Proper Way to Splice Into Existing Electrical Line
In addition the connection with the short "jumper" cable needs to have two junction boxes. This may seem less than elegant but it is a normal thing. 

If the cable to be cut is a long one it may be possible to make the cut in such a way that a part of the existing run can be re-routed to the first added lighting fixture box. Then the wires can be spliced there and the "jumper" can be routed over to near the cut location and spliced into existing wiring at the added junction box.