question will assuredly awaken. 

 Believe me, 



Dear Mr. Faraday, 

 Royal Institution, Yours very faithfully 



Febmar - v1855 - John Tyndall. 



^cft}on h nf°r mS m tJ % Q ft atW ' e °f Sur M<" <** the Rectifi- 

 ed on of Curves By the Rev. Robert Carmichael, A M 

 fellow of Tnmty College, Dublin, and Examiner in Math;'. 

 mattes for the year 1854 in the Queen's University in Ireland*. 

 1. TT is well known that there are many plane curves whose 

 ™~T f^ions are more easily expressed in polar than in 

 theSula C ° 0rdmateS ' aud for whos " rectincatiLweemplo? 



Of this class are, the spiral of Archimedes, 



it v r = ae > 



the ntuus, 



the lemniscate, 



the logarithmic spiral, 



and the cardioid, 



-m 



?- 9 = « 2 COs20; 



r=ce'<; 



•=a(l — cos 6). 



trace' tlu" „ ? a "' are 1 that an y mathematician has attempted to 



u'ch ■ 7 ana ° g ° US to these ' but for tbc quadrature of 



2 1 ZfF' *««»**> ifc j * absolutely necessary that 



We sh ild have a general expression in polar coordinates for the 



1 1 oent of any surface. Such an expression is not found in the 



* Communicated by the Author. 



/ hi. May. S. 4. Vol. 9. No. 58. March 1855. p 



