"52G REPORT— 1902. 



5. Oti'the Newtonian Potential. By Professor A. C. Dixon, Sg.D, 



A. vector jX +,/Y + IcL in wliich X, Y, Z are functions of X, Y, Z, and which 

 is reduced to zero by the operator v, appears to be at least one of the analogues 

 for three-dimensioned space of a function of a complex variable. 



If such an expression is called a Hamiltonian function, then triply periodic 

 Ilamiltonian functions can be readily constructed. The expression 



]^ S ^ /(■*-■ -p^ y-Q>~~i'> >'h> »^2, »h) 



-/(•t'o-i^. !/^~<h ~o-''> '«P '»-' '«:i) -/('!■■ --i'o) y - !/u. =-'o. »«l. '«2. '"a) 



■where 



/(.v, y, z, m„ VI.., m.) 

 denotes 



V (.(•■ - )«,«, - wi;,o.j - rn^aS)' + {i/- Wj)3, - • )?;o/3.. - m.^B-^^Y 



+ {z- m,y, - 711, y., - )«;,y,)^J ~ - 



is such a function. It has two isolated affinities of the simplest kind at {p, q, r) 

 (i'lii !7oi '■()) ii"^ ^'^ arbitrary zero (.r„, ?/,„ ;„), and by means of it other functions 

 with more complicated singularities can be constructed. 



In the theory which corresponds to that of the functions on a Riemann surface 

 the use of Hamiltonian functions does not appear to lead to the simplest results, 

 the reason being that v is not invariant in regard to inversions. It seems better 

 to take as analogous to a pair of conjugate functions two things of different nature, 

 •one being a simple and the other a double integral. Constant quantities lose their 

 ■unique position and have to be considered as having a simple singularity at 

 infinity in each sheet. The sheets are infinite coextensive .spaces of three dimension.<i 

 •and are connected by doors which may be of various shapes. Circuits are of two 

 'kinds, one-dimensioned and two-dimensioned ; to bar one of either kind one of 

 ■the other kind must he added to the boundary. 



On account of the disappearance of constant quantities there is nothing 

 'properly corresponding to the Abeliau integrals of the first kind. The normal 

 ■elementary potential of the second kind exists and will doubtless be the foundation 

 •of the theory, much of which is still to be worked out. 



tJ. The Di6criiiii)ianC of a Famih/ of Curves or Surfaces.^ 7?y Professor 

 Bkomwich, M.A., and R. W. H. T. Hudson, M.A. 



This investigation contains all the simple results relating to the discriminant 

 ■of a family of algebraic curves or surfaces which can be obtained without unduly 

 specialising the family. The chief new results are connected with the question of 

 osculating contact between the discriminant and some or all of the members of 

 the family of curves. The ease with which properties of the discriminant may 

 be demonstrated depends on the use of a notation which, as it does not involve 

 the coordinates explicitly, is applicable , to both two and three dimensions. The 

 ■equation of the variable curve or surface is expanded in terms of the parameter t 

 in the form 



^|, = A + Bt + Ct•^■ . . . +Ki'" = 



•and each of the polynomials A, B ... is subdivided into homogeneous polynomials 

 ■of degrees indicated by their suffixes ; thus A = A„-i- A, + A, + .... The origin 

 •being taken on the discriminant we have A,j = 0, B,j = U. The following theorems 



' Quarterly Journal, vol. xxxiv. p, 08. 



