556 EEroRT— 1901. 



.and it is shown that the isothermals may in certain defined casas be taken as the 

 axial paraboloids of revolution 



>•- = 



• 2 V-^La [x + ^^y log, 4L!;/(4L + «) V } 



and the lines of flow as the logarithmic curves 



Drawings of the curves and numerical examples are given in illustration of 

 these results. 



12. ^o(e on the Potential of a Surface Distribution. 

 By T. J. I'A. Bromwich, St. John's College, Cambridge. 



The problem is the determination of the discontinuities (at the surface) of the 

 second differential coefficients of the potential : the results are familiar, but the 

 method seems, easier than any other lam acquainted with. The same method 

 has beerx used by Weingarten ("Acta Mathematica,' Bd. x. 1887, p. 308; ' Archiv 

 d. Math. u. Phys.' (3), Bd. i. 1901, p. 27) to find the discontinuities in the second 

 differential coefficieuts of the potential of an attracting mass at the boundary of 

 the space which it occupies ; also for some kinematical conclusions in connection 

 with vortex motion. 



Take the origin on the surface at an ordinary point of the surface and let the 

 axis of s be normal to the surface. If the surface is closed the positive direction 

 of s will be from the inside towards the outside of the surface ; if the surface is 

 not closed the direction of z can be taken arbitrarily. The side for which = is 

 positive will be denoted by the suffix 0, the other side by the suffix 1. The 

 equation to the surface then takes the form (near the origin) 



s = ^(a,r- + 2hxy + bi/-) + . . . 



Let (T be the surface density at (.? , i/, z), supposed to be finite, continuous, and 

 diff'erentiable, and let s be the value of o- at the origin, .'.r, %, ^: being the first 

 differential coefficients there. Then we may write 



(r = s + .vSj: + ySy + zs- + er 



where >•- = x~ +y' + z" 



and t may be made as small as we please by sufficiently diminishing r. 



The potentials on the two sides of the surface are denoted by V,j, Vi, and we 



write 



""' W ' dx ' " d^ "D.r^- ' •'■" ~ my dWy' *"'" 



where the values of the diflTerential coefficients are to be taken at the origin. 

 Thus 



3V, avo 



-r.]- -y - = J<x + .rK,,-,i. + yu,y + zii,f, f ej; &c, 



where the quantities e' may be made as small as we please by sufficiently dimi* 

 nishing ;•. 



But at points on the given surface 



s = h{ax' + 2/ixy + by-) + . . . 



and so we may write 



0- = 5 + xs^ + ys,j + e"r, -^ - ^ = u^ + xii,rx + tjUry + e'"'') &c. 



