26 ART. 6.— S. KUSAKABE : 



a 2 ? _,. , r {^+4(;.+i)j[^-2tj--'{9î+4(;.+ i)-.sot+i)r j 



"A« Ap ^L ^{^ + 4(/+l)-(9(+D}1^-ö2t}--' 



...[p+i+3ai}'{/>+i+ac+i} a H >0 , lq) 



...[^+1+3(21 + l)} s {p + l + 9l} a J v 



which is always positive since we have 



M + 4W+1) 



3t 



1; 



f9î-3l}ffi + 4(;. + l)-3(9I + l)} _ G3I-(jP + l) i . 



{3î + 4(;.+ l)-(9I+l)]{9 f f-32l} _i+ {3l-2l + 4;. + 3}{«R-33l} ' 



etc. etc. 



Proposition V. When the centre of cycle is fixed, the hysteresis 

 curve at any stage p after X cycles becomes more and more steep when the 

 amplitude of the cycle becomes greater and greater. 



Let the number of cycles be increased from A to A + l, then we 

 have 



A 2 ? =/ 7 { (4^+6)3t+y+l }{[ (4^ + 6)3t+j>+lJ--3^ } +22l 3 

 A*&P [<4A+6)a+j»+l}{[(4Jt +6)21 +2>+l) 2 -33l 2 } -231 s ; ' 



which is necessarily positive. Again take a further differential : — 

 A!5 _,, {Sff + 8aMH9F + 8gM + 153Ty <0 , 21) 



{A^} 2 A/> ^ {^ 3 + 8^ + 7â 2 ] s {ft 3 + o2l9i + 1691 s ) 2 



which is evidently negative. 



From what has been just proved, (20) and (21), follows 

 Proposition VI. Provided the centre of cycle is fixed at origin, 

 the hysteresis curve becomes more and more steep when the cycle is repeated 

 over and over again and the carve asymptotically approaches a closed one. 



Suppose that after ^ cycles of the amplitude 2Ï about the neutral 

 state, another smaller cycle of the amplitude a, whose centre is situat- 

 ed at j, is completed A times, and then it is just in the stagey. 

 Then, the yielding at this instant is, by the general equation ((>), 

 writing 2) for ±A 2t + A{1 + 1 )a -r 1 , 



