TRANSACTIONS OF THK SECTIONS. 9 



iiivestig-atiou led to Dr. Jacobi, the eminent plij^sicist of St. Petersburg, who was 

 present at the delivery of the address, favouring me with the annexed anecdote 

 relative to his illustrious brother C. G. .T. Jacobi. 



" En causant un jour avec mon frere defant sur la necessite de controler par des 

 experiences reiterees toute observation, meuie si elle confirmo I'hypothese, il me 

 raconta avoir decouvert un jour une loi tres-remarquablo de la tlieorie des nombres, 

 dont il ne douta guere qu'eile fiit generale. Cependant par un exces de precaution 

 ou plutot pour faire le superflu, il voulut substituer un chiffre quelcouque reel aux 

 termes generaux, chiffre qu'il choisit au hasard ou, pent-etre, par une espece de 

 divination, car en eftet ce chiffre niit sa formule on defant ; tout autre chiflre qu'il 

 essaya en confirma la generalite. Plus tard il reussit a prouver que le chiffre 

 clioisi par lui par hasard, appartenait a un systerae de chift'res qui faisait la seule 

 exception a la regie. 



" Ce fait curieux m'est reste dans In. mi'moire, mais comme il s'est passe il y a plus 

 d'une trentaine d'annees, je ne rappelle plus des details, 



" M. 11. Jacobi." 



"Exeter, 21. Ao'.U, 1869." 



Mathematics. 

 On the Thconj of Distance. ByW. K. Clifford. 



This communication relates to the following two theorems on the foci and asym- 

 ptotes of curves. 



Theorem I. — L, INI, N, . . . are the m tangents from a point a to a cur^o Cm of the 

 ))jth class ; B is any lino tlirough a, meeting the curve in m(m — l) points ; /, m, n, . . 

 P, Q, 11, ... are the ?«(m — 1 ) asymptotes of the curve, and p, q,r, . .. are a set of 

 }}i foci. 



sin= LM . sin- LX . sin- MN.frty;' . an' . ar' . . )"'"* —2 -2 —2 



III . am . mi ... sin BP . sin PQ . sin BR . , 



Theorem II. — I, m, n, . . are the n intersections of a line A with a curve Ca of tlie 

 wtli order ; h is any point on A from which are drawn the n{n—V) tangents ; L, M, 

 N, . . .p, q, r, . . . are a set of m(m— \) foci, and P, Q, E, . . . are the n asymptotes. 



/V.^^^?7^'^..(sin-AP. sin^AQ.sin^AR...)"-! • o -n,^ . „ ^-o • „T,-r> 



^-pf --Vv? — '■ — nvf ^^7 — 1 — 7 — =s"i TQ sm2 QR . sin2 PR. . . 



sm AL . sin AM . sm AN ... Ip .Iq. Ir . .. ^ ^ 



The numerator and denominator of the fraction on the left-hand side of the equa- 

 tion in Theorem I. are quantities either of which I call the distance of the point a 

 from the curve Cm- The coiTesponding quantities in Theorem II. I call the Distance 

 of the lino A from the curve C„. The reason of this is iu the similarity of the ana- 

 lytical expressions for the distance of two geometrical forms in all cases, viz. the 

 distance vanishes when the two forms have contact, and is infinite when either of 

 them has contact witli the "absolute." The "absolute" in plane geometry (so 

 called by Professor Cayle}-) is the two circular points at infinity. 



I also consider the modifications undergone by these theorems in the case of 

 spherical curves. 



The method of investigation employed is an extension of the "' geometric analy- 

 sis " of Grassmann, itself a development of a remark of Leibnitz. 



On tlie Umhilici of Amillac/inatiG Surfaces. By "W. K. Clifford, 



On tlie Common Tangents of Circles. By M. Collins. 



Neumayer) took frequent part, the opening and concluding papers (each of surpassing 

 interest, and a letting-out of mighty waters) were on Obscure Heat, by Prof. Magnus, and 

 on Stellar Heat, by Mr. Huggins. 



