Page:Scientific Memoirs, Vol. 2 (1841).djvu/425

 Rh then Thirdly, since $$C D = K K'$$ and $$F X$$ is equal to the part of $$K K'$$ which extends from $$K$$ to the line $$X Y$$, we have whence or, since $$K F = K I + I H' - F' H$$ and $$F' H = G H - G F'$$, If now for $$L K'$$, $$I H'$$, $$G F'$$ we substitute their values and if by $$x$$ we represent a line such that we have These values of the ordinates, belonging to the three distinct parts of the circuit and different in form from each other, may be reduced as follows to a common expression. For if $$F$$ is taken as the origin of the abscissæ, $$FX$$ will be the abscissa corresponding to the ordinate $$X Y$$ which belongs to the homogeneous part $$AB$$ of the ring, and $$x$$ will represent the length corresponding to this abscissa in the reduced proportion of $$A B : \lambda$$. In like manner $$F X'$$ is the abscissa corresponding to the ordinate $$X' Y'$$ which is composed of the parts $$F F'$$ and $$F' X'$$ belonging to the homogeneous portions of the ring, and $$\lambda$$, $$x'$$ are the lengths reduced in the proportions of $$A B : \lambda$$ and $$B C : \lambda'$$ corresponding to these parts. Lastly $$F X$$ is the abscissa corresponding to the ordinate $$X Y$$, which is composed of the parts $$F F'$$, $$F' F$$, $$F' X$$ belonging to the homogeneous portions of the ring, and $$\lambda$$, $$\lambda'$$, $$x$$ are the lengths reduced in the proportions of $$A B : \lambda$$, $$B C : \lambda'$$, $$C D : \lambda''$$. If in consequence of this consideration we call the values $$x$$, $$\lambda + x'$$, $$\lambda + \lambda' + x''$$