p' 



TRANSACTIONS OF THE SECTIONS. 23 



If T = (X2a?3 - ;u2y2) (p2a;2_ (rY),^^^^^ are fivo pair of oblique asjTnptotes, \-x'^=fj?y'^, 

 a?=(T-y'^ ; and by combining either of them with the second equation, 



[=2E, -1 

 [=-2F,/ 



we decide on Paratomy. 



When the general equation is given to us in this form and we desire to find the 

 Proximate Conies, the most direct method is to assume /i^ = l, o--=l, and 



(r-XV-M)(/-pV-N) = T+2(Ea;2+Fy)+MN; 

 whence 



N+ M: 



or 



_N=2(E+ry) 



Then the Proximate Conies are 



These are closer indications of the infinite branches than the asymptotes. PutMN = C' 



ADE 

 DBF 

 EFC 



But in general it is expedient to put ls.=gx^+2hx^+k, and study the variations 

 of X in the equation By2+Da;2-t-F= + ^X. In many cases the lower sign is in- 

 admissible ; in most it is more restricted than the upper. When we have only the 

 upper, evidently there is no undulation across the axis of x ; for y^ has then but one 

 positive value for any given value of x. 

 The forms of X are as follows : — 



■yi^ = n'{m^-x% -, X,=«V +«»'), 1 X, =n\x'-m'^), ^ 



X^={m'-x'){x'-7i',) [_ ^, = {x'+m^){x^+n\ [ X,„ = (:r^-,«=)(^2+«2), I 



■X^ = {m^-x^)x\ [ X, = (a;=+»i2)^2^ r yi,, = {x^-7n')x', Y 



X,^{m^--x'){x'-n% J X^ = {x''+m2f+n\ J X,, = {a?-m-'){x'-n')^ 



=0. 



RemarTcs on Ncqner's original Method of Logarithms. By Professor Pfrseb. 



On Linear Differential Equations. By W. H. L. Russell, F.R.S. 



The object of this paper was to explain the progi-ess the author is making in his 

 theory for the solution of Linear Differential Equations, especially when the com- 

 plete integral involves logarithmic functions. 



On llacOullagh's Theorem. By W. H. L. Eitssell, F.R.S. 



This paper was intended to simplify the process given by Dr. Salmon to prove 

 MacCullagh's theorem relative to the focal properties of surfaces of the second order. 



Note on the Theory of a Point in Partitions. By J. J. Sylvester, F.R.S. 



In writing down all the solutions in positive integers of the indefinite Equation of 

 Weight, x+2y-\-^z-\- . .. = «, or, in other words, in exhibiting all the partitions 



