430 DR. T. R. LYLE ON THE SELF-INDUCTANCE OF 



APPENDIX I. 



In order to determine L to the fourth order we have seen that it is necessary to 

 evaluate the definite integral 



fie rift /'"-* rjA-x 



MdxdydXd\ T 



J -! JC . I _?.A J-Jc-Y J-^A-X 



where 



M = P + QY+RY'+SY-'+TY 4 , 



P, Q, R, S, and T being functions of x and y. 



If we proceed in the ordinary way by putting in the limits after each integration 

 the expression becomes very cumbrous on account of the nature of some of the 

 functions (log and tan" 1 ) with which we have to deal. 



By the method to be explained below all the integration will be carried out first 

 and the limits introduced in an easy and symmetrical way at the finish. 



1 . Dealing first with P, the term independent of Y, if 



y = 6 (xy), 

 the result, with limits introduced, of the integrations with respect to x and y will bo 



where 



y = I C _Y y. = fc Y. 

 We have now to evaluate four definite integrals of which the first is 



Changing the variables to x t and /i and the limits accordingly, this integral is 

 equal to 



e(x l y 1 )dx l dy i 



Jc >!> 



where 



<j> (xy] = |j 6 (xy) d.c dy = Jjjj P 



Dealing in the same way with the three remaining integrals 

 -f" T 0(xM)dXdY, -\* ['' 9(x. 4h )dXdY, and T f"* 6(x 2 y 2 



J_i<:J_iJ J_J c J_i6 J_; C J_JJ 



