Integration of Linear Partial Differential Equations. 47 

 duct for the images of the corner pole will be found to be 



p 2 =^!nf C oth^coth^ 



7T V * 4 



'coth 



nm 



where x is to take the successive values 4, 8, 12 ; . . . , and af 

 the values 2, 6, 10, .... 



Writing this product P 2 in the form 



- tanh -^- . tanh ~ . . . 



. ^ . COth 7T . COth 27T . COth 37T . . . , 



77 tanh -jr-. tanh -^- . . . 



it is brought into connexion with (8) and (e) (§ 23); and its 

 value is thus found to be 



2v2 /V" /~^r~ 21 



Hence 



2K V 2*K -K 



Q5 =| OTQ =f 



In other words, the resistance of a square plate with the poles 

 arranged as in fig. 11 is given by (21), if for m-\-n we read its 

 value |-, and consider a equal to \. 



One more distinct case may be mentioned, pjg, 12. 

 namely the right-angled triangle shown in fig. 12. j$ 



This also requires two factors raised to different 



powers ; and the result is the expression (21) £■ — »- 

 with « = -g-. 



[To be continued.] 



V. Some Remarks on the Finite Integration of Linear Partial 

 Differential Equations with constant Coefficients, By the 

 Kev. S. Earnshaw, M.A. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



IN your Magazine for June 1849 you kindly printed a short 

 paper of mine " On the Transformation of Linear Partial 

 Differential Equations with constant Coefficients to Funda- 

 mental Forms," in which I promised to make "in a future 

 communication a few remarks on the finite integration " of 

 equations of the second order with two or three independent 



