Problem in Two Dimensions. 483 
In the double sum 
ea tkr 
== 
which we have now to evaluate, each pair of elements is to 
be taken once only, and the product is to be summed after 
multiplication by the factor r—1e—“””, depending on their mutual 
distance. The best method is that suggested by Maxwell for 
the common potential. The quantity (3) is regarded as the 
work that would be consumed in the complete dissociation of 
the matter composing the plate, that is to say, in the removal 
of every element from the influence of every other, on the 
supposition that the potential of two elements is proportional 
to r—te-“", The amount of work required, which depends 
only on the initial and final states, may be calculated by 
supposing the operation performed in any way that may be 
most convenient. For this purpose we suppose that the plate 
is divided into elementary strips, and that on one side 
external strips are removed in succession. 
To carry out this method we require first an expression for 
the potential (V) at the edge of a strip. Here, 
Xs » 
FSULOR: dub siti acne ate) 
+o ew ihr 
Ves rend sites te, =) ines a(4) 
aden r 
where r=,/(27+y") ; and therein 
om en ikr a eaikr _(° enikxo 
Oe —— dp=2 =p CUNT aa) 
pas et oY Ji Vv—-l 
representing the potential of a linear source at a point distant 
« from it. Convergent and semi-convergent series for (5), 
applicable respectively when w is small and when z is great, 
are well known. 
We have 
BGO) ye 
bk Oe 2h Aes ) ted 1 whee baat + i 3 eae 
{ eb aes 1. Sie "1.2. (Gikoj? 377 
] Dateat J hea hAyt 
=—(v+ ey (ar 22 Weer —,.. 
—2 
where eee ime... 
and y is Huler’s constant (5772...). A simple method of 
derivation, adequate so far as the leading terms are concerned, 
will be found in Phil. Mag. xliii. p. 259, 1897 (Scientific 
Papers, iv. p. 290). 
2L2 
