Forced Vibrations of Electromagnetic Systems. 363 



/starting when £=0, we have 



p-Kfi»)=fp- 3 (e*-l-pt-i P H*)=p-*f(e'% S! i r , . (218) 



&c. &c. Next, operating with the exponential containing^? in 

 (217) turns t to t—(r—a)/v, and gives the required solution 

 in the form 



+ same function of —a , . (219) 



where t 1 = t—(r—a)/v ; the represented part beginning when 

 t x reaches zero, and the rest whea t—(r + a)/v reaches zero. 



36. Fuller development in a special case. Theorems invol- 

 ving Irrational Operators. — As this process is very complex, 

 and (219) does not admit of being brought to a readily inter- 

 pretable form, we should seek for special cases which are, 

 when fully developed, of a comparatively simple nature. 

 Write the first half of (212) thus, 



-n- cv va 



-tL = —7Z € 



» -^:lXz + ^[(^'*-^M-« 



Now the part in the square brackets can be finitely integrated 

 when feP* subsides in a certain way. We can show that 



(lliy^^i'-^oi^-" ) 2 -^]*}, (221) 



in which, observe, the sign of <r may be changed, making no 

 difference on the right side (the result), but a great deal on 

 the left side. 



The simplest proof of (221) is perhaps this. First let r=a. 

 Then 



(^f)V")=e-(l-f )"*(!), • • K222) 



by getting the exponential to the left side, so as to operate on 

 unity. Next, by the binomial theorem, 



-*++JHs(yy+-"]» ■ <-> 



Now integrate, and we have (/commencing when £=0), 



-**i 1+ « + & «*+ 4Hr rt+ "-M -(224) 



= €-*'€*' J (<rtO ; * 



