24 EEPORT — 1872. 



'17. )/=y'X^ (Quartohyperbolie Group). 



18. y^±A==VXi. 



19. ?/"+Xj = VXj (oue Hyperbolic asymptote at most). 

 V.<( 20. ?/'+X2»/^=X3 or X^ or Xj (perhaps oue Parabolic aud oue Hyperbolic 



asyuiptote). 

 21. 2/'+X,y^ = X^ or w-+X2= VX'^ (perhaps two Hyperbolic asymptotes, 



all difierently directed). 



The mode of analysis used in the most difficult cases is as follows : — - 



It is assumed that if (vx) = 0, and/ (?« a;) = are Imown curves, and ?/'^ = r- + (r, 

 y"2=i;2 — j<2^ ^]jg ciii've F (y^ x)=^0 can thence be traced, «/'- and y"- being the two 

 positive roots of y-, when such are real. Practically it is uot difficult to decide on 

 the course of (ya;), if the constants which enter the two auxiliaries are fixed ; but 

 the number of hypotheses concerning the relatious of the coustants in (f> to the 

 constants in / are embarrassing. 



Thus, to trace (y, .i') from the eq^uation ?/- + X^,= + \/X|, which is the largest 

 case, we put X„= — y/^, or else X.^=+y(,-, according to the sign which X^ may 

 assume within ditferent limits, and Y-=v'X^. Then either y-=Y^ — y„-, giving 

 at most only one positive value to y^ ; ory^=yj2+Y^, giving in some cases two 

 positive values to y'^. 



This assumes that we know not only y, and y„, which define Conic curves, but 

 also Y-= is/X^. If Xj degenerate, Y^=; V^i is a Quartic Parabola. Y'^= ^X., is 

 a Doiihhj Diametral Quartan, which is hero assumed to be known ; Y^= v^X., is 

 the primary Quartotertian (9th Class of Quartans) ; Y-= \/Xj is the primary Quarto- 

 hyperbolic of the 17th class. Thus the 9th class becomes auxiliary to the 10th, 

 13th, and IGth ; and the 17th is auxiliary to all which follow it. The 1st class 

 (Quartic Parabola) is auxiliary to the 11th and 14th. 



It is believed that in the 8th class alone there are in strictness as many as 260 

 species. This makes it impossible to undertake to draw them all, which multiply 

 more and more in the higher classes, as the number of constants increase. Never- 

 theless many diagrams are laid before the Association, nearly exhausting the forms 

 of the earlier classes. The Semicubical and the Quartotertian are notable as pecu- 

 liarly novel and most remote from the Doubly Diametral. 



Many of the forms might be conjectured beforehand from the Doubly Diametral 

 by merely introducing inequality, as in place of two equal, two unequal ovals. 

 Nevertheless there is much that could never be so conjectured, just as in the Doubly 

 Diametral we could not conjecture the forms of the inferior classes from knowing 

 the superior forms. 



On the Circular Transformation nf 2IdbiHS. 

 Bij Prof. H. J. Stephen Smith, F.B.S. 



GenebaTj PnYsics. 

 - On Symixitliy of Pendulums. By Professor P. G. Tait, F.E.S.E. 



On Relations between the Gaseous, the Liquid, and the Solid States of Matter. 

 By Prof. James Thomson, LL.B., Queen's College, Belfast. 



The object of this paper is to submit some new theoretical considerations which 

 constitute a further development of one portion of the views offered, at last year's 

 Meeting of the Association, by the author, in his paper entitled " Speculations on 

 the Continuity of the Fluid State of ISIatter, and on Relations between the Gaseous, 

 the Liquid, and the Solid States." He has now to make reference to the abstract 



