The delay line is used to delay the incoming signal vertical
information and sync it with the horizontal. Old scopes used a
series of LC combinations tuned to get a good waveform
representation.
The HP scope I had used a very very long coil of coax cable as
the delay line and got much better waveforms than the LC delay line
scopes could, but the coil of coax was bulky (about a foot long by
five inches wide and 1.5 inches thick).