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



Hence the definite integral above is equal to 



s , -\ V i &2 + e2 <? ^ & a + c " 25 III. , < < .l\~] 



+<r) - 6? log IT - w log -T + a (c tan i + b tim ;)j 



The above is the well-known definite integral used for determining the G.M.D. of 

 a rectangle from itself. 

 3. To determine 



between the given limits. 

 If 



6(xy) = II Qdxdy, 

 the result of the integrations with respect to x and y will now be 



Y{o(xM)-e\ 



where x u y l} x 2 , y% have the same significations as before. 

 We have now to evaluate four integrals of the type 



T T Y 



J -\c J -|i 



Proceeding as in 1, these, affected by their proper signs, become 

 (ic-f/OH^O^i^-f f ($c+y a ) 6(x,y 2 }dx^dy., 



Ic Jb J c Jli 



+ (^c-t/i)0(x 2 y 1 )dx 2 di/ l -\ \ 



>c J -b J - c J -b 



so that, if in this case 



*' M = J V dy jjj Q dx- dy, 

 and 



-<!> (-6, -P) + 20 (0, -c)-20 (0, c), 

 then 



between the given limits is equal to 



where 2 has the signification given to it in 1 . 



