﻿X U — r~ Z 



the twofold plane y <2 = } and the spheroid -% + g n _ ^ =1- 



360 Sir James Cockle on Fractional Criticoids. 



(a is an arbitrary parameter, since only the value of ae has been 

 defined), we have 



s=a + e%j r=a—ex; 

 and thence 



so 4- ae 4- 5 = ( 1 + e) (x + a), 



x — ae + r=(l — e)(x + a), 



1 + e 



and the quotient is = - , a constant value, as it should be. 



\ — e 



The equation V= const, may in fact be written 



\ + e_x + ae + S t 

 1 — e x — ae + r' 



viz. this equation, apparently of the fourth order, breaks up into 



y*+** 

 , \ 



The foregoing results in regard to the attraction of a line are 

 not new. See Green's ' Essay on Electricity/ 1828, and Collected 

 Works, Cambridge, 1871, p. 68; also Joacbimsthal, "On the 

 Attraction of a Straight Line," with Sir W. Thomson's Note, 

 Camb. and Dub. Math. Journ. vol. iii. (1848) p. 93 ; but it does 

 not appear to have been noticed that they are, in fact, included 

 in the theory of the attraction of ellipsoids. 



The like considerations show that the attractions of the ellip- 

 soidal shell (a, b,c; e) upon an exterior point are equal to those 



x u 



of an elliptic disk ^ = 0, -^ ^ + j^- — ^ = 1, the mass of which 



is equal to that of the shell, and which has the density at the 



point (x, y) proportional to [1 - -^—^ - ^T^J ' 



Sir W. Thomson informs me that the foregoing results have 

 long been familiar to him. 



XLVIII. On Fractional Criticoids. By Sir James Cockle, 

 F.R.S., Corresponding Member of the Literary and Philosophical 

 Society of Manchester, President of the Queensland Philoso- 

 phical Society, tyc* 



1. r INHERE is a conformity between the rational and entire 



-*- algebraical expression (or, as we may call it for the sake 



of brevity, quotic) <p(x, y, . .) and the corresponding differential 



expression (or, as we may call it, quotoid) <p( -j-y -?-, . . ). Coeffi- 

 * Communicated by the Rev. Robert Harley, F.R.S. 



