40 Mr. 0. J. Lodge on some Problems connected 



w.w.wi -j/* .... m 



l + q 1 + ( f ! + ,/••• k > ^ 



n-^ 2 i+ ? 4 i+</ — v - ir~) ' * w 



_ n 



<7 stands for <? *" k ? K having its ordinary meaning of 



eld 



J 



, v/(l-Psin 2 6>) 



Taking the modulus-angle 45° so that k = k / = — -=■ and 



K = K7 = K say, and then dividing (8) by (e), there results 

 the product required, 



Hence Q= ^-, and the resistance of a triangle with each elec- 





 trode on an angle of 45° is 



-iW? ■%)■ ■■-■ <"> 



The numerical value of K from Legendre's Tables is 

 1-854 074 677 301. 

 § 24. If one writes out (17) in two parts, 



8 . AB 8 1 fi dO 



^ lo ^-^ lo sJ V (i-isin^y • ( 17 > 



the first term is the resistance of a segment of a circle with 

 base AB and with its arc touching the two sides of the triangle 

 at A and B ; and the second term may be called the effect of 

 the " coquadrant " * by which the triangle exceeds the seg- 

 ment in area. But one must not imagine that this term ex- 

 presses the resistance of this figure. The fact is that its resist- 

 ance cannot be found by any consideration of (17y, for reasons 

 which shall be stated, § 30, footnote. It is worth while to 

 notice the arrangement of poles and flow-lines which would 

 produce this coquadrant figure in an unlimited sheet ; but they 

 will come better later. 



* I call it a coquadrant because it is the excess of a square over the qua- 

 drant whose radius is a side of the square : more generally the figure en- 

 closed by any circular arc and two tangents might be called a cosector) it 

 would be the excess of an isosceles cirquadrilateral over the inscribed sec- 

 tor, and it only differs from a sector in haying the equal angles 0° instead 

 of 90°. 



