292 A. TANAKADATE 



which at the center of the coils (^, V> ^ being a, b, c) reduces to 



î)a'' {a^ + b' + c']i{a' + c'yl ^^ ^ 



+ {6[b^ + c''J-2bY')a'-c\b^ + à) {Sc'' + 2P) \ =0 



Unless these two equations are simultaneously satisfied, the three 

 partial differential coefficients will not vanish. Eliminating a between 

 the two equations we find that the only admissible cases are when 

 h — c, and either h or c = oo. Thus it seems that ^'^F can not be made 

 to vanish entirely except for the case of a square, and four infinite 

 parallel straight currents. But when either h or c is great compared 

 with the other, the partial differential coefficient with respect to the 

 greater will be comparatively small as may be judged from the equa- 

 tions (A). Hence for such cases, if IW/iî^x = 0, both the others must 

 be small. But since ^ ^ = — (y^ + ^2")* we find from the above 

 two equations 



^ oc 2ia'^ + 72 {b' + à) a'' + [93 {b' + c') + 146 fc-c^] a« 



+ 8 [b' + C-) [9 (i' + c') + 4by'] a« 



+ 3 {Ij' + c^)^[ll {b' + c') - 10U'(r] a' 



+ 2 (6^ + cy[3 {b' + e') - 76'V^] a" 



- b^"' {W + âf\:2 {¥ + c') + UY-) ]^ 



from which a is to be found for any given value of b and c, the sides 

 of the coils. Since the equation is homogeneous, we may take either 

 b or c as unit of leno-th and measure the other lengths in terms of it. 

 Thus if we take b (half the height of the coils) as the unit, and express 

 a and c in terms of it, we have all the possible cases brought out by 

 varying c from 1 to co. Examining the above equation, we find that 



* Tlie equcatiou y F/'èl/^ + y Fi'hz' =0 shows that at the origin i'', cousiJered as a fiiuctiou 

 of ij and z, is a miüiiuax with respect to those variables ; wheu b = c, this becomes what may be 

 cillcd njlit puiiit, aud for & = x, a //(if Une. 



