486 KBPOET— 1880 



(3) . . . . 



s/v^k^' = (-1) '"'"'' Vi-rv* 



are agreeable to both the transcendental equations (2). 

 I then consider the transcendental sj'stem : — 



(4) . fy{z)dz + r^f{z)dz + 2l<^ + 2nS + 2\ia + 2/x«* 



y"' sy (z)dz +/ "' =y(=)o's + 2/^' + 2mS' + 2\ia' + 2tii^' 



= r''z^f{z)dz + f'''z'f{z)dz 



J J J . . <-^ *y 



and deduce its equivalent algebraic system. 



Finally I put U = / ""V (s)ffc + /" "\f {z)dz 



\ = /'''z"-f{z)dz + r"'z-'-f{z)dz 



and I write z^z.^ = F (U, V), and show that F (U, V) is a periodic function of two 

 variables U and V, each of which has four periods, two real and two imaginary ; 

 the nature of the periodicity of which I discuss in the investigation of the general 



values of the integrals / "f{z)dz and / "zy{z)dz. 



«.' «-^ 



17. On the Integral of Laplace^ s Equation in Finite Terms. 

 By the Rev. S. Earnshaw, M.A. 



The Integkal of Laplace's EciTrATioN. 

 The equation to which this title refers is the following : — 

 (Pu ^u d'-u _ fx ^1, 



and I am desirous of the three following propositions being communicated to Sec- 

 tion A, at the meeting of the British Association at Swansea. 



Prop. A. — The independent variables .r, i/, z are in this equation not necessarily 

 the coordinates of some point P, in space referred to a fixed rectangular system of 

 coordinate axes O.r, Oy, Oz. We shall, however, h)^othetically treat them as 

 such ; and therefore we say that 



0P2 = r- = .r'2 + if + s^ 

 Now from draw any two lines, OA, OB, at right angles to each other, and let ^, 

 T) represent the lengths of the rectangular projections of OP upon these two 

 arbitrary lines ; then vsrill the following be a general integral of Laplace's equation 

 given above, 



u = A/'^ cos (ar] + 6) .... (2) 



in which A, a, b are arbitrary constants which have no reference whatever to the 

 ajbitrary positions of OA, OB. 



